Logic in Theology

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he reflection over the role of logic in theology has a long and rich tradition. Medieval thinkers investigated complex theological problems, utilizing the best logical tools at their disposal and often inventing new ones. The evidence is to be found in the writings of almost all major medieval thinkers, from St. Anselm to John Buridan, dealing with various theological issues, such as existence of God, His omniscience or the mystery of the Holy Trinity. In the 20th century, two different approaches were developed to applying logical tools to theological problems. The first may be called ‘incidental’, and includes numerous attempts to logically analyze some particular theological ideas, such as the ontological argument or the properties of God. This trend includes works by famous logicians and philosophers such as Kurt Gödel or Alvin Plantinga. The second strategy is ‘systematic’; it is an attempt to reconstruct the entirety of theological discourse with the use of modern logical techniques. It was the goal of the Kraków Circle, a group of theologians, philosophers and logicians (including Jan Salamucha, Józef Bocheński, Jan Drewnowski and Bolesław Sobociński), which was formed in the 1930s. After World War II, the programme of the Circle was kept alive by J. Bocheński, and culminated with his The Logic of Religion. Following in the footsteps of the Kraków Circle, we believe that there are a number of intriguing theological problems which may be analyzed, and illuminated, from the logical perspective. On a minimal reading, theology is any theory which embraces the thesis that God (or the sphere of the divine) exists and provides us with some description of God (or the sphere of the divine) and the relationship



between God (the sphere of the divine) and human beings. In some theologies, e.g. in Catholicism, things are more complicated, as they include the content of the revelation – a body of knowledge which has a special epistemic status – as well as all the theories that serve to explain and develop it in more detail. Crucially, the assumption here is that both revelation and theological theories are expressed in human language and, as such, may be subject to formal scrutiny. As in the case of any set of sentences, one can investigate logically a number of theological issues, including – but not being limited to – syntactic (the structure of theological sentences and theological theories), semantic (truth, evidence in theology), pragmatic (analysis of the propositional attitudes in theology), conceptual (definitions of and interrelations between theological concepts) and methodological (types of arguments and criteria of justification in theology) ones. Thus, we understand logic broadly to include not only semantics, syntax and pragmatics, but also the study of the theological conceptual scheme, as well as its methodological aspects. Importantly, in all these dimensions one can analyze not only theological discourse per se, but also its relations with other discourses, such as scientific, philosophical, etc. Our second assumption concerns logic: we believe that a fruitful formal study of theological discourse cannot be limited to the application of classical logic. Rather, any logical tool may prove useful, and examples include many-valued logics, modal logics, non-monotonic logics, and other formal mechanisms such as belief revision or the theory of circular definitions. Given this broad scope of the theological problems accessible with logical tools, and the number of logical techniques one can utilize while analyzing theological discourse, we prefer to speak of logic in theology rather than logic of theology. The latter expression seems to suggest that there is one and only logical system suitable for theological reflection or that the logical aspect of theological thinking is somehow external to the content of theology. In our eyes, both these statements are false. On the one hand, there is no reason to believe that the different theological problems can be handled with



one formal recipe; on the other, many theological questions are so intertwined with the underlying logical problems that they cannot be properly comprehended without some logical analysis. Thus, we believe that speaking of logic in theology is more appropriate. This small change of prepositions means an important change of approach: one is not limited to the question of what is the ‘proper’ logic of theology, but is free to investigate any theological problem which gives rise to logical analysis, or to consider loci theologici which may inspire the development of new logical instrumentarium. Thus, the present volume is intended as an exercise in logic in theology. It puts together contributions pertaining to various aspects of the relationship between logic and theology, from historical essays, through the formal reconstruction of theological concepts and the structure of theological discourse, to more methodologically oriented papers, examining the criteria of theological justification and the interplay between theology and other disciplines. The papers collected here have been written within the research project entitled The Limits of Scientific Explanation, carried out at the Copernicus Center for Interdisciplinary Studies in Kraków, and sponsored by the John Templeton Foundation.

Bartosz Brożek Adam Olszewski Mateusz Hohol

Table of Contents

Jan Woleński Theology and Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Jerzy Dadaczyński What Kind of Logic Does Contemporary Theology Need? . . . . . . 39 Antonino Rotolo, Erica Calardo God’s Omniscience: A Formal Analysis in Normal and Non-normal Epistemic Logics . . . . . . . . . . . . . . . . . . . . . 61 Kazimierz Trzęsicki Problems of Omniscience and Infallibility. A Temporal-Logical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Bartosz Brożek, Adam Olszewski Miracles: A Logical Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Damian Wąsek The Role of Dialectics in Peter Abelard’s Concept of Theology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Marek Porwolik Formalizations of the Argument Ex Causa Efficientis Presented by Fr. Bocheński . . . . . . . . . . . . . . . . . . . . . . . . . . 159


Table of Contents

Marie Duží Ambiguities in Natural Langauge and Ontological Proofs . . . . . . 179 Kim Solin Mathematics and Religion: On a Remark by Simone Weil . . . . . 219 Pavel Materna Science – Logic – Philosophy. An Old Problem Resuscitated . . . 237 Jan D. Szczurek The Rationality of Theology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 Mieszko Tałasiewicz Science as Theology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Wojciech P. Grygiel Physics in the Service of Theology: A Methodological Inquiry . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

Jan Woleński University of Information, Technology and Management, Rzeszów

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logic applicable to theology? If so, is logic basically relevant for theology or does only the former have a secondary importance for the latter? Answers to both questions require several preliminary explanations concerning logic as well as theology. Let me begin with the scientia divina. First of all, there are many theologies, not only because many religions exist. Even if we restrict our attention to three great monotheistic religions (Islam, Judaism, Christianity), we do not find a uniform idea of theology. The situation does not change very much if we make further restrictions, for example to Christianity only. The theological tradition of this confession does not appear as uniform. On the contrary, we encounter various conceptions of theology. Two of them are particularly relevant for my further considerations. Firstly, we have the negative or aphophantic theology as represented by Nicolaus Cusanus (Nicholas of Cusa), which is very popular in the Orthodox Church. This theology says that we cannot say anything positive about God and His attributes. We should abstain from positive assertions and limit ourselves to statements like ‘I do not know what God is like or God is not…’. According to this kind of theology, the cognitive gap stemming from such assertions is sufficiently filled by belief as faith. If we believe, we do not need to be bothered by apparent inconsistencies in the body of theological statements. As This contribution was made possible through the support of a grant “The Limits of Scientific Explanation” from the John Templeton Foundation. *


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Cusanus said, docta ignorantia sufficies and coicidentia oppositorum are to be accepted. Clearly, logic plays no essential role in negative theology, which is not particularly interested in arguments. Secondly, there is theology, which is called natural, rational or philosophical (this last label is not quite correct, because negative theology has an explicit philosophical dimension). This version of the scientia divina1 admits and recommends logical arguments in theological debates. Although its particular representatives, for example, Augustine of Hippo, Anselm of Canterbury, Thomas Aquinas, Duns Scotus or Leibniz accepted the formula fides quarens intellectum, they differ considerably as far as the matter concerns the scope of logic as a tool of theology. The standpoint of the official Catholic theology, confirmed by decisions of several councils, follows Thomas Aquinas, who distinguished theological truths (for example, the existence of God) accessible to human reason and subjected to rational or logical argumentations and theological truths, which exceed human understanding and are grounded in the revelation (for example, the doctrine creatio ex nihilo). Moreover, there cannot be any inconsistency between both kinds of truths, because both are given by God. Thomas’ picture, for its compromising character, provides a very good point of orientation. Augustine ascribed more to faith than Thomas did, Anselm and Scotus were more rationalistic than the Doctor Angelicus, but Leibniz and the Deists of the 18th century became the extreme rationalists in theology. The rational scientia divina forms the proper environment for investigating the role of logic in theology. But what is logic? The noun ‘logic’ co-occurs with various and mutually different qualifications. Omitting several metaphorical denominations, like ‘the logic of history’ or ‘the logic of politics’, we read about formal logic, transcendental logic, hermeneutic logic, dialectical logic, material logic or philosophical logic. My further considerations are directed to formal logic,

See Th. Flint, M. Rea, The Oxford Handbook of Philosophical Theoology, Clarendon Press, Oxford 2009 for a comprehensive survey.


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because I think that it is the only legitimate kind of logic. We can eventually distinguish logic in the narrow sense and the logic in the broad sense. The latter covers semantics or semiotics (roughly speaking, logical theory of language), formal logic and the methodology of science, but the former restricts the scope of logic to its formal part. Since I will touch upon semantic, formal-logical and methodological problems, it is convenient to work with logic in the broad sense. However, formal logic is the core of logic in any legitimate sense, because it provides the basic criterion of logical correctness. Assume that A1, …, An, A is an argument with A1, …, An, as its premises and A as its conclusion. We say that this argument is formally correct or valid, when A logically follows from the set of premises consisting of A1, …, An (symbolically A1, …, An ├ A), but it is materially correct, when its premises A1, …, An are true. This definition entails that if A1, …, An ├ A and premises are true, and the conclusion must be true as well. This statement exhibits the main property of deduction, namely its soundness or truth-preserving. In general, formal logic is a collection of systems consisting of principles being special cases of logical entailment. Of course, we can try to weaken the concept of logical validity and admit partial or inductive correctness, but it has to be done very carefully. In particular, clear features of generalized validity are required. In fact, logic called dialectical, material or transcendental completely lacks, at least until now, precise criteria of the validity of its arguments. I include a logical analysis of various problems into the scope of logic. Logical analysis has a special importance for philosophy. In my view, rational theology is more a kind of philosophy than the scientia divina in the traditional sense; this opinion holds to some parts of theology at least, but I will not enter into details. Consequently, natural theology is comparable with philosophy with respect to their status as academic disciplines. This statement has very far-reaching consequences, because if a field has an academic character, it has to fulfil several methodological requirements. Thus, the applications of logic to theology should be considered as analogous to those performed in general philosophy, if one is inclined to agree that both


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are similar. I do not claim that logical analysis exhausts the entire scope of philosophy or per analogiam theology. Yet several theological problems are suitable for logical treatment. Anyway, the phrase ‘Logical analysis in theology’ could be another title of this paper. Some authors, like Jan Salamucha or Joseph M. Bocheński, Polish theologian-philosophers, claimed that rigorous logical analysis is indispensable in order to modernize theology;2 other were less ambitious, but there is no doubt that the place of logic in rational theology is important. Thus, I do not make programmatic postulates, but assert the existing state of affairs. I am fully conscious that there are several specific problems in theology, which should be taken into account, when methods of theology become a discussed topic. The typical list of problems included into a survey of rational philosophy3 covers the following problems: (i) Theological Prolegomena; (ii) Divine Attributes; (iii) God and Creation; (iv) Topics in Christian Philosophical Theology; (v) Non-Christian Philosophical Theology. Logical analysis can be used in dealing with any topic taken from the above list. The point (i) has a special character. It concerns, among others, the authority of the Scripture, tradition and the Church, and the revelation and inspiration. Although particular Christian confessions differ in their relation to the Scripture, the revelation, the Tradition and the Church as authorities, we can assume for the sake of argument that there is the ultimate core of theological opinions. They concern further problems, like divine attributes, God and creation, and several other theological questions. Believing Christians are obliged to accept theological preliminaries and their consequences for special issues. A secular thinker can complain about this situation and say that theological limitations are at odds with rationality of arguments, particularly, because these restrictions unjustifiably mix natural and 2 For details of this movement, see J. Woleński, Polish Attempts to Modernize Thomism by Logic (Bocheński and Salamucha), “Studies in East European Thought” 2003, vol. 55, p. 299-313. 3 See Th. Flint, M. Rea, The Oxford Handbook, op. cit., p. 5.

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supra-natural factors. On the other hand, a Christian thinker might argue that if someone does not accept theological preliminaries, he or she unjustly excludes supra-natural entities from the scope of sound argumentations. This controversy results in the conclusion that any rational exchange of views between believers and unbelievers is impossible. I do not agree with this diagnosis. First of all, it appears as unclear why the line of division separates believers and unbelievers. In fact, although there are various dissimilarities between Catholics and some Protestants as far as the matter is concerned, theological preliminaries force completely different views concerning divine issues, but this fact does not exclude discussions between rival Christian confessions. I would like to propose a debate without special assumptions, in particular, accusing the opposite side of irrationality. In order to give a concrete example, I employ a well-known meta-logical fact. Assume that X is a consistent set of sentences. By the Gödel-Malcev completeness theorem, X has a model in which its elements (sentences belonging to it) are true. We do not need to think that these sentences are true in the real world. The model of X is a mathematical entity, philosophically speaking, an intentional object correlated with this set by the content of propositions in question. Assume now that these sentences are about God and follow theological preliminaries. According to our assumption, we initially consider X as a consistent set of sentences. These assumptions can be correct or not and one of the uses of logic in theology consists in checking which answer should be accepted. Even if both parties of the debate, that is, believers and unbelievers, propose conflicting answers to the problem of consistency of theological discourse, they can understand each other perfectly. Similarly, we can investigate several other questions, for example, the problem of the form of propositions about God, the logical status of the term ‘God’ or the relation between naturalistic and supra-naturalistic language. Step by step, our theo-logic, to use a convenient label, might be extended to include discussions about the argument invented to demonstrate the existence of God (are they


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proofs or only persuasive strategies?) or the relation of God to evil (do the attributes of God cohere with the existence of God?). A skeptic might observe that such debates are dramatically inconclusive. Well, but this remark concerns every philosophical topic logically analyzed. Thus, philosophical theology is not an exception in this respect and the opposite situation would be surprising. Similarly, an argument that some theological problems, for instance, the Resurrection or the Incarnation exceed the tools of logical analysis does not seem to be as convincing, because this is typical for philosophers to maintain that some topics transcend given methods. My position is rather modest and consists in a claim that logical analysis can help in clarifying one’s own positions, defending them and arguing against opposite views. I will give some examples of such analytic arguments. And one additional remark is in order. I think that any participant of a discussion concerning theology should openly declare their own relation to religion. I consider myself as an unbeliever (this is temporary denomination). This attitude determines to some extent that I am interested in the place of logic in arguments for or against God’s existence. Otherwise speaking, I consider this problem as the most central question of religion and theology. The first problem related to the ‘model-theoretic’ account of theological discourse pertains to the status of the term ‘God’. Assume that we work in the standard first-order logic. If we take the sentence (*) ‘God exists’, its subject-word looks like a proper name (individual constant). Now, first-order logic excludes empty proper names. This is the reason why some believers say that unbelievers cannot discuss the main theological issue. However, it is not true. The unbeliever has no reason to maintain a priori that theology is inconsistent, he or she can, without violating own convictions, accept models in which (a) is true, that is, God exists. Yet it must be emphasized that ‘to exist in a model’ does not mean ‘to really exist ‘. If a believer and his/her opponent agree at this point, they can argue for their views. We have another device enabling us to achieve the same task. In fact, the logical form of (*) exceeds beyond the syntax of first order-logic, because

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the verb ‘exists’ functions as a predicate. In order to work with a form which is coherent with first-order syntax, we can employ: (**) ∃x(x = God), in which the word ‘God’ serves as an individual constant, but the existential mark is hidden in the quantificational prefix. Yet (**) assumes that the term ‘God’ is non-empty. This situation seems be problematic for some unbelievers, because they deny that God exists. Fortunately, there is a very simple manoeuvre in order to read (**) in a way which satisfies all parties. We can change (**) into: (***) ∃x(x is God). The word ‘God’ occurs in (***) as a part of the predicate ‘is God’. This predicate has a set as its denotation. Since denotations of predicates can be empty sets, the difficulty noted above simply disappears. Denote the reference of ‘is God’ by G. Roughly speaking, believers maintain that G is a non-empty set, but unbelievers think that G = ∅. Obviously, the believer should justify that there is an x such that x ∈ G, but the unbeliever ought to argue that there is no x such that x ∈ G. The division into believers and unbelievers is too simple in order to cover all religious (theological) possibilities. Some people declare that they know that God exists, other say that they know the opposite. The statements ‘I believe that God exists’ and its negation ‘I do not believe that God exists’ also render attitudes towards the existence of God. The notorious ambiguity of ‘I know’ and ‘I believe’ causes considerable difficulties in establishing a map of possible positions in religious matters. Ordinarily speaking, any motivated belief can be considered to be a piece of knowledge. Motives for believing can vary from astrology and superstitions to mature arguments taken from science. They are psychological, cultural, practical, political, etc. This approach to knowledge makes it indistinguishable from merely having beliefs. On the other hand, we have the defini-


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tion of knowledge as justified true belief. For many reasons (see, the Gettier problem, for example), it claims too much. Let us take scientific knowledge as a paradigmatic case of knowledge. We can say that I know that A if and only if (a) I believe that A; (b) A is true according to the present shape of science; (c) I am ready to defend A using arguments taken from science. The point (a) treats belief as a mental state, the point (b) suggests that truth is approximate, not ultimate, but the point (c) treats defending as something dispositional. I assume that common knowledge about empirical matters can be regarded as satisfying points (a)-(c) and call both specimens of knowledge cognitively rational. The last qualification has nothing to do with evaluation, positive or negative. Rational knowledge in the outlined sense can be much less valuable from a practical point of view. Rational cognitive knowledge is to be contrasted with beliefs based on the act of faith. Religious beliefs are paradigmatic in this respect. Some believers say that they believe in God and they do not need any rational arguments for their faith. I will consider beliefs based on faith and knowledge (or cognitive beliefs, if you like) as ideal types. Actual mental states are usually a mixture of various and different factors. I propose a map of attitudes towards religion.4 It consists of theism, fideism, atheism and agnosticism as attitudes related to the statement (***) (or (*) or (*). Theism and fideism are frequently considered equivalent (or almost equivalent) standpoints and the same concerns atheism and agnosticism. My analysis proposes to distinguish theism and fideism on the one hand, and atheism and agnosticism on the other hand. I guess that the distinction proposed below might help in clarifying discussions between philosophers and ordinary people affirming or denying the existence of God.

4 I follow J. Woleński, Theism, Fideism, Atheism, Agnosticism, [in:] L. Johannson (ed.), Logic, Ethics, and All That Jazz. Essays in Honour of Jordan Howard Sobel, Department of Philosophy, Upsala University, Uppsala 2009; some fragments presented are taken verbatim from this paper.

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Using contemporary terminology, the theist in Aquinas’ style considers (***) as a piece of knowledge, that is, the statement satisfying (a)-(c). Hence, the theist of this kind can express his or her position by: (1) I know that God exists. What should the atheist say? If he or she replies ‘God does not exist’, the theist might ask ‘Well, but do you know that?’. The atheist has a problem, because if his/her position is considered the negation of theism, the answer should be expressed by: (2) I do not know that God exists, but this is too weak. The atheist is rather inclined to say: (3) I know that God does not exist. However, (2) and (3) are not mutually contradictory, but contrary. This simple reasoning shows that there are other possibilities in attitudes towards (***), then (1) and (2), if the latter is taken as an adequate expression of atheism. The matter is still more complicated if we observe that: (4) Although I do not know that God exists, I believe that He exists. Since many historical fideists accepted (4), this possibility is not abstract. Thus, a closer analysis of theism, atheism, etc. is required. In particular, we should take into account knowledge and belief as parameters important in attitudes related to (***). How are they related? I accept the following dependence (KA – I know that A, BA – I believe that A):

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(5) KA ⇒ BA. This postulate is not a logical axiom but rather the result of an analysis of the meaning of ‘to know’ and ‘to believe’. It can be justified by the account of knowledge as justified true belief. Even if we consider ‘true’ and ‘justified’ in the above definition of knowledge to be somehow problematic, (5) can be taken as fully granted for any kind of belief, including religious faith. The reverse dependence, that is BA ⇒ KA, does not hold. Speaking metaphorically, belief and knowledge are half-separated: although knowledge implies belief, the latter does not imply the former. In order to go further, I will establish some formal properties of K and B. The following diagram (D) is instrumental in this respect:5 ν









See J. Woleński, Logical Squares: Generalizations and Interpretations, “Logica” 1995, Proceedings of the 9th International Symposium, T. Childers, P. Kolár, and O. Majer (eds), Filosofia, Praha 1996, p. 67-75; Applications of Squares of Oppositions and Their Generalizations In Philosophical Analysis, “Logica Universalis” 2008, 2 (1), p. 13-29; reprinted in J. Woleński, Essays..., op. cit., p. 255-269.

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This diagram is a generalization of the well-known logical square for categorical and modal propositions. Let symbol ■ denote K or B. Understanding α as ■A, β as ¬A, γ as ♦A and δ as ♦¬A, we have the following facts, among others: (6) ¬(α ∧ β) (α and β are contrary) (7) (α ⇒ γ) (α entails γ; γ is subordinated to α) (8) (β ⇒ δ) (β entails δ; δ is subordinated to β) (9) (α ⇔ ¬δ) (α and δ are contradictory) (10) (β ⇔ ¬γ) (β and γ are contradictory) (11) (γ ∨ δ) (γ and δ are complementary) (12) (■A ⇔ ¬♦¬A) (■ is definable as ¬♦¬A) (13) (♦A ⇔ ¬■¬A) (♦ is definable as ¬■¬) (14) (β ⇔ ■¬A) (β is definable as ■¬A) (15) ν ⇔ (α ∨ β) (ν ⇔ (■A ∨ ■B) (16) µ ⇔ (γ ∧ δ) (16) µ ⇔ (♦A ∧ ♦¬A) (17) ¬(ν ⇔ µ) (ν and µ are contradictory) (18) α ∨ β∨ µ (α, β, µ are mutually exclusive and jointly exhaustive) (19) κ ⇔ A (20) λ ⇔ ¬A) (21) ¬(α ⇒ κ) (¬(■A ⇒ A)) (22) ¬(κ ⇒ α) (¬(A ⇒ ■A)) (23) ¬(λ ⇒ β) (¬(■¬A ⇒ ¬A)) (24) ¬(β ⇒ λ) (¬(■¬A ⇒ ¬A)). These dependencies (in particular, (21)-(24)) normalize the operators K and B as functioning in non-normal modal logic. It is important to observe that knowledge that A does not implies that A is true, but this fact is not at odds with (a)-(c) (see above). The main reason is that ‘to know’ and ‘to belief’ is a strongly intensional word. A well-known criterion of the sentence I∃xPx implies neither ∃xPx nor ¬∃xPx, if the letter I stands for a strong intensional operator. By analogy, neither KA nor BA implies A. Note also that (5) has no justification in (D). However, we can say that accepting KA ∧ ¬BA would not be rational,


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because if someone knows that A he or she should (is cognitively rational to) believe that A as well. Of course, there are always problematic cases of assertions like ‘I know that A, but I do not believe that A’ (this example puzzled Moore and Carnap), but if we distinguish consistency and rationality, (5) is defensible via an appeal to the latter. Some special views concerning the plausibility of BA ⇒ KA and it its strengthening to the equivalence KA ⇔ BA will be discussed later. If we accept (5) as a conceptual truth or an additional axiom, we have the following consistent possibilities: (25) KA ∧ BA (26) ¬KA ∧ BA (27) ¬KA ∧ ¬BA. Thus, I can know that A and believe that A or not know that A and believe that A or not know that A and not believe that A. Further, not knowing that A is consistent with believing that A as well as not believing that A, but not with both simultaneously. Consequently, not knowing that A does not imply anything definite about believing. Some additional possibilities appear when we consider: (28) K¬A ⇒ B¬A. This generates the following coherent cases: (29) K¬A ∧ B¬A (30) ¬K¬A ∧ B¬A (31) ¬K¬A ∧ ¬B¬A. By combining, (25)-(27) and (29)-(31), we obtain: (32) ¬K¬A ∧ ¬KA ∧ BA (33) ¬K¬A ∧ ¬KA ∧ B¬A (34) ¬KA ∧ ¬BA ∧ ¬B¬A

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(35) ¬K¬A ∧ ¬BA ∧ ¬B¬A (36) ¬KA ∧ ¬K¬A ∧ ¬BA ∧ ¬B¬A. Now, if we skip the formulas which are proper parts of others or follow from others, there remain (25), (29), (32), (33) and (36). If we apply the above considerations to (*), we obtain: (37) I know that God exists and I believe that God exists (38) I know that God does not exist and I believe that God does not exist (39) I do not know that God exists and I do not know that God does not exist and I believe that God exists (40) I do not know that God exists and I do not know that God does not exist and I believe that God does not exist (41) I do not know that God exists and I do not know that God does not exist and I do not believe that God exists and I do not believe that God does not exist. Combination (37) expresses the standpoint of the Catholic Church on which knowledge and faith (religious belief) are complementary and mutually irreducible; the famous formula fides quaerens intellectum displays this view. (38) is characteristic for atheism popular in the French Enlightenment or in Sartre. (39) is the credo of fideism. (40) defines agnosticism and (41) describes the radical epistemological-theological skepticism (I will skip this view, because I consider it not very interesting). Theism and fideism contain ‘I believe that God exists’ as their common part and this statement can be taken as expressing theism (or fideism) in a wider sense. Similarly, atheism, agnosticism and radical skepticism (as given by (41)) agree with respect to ‘I do not believe that God exists’ and this assertion characterizes atheism in a broader sense; this picture is historically faithful, because all three views were and are pointed out as atheist. Atheism and agnosticism accept the statement ‘I do not know that God exists and I believe that God does not exist’. I propose to count this attitude


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as indirect atheism. If we take the set {theism, fideism, atheism, agnosticism}, all of the standpoints listed in it are mutually independent and contrary, but not all are mutually contradictory. Theism in the broader sense and indirect atheism are also contraries, but the former is inconsistent with atheism in the broader sense. The above analysis is based on a purely formal criterion. This does not preclude that particular views differ in their account of how knowledge and faith are related. Now, I can supplement remarks (see above) about various motives of religious belief. Let me start with fideism. The fideist will probably say that he or she accepts (5), but providing that knowledge as applied to religion is of a special kind and, therefore, ‘I know’ in the religious discourse has a specific meaning, different from that in the statement ‘I know that 2 is the first prime number’. Thus, when the fideist announces that we have no rational knowledge of God’s existence, that is, based on intersubjectively accessible empirical evidence, this does not exclude another kind of experience, for example, mystical experience. A more ordinary understanding of fideism is perhaps given by: (42) I believe that God exists, although I have no rational support for my belief and I claim that there are no rational reasons for denying God’s existence. Yet I think that (39) gives a better account of fideism, at least for its logical analysis and provided that ‘I know’ refers to rational knowledge. Logical reasons also dictate that I prefer ‘I know (do not know) that God exists’ than ‘I know (I do not know) whether God exists’. The theist of the Thomistic brand (or more generally, recommended by the Catholic Church) accepts the equivalence (i) KA ⇔ BA and justifies this by the above quoted formula fides quaerens intellectum, at least so far as the matter concerns the existence of God. Pragmatically speaking, faith (religious belief) is the starting point and its equivalence with knowledge is dictated by ratio recta. Since (i) holds for ¬KA and ¬BA as well, the theist has to accept it, too. According

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to standard theism, (ii) ¬KA ⇔ ¬BA describes unbelievers. They do not believe if and only if they do not know. Their lack of knowledge and belief may be unconscious and non-culpable, but if it is deliberate, they are deprived of ratio recta. Paradoxically, the atheist also shares (i) and (ii), but the motivation is different here. First of all, the atheist strengthens (ii) to (iii) K¬A ⇔ B¬A, because atheism consists in a positive view about God’s non-existence, not only in expressing the lack of knowledge and faith concerning His non-existence. Thus, the atheist asserts (iv) K¬A ⇔ ¬KA, (v) B¬A ⇔ ¬BA, and (vi) K¬A ⇔ B¬A, and argues that the theist abuses the concept of knowledge. In other words, the atheist maintains that belief and knowledge as attitudes concerning (***) should be entirely rational. This position is best expressed by (vi), but (ii) holds only because KA and BA are false. Finally, the atheist can agree that KA and BA are true, but knowledge, belief and truth are understood differently than in rational discourse. Since (i)-(vi) do not follow from (D), their acceptance or rejection has their source in decisions motivated by various factors. The agnostic proceeds similarly to the fideist in the matter of the relation between knowledge and faith, namely he or she derives some consequences from the lack of knowledge. However, agnosticism insists that knowledge should be taken as rational, but belief as rational if it is grounded by knowledge, otherwise it is irrational. Thus, there are obvious affinities between atheism and agnosticism. The skeptic, according to his or her general epistemological position, is completely neutral about the above issues. Skepticism is not particularly interesting, because it consists of merely negative assertions. The remaining views propose at least some positive beliefs, but theism and atheism additionally insist that their adherents know something about God’s existence or non-existence. A positive element in belief or knowledge seems to be of the utmost importance for world-view matters, because negative declarations do not suffice in this area. Although, on the one hand, every view (I omit skepticism) in the set {theism, fideism, atheism, agnosticism} is contrary to the rest, on the other hand, theism and atheism operate on a different epistemological level than


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fideism and agnosticism. In particular, the latter assume epistemological skepticism concerning the knowledge of God’s existence, but reject theological skepticism in this question. This positive content, directly related to theology in the case of fideism and indirectly in the case of agnosticism, is contained in statements ‘I believe that God exists’ and ‘I do not believe that God exists’, respectively. Similarly as in the case of theism and atheism, the agnostic or fideist approach to (i)-(vi) exceeds the logical space. If we start with ‘I do not know whether God exists’, we have two choices concerning God’ existence, namely we can either assert or deny it; fideism chooses the first possibility, agnosticism opts for the second one. Saying that decisions in world-view matters are practical and motivated by many factors, I do not deny that defenders of attitudes towards religion support their own positions and criticize those of their opponents. Yet, due to the variety of evidence involved in the proposed arguments, we have an actual mosaic of issues and their epistemological qualifications. Let me appeal to historical examples. The theism of the Aquinate and similar thinkers looks for deductive arguments for God’s existence, eventually supplemented by data from physics and metaphysics. Anselm of Canterbury and other theological rationalists formulated ontological proofs, which are deductive and entirely conceptual. Circumstantial proofs or arguments for God’s existence invoke very different circumstances. For example, the defenders of the idea of Intelligent Design frequently say that the mathematical nature of physical reality cannot be explained without an appeal to God. Other defenders of religion point out miracles, historical facts (for instance, Jesus’ life), morality or mystical states. It is difficult to say something more about this variety than point out that its elements have a very different argumentative force. The situation is additionally complicated by the fact confirmed by almost every world-view discussion, that the use of arguments pro and contra (*) is very deeply determined by the positions of people involved in such debates, much more than in scientific or even philosophical controversies.

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Nevertheless, the difficulties in assessing and collecting possible arguments for God’s existence or against this belief do not preclude some general observations about the strategies of theism, fideism, atheism and agnosticism; I exclude skepticism once again. In particular, we can say something about onus probandi in related proofs or disproofs. The onus probandi (the burden of proof) indicates which part in a discussion should prove or justify its thesis. In general, the onus probandi belongs to the party which asserts something positively. Clearly, the issue primarily concerns (***) (this form is the most suitable for a logical discussion), because proving this sentence motivates the theist to take his or her position captured by (37). Hence, the atheist is not obliged to prove the negation of (***). Thus, the demand, frequently directed to the atheists, that they should prove God’s non-existence, is simply misguided for logical reasons. This does not mean that the atheist has nothing to do with justifying his or her own standpoint. In general, the atheist asserts that the set {God} is empty. One argument consists in objecting to given proofs that this set is non-empty. This way is of limited force, just because the theist can always say that other proofs are possible. Other atheist arguments try to show that no empirical data suggest that the predicate ‘is God’ refers to a non-empty set. For example, many atheists claim that no empirical support can be given for assertions about beings devoid of empirically graspable arguments. Further analysis leads to a substantial discussion concerning reasons invoked by defenders of theism and atheism. Since the fideist does not appeal to knowledge (e.g. scientific knowledge) of God or His existence, he or she limits his or her arguments to personal ones. Thus, fideists consider the onus probandi as related to ‘I believe’, not to ‘I know’. Similarly, the agnostic stresses ‘I believe that God does not exist’ and claims that there are no sufficient reasons for believing that God exists. Fideists and agnostics, contrary to theist and atheists, are more interested in believing than in knowing, both related to God. They agree that knowledge (in the rational or scientific sense) does not apply to God, but they can disagree whether there is another kind of knowledge or


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what the word ‘knowledge’ means. Historically speaking, atheism and agnosticism usually point out the lack of scientific knowledge of God and insist that other types of knowledge are merely personal and subjective cognitive states, and thereby cannot pretend to objectivity. This last observation shows well how the whole issue is complicated and what the consequences are of the fact that theism and atheism operate on different epistemological levels, although, on the other hand, fideism is closer to theism and agnosticism is closer to atheism, in particular, if we take into account the practical consequences of discussed views. Demonstratio appeared in mathematical texts in the 17th century as regards speaking about deductive proofs (in our sense). The letters QED (quod erat demonstrandum, which was to be proved) remind us of this terminological custom until now. It is interesting that the word deductio had a different meaning at that time. Its employment in medieval Latin was related to law and applied to deriving a claim or right before a court. The process of such a derivation was divided into two parts, namely factual and legal demonstration. Assume that a person Y borrowed money from a person X and did not return the loan. If X put the case before the court, he had to prove that the loan actually took place and that there is a legal rule, which generated the claim in question. In other words, one problem concerned quid facti?, and the second one, quid juris? Kant used this sense of deductio in explaining his idea of the transcendental deduction, pointing out that every case of this reasoning has a quid facti? part and a quid juris? part. When Kant spoke about deduction in our sense, he used the term Ableitung (derivation by logically admissible steps). Of course, Ableitung is analytic, apodictic and, thereby, a priori. For Kant, the transcendental deduction has the same nature, except that it is analytic. Thus, Kant argued that some, in fact, the most important pattern of reasoning, is apodictic and a priori, but is not reducible to purely logical steps. A less sophisticated philosophically usage occurred in Newton, who also employed deductio in a very loose sense as deriving hypotheses on the basis of some evidence, but who proved the

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theorems in the Principia Philosophiae Naturalis by rigorous demonstratio. The development of logic and meta-mathematics in the 19th and 20th centuries resulted in, among other things, establishing the meaning of ‘deductive proof’ in the sense of Kant’s Anleitung in the following manner. The deductive proof of a sentence (more generally, a formula; but we can speak about sentences in the present context without any loss of generality) A on the basis of a set X of sentences (notation: Pr(A, X)) is a finite sequence of sentences S = A1, …, An-1, An such that An = A and for any 1 ≤ i ≤ n, Ai is either a logical axiom (Ai is provable from the empty set, that is, Pr(Ai, ∅)) or An ∈ X or An is provable (derivable) from the earlier items of S by the applying merely logical rules of inference. By a logical rule of inference we mean a rule which preserves truth, that is, never leads to false conclusions from true premises. Accordingly, the concept of deductive proof essentially uses the notion of logical consequence or logical entailment. Thus, we can say that A is a logical consequence of X or X logically entails A instead of saying that A is provable from X. If X is an axiomatic system, we speak about axiomatic proofs, that is, proofs based on some sentences that are adopted as axioms. Traditionally, one says that axioms are accepted without proof. However, if one writes axioms, one automatically constructs one-term sequences and any such sequence can be considered a proof for an axiom which constituted this sequence. In the case of axiomatic proof (as in geometry), every member of S is either an axiom of a consequence of axioms obtained by the logical rules of inference. By conditional proofs or proofs from assumptions, we understand derivations in which S consists of assumptions, not necessarily considered to be axioms. If A is an axiom, we think of it as a true sentence, unless it is demonstrated incorrect. This presumption does not concern conditional proofs, because these assumptions are only accepted, but not accepted as true. For example, we can derive deductive conclusions from falsehoods or sentences temporally or hypothetically accepted. Since the general concept of proof does not require that all elements of


Jan Woleński

S are true, one traditionally distinguishes formal and material correctness of proofs. A proof Pr(A, X) is formally correct if A is logically entailed by X, but it is materially correct if it is formally correct and its premises are true. It is taken as granted that axiomatic proofs are materially correct. This allows us to say that proving A consists in proving that A is true, but this is an oversimplification. Anyway, the concepts of proof and deductive proof can be considered as equivalent under the above analysis. Not all arguments are deductive. Consequently, if we accept the equivalence of proofs and deductive proofs, not all arguments fall under the form Pr(A, X). This looks trivial, but it is not, because we should distinguish between worthless non-deductive arguments and good non-deductive arguments. If I say that tomorrow it will rain because one week ago it was Friday, my argument fails entirely, but if I say that tomorrow it will snow because today it is cold, this reasoning has some credibility. Now, there are well-known problems with good non-deductive arguments consisting in difficulties in defining non-deductive goodness, soundness, correctness, etc. Let us assume, without attempting to state how the qualities of non-deductive arguments might be measured, that we have intuitive criteria in this respect. The tradition of using such words as apoidexis, demonstratio or ‘proof’ suggests broadening the concept of truth in order to apply it to good non-deductive arguments. In general, such arguments consist in acquiring empirical evidence to support the proposed statements or hypotheses. Legal arguments, related to the question quid facti? and empirical reasoning in science are paradigmatic examples of proofs in the presently considered sense. Let us observe that the word ‘proof’ has a definite rhetorical flavour. One could propose that we should use the generic term ‘justification’ and divide justifications into deductive (proofs) and inductive (in the broad sense), and eventually the latter into good and worthless. Under such a convention, every proof is a justification, but not reversely. However, it is difficult to expect that this proposal would be accepted, perhaps except by philosophers of science. For example, judges will say ‘We do

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not merely justify our statements concerning that someone is guilty, we must prove them’. Similarly, when Newton said hypotheses non fingo, he had in mind that he proposed something proved, not merely justified. The reason for such an attitude seems to be related to an evaluative or normative aspect of the meaning of ‘proof’, suggesting that if something is proven, it is beyond a reasonable doubt, at least subjectively. This observation decides that apodeixis is understood as apodictically convincing (the same concerns de-monstratio), although etymology does not suggest this. Anyway, according to common opinion it is much better to prove than to merely justify. Deductive proofs and proofs from various empirical circumstances are closely related. As it has been observed many times not everything is deductively provable. Even if we agree that logical tautologies are derivable from the empty set and proofs of axioms are one-term sequences, it remains to justify that these, but not other axioms, are adopted. The issue is particularly important when the postulates of empirical theories are proposed, but also mathematical axioms cannot spring from nothing. I will illustrate this issue by an example relevant for the main topic of my paper.6 The proofs of the existence of God ex motu and ex causa efficiendi fall under the following scheme: (****) if ∀x∃yP(x, y), then ∃y∀xP(x, y). The following arguments are instances of (****): (a) if every moving thing is moved by its mover, then there exists a mover of all things; (b) if everything is an effect of a cause, then there exists a cause of all things. Clearly, (****) is invalid. As counter-example we can take a the fallacious argument saying that if for every natural number n, there is a number greater than n, then there is the natural number greater than all natural numbers. Russell in his famous conversation See H. Sobel, Logic and Theism. Arguments for and against Beliefs in God, Cambridge University Press, Cambridge 2004 for the comprehensive survey of proofs of God’s existence. 6


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with Father F. Copleston used another example, namely that the assertion that there exists the mother of all people does not follow from the assumption that every man has a mother, because the former is false and the latter true. Thomas Aquinas supplemented (****) in his proofs by reasoning to the effect that infinite sequences of movers or efficient causes are impossible. I do not evaluate whether his demonstrations are sound or not. He appealed to Aristotelian physics, while his followers are rather inclined to rely on metaphysical necessity. Anyway, Aquinas intended to prove the existence of God. He used the label demonstratio, applied syllogistic forms and proceeded by reductio ad absurdum. Although his reasoning also appeals to empirical circumstance, the Five Ways are deductive and have the structure of syllogisms. Thus, interpretations of Aquinas that point out7 that he did not offer proofs, but only suggestions for unbelievers or defenders of the faith, are historically unfaithful. There is still another reason to think that Thomas Aquinas considered the Five Ways to be proofs. His idea of religious belief regarded every act of faith as veridical, if it comes from right reason (ratio recta). This means that ratio recta produces knowledge, independently of whether it concerns religion or not. There can be a problem with revealed truths that transcend human cognitive capacities (for example, the Trinity Dogma), but (***). This statement is subjected to a normal demonstratio by syllogisms and reductio ad absurdum employing factual information. The Aquinate, a very faithful follower of Aristotle, could not accept knowledge without its being proven. Anyway, there is no obstacle to consider Aquinas’ and other proofs to be arguments for theological models in the sense exhibited at the beginning of this paper. The role of considering theological argument as concerning some model is particularly impressive in the case of ontological proofs of God’s existence. I shall comment upon a proposal made by the late

See for example J. Schmidt, Philosophische Theology, Kohlhammer Verlag, Stuttgart 2003 in the introduction to Part I.


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Jerzy Perzanowski.8 Perzanowski offers a reconstruction of the reasoning proposed by Charles Harthshorne. It is based on two axioms: (a) if a being is the most perfect, its existence is necessary (b) the existence of the most perfect being is possible (Leibniz lemma). Using modal system S5, we can obtain: (c) the most perfect being exists. Perzanowski simplified Harthshorne’s argument by proving (b) in a stronger modal logic equating truth and necessary truth. Moreover, this logic proves (this is another version of the Leibniz lemma): (d) if the most perfect being is possible (its existence is possible), it is necessary (its existence is necessary). Hence, via modus ponens, we have: (e) the most perfect being exists as necessary (necessary existence implies existence). The last step ends Perzanowski’s reconstruction. If the matter concerns (a), one might observe that it is a conditional assertion having an existential antecedent. Hence, any further application of this axiom essentially depends on the truth of the sentence ‘the most perfect being exists’. If (b) is applied in proving that the most perfect being exists, the entire argument is burdened by petitio principii. This is quite evident if we consider the equivalence:

See J. Perzanowsaki, O wskazanych przez Hartshorne’a modalnych krokach w dowodzie ontologicznym Anzelma (1), unpublished. 8


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(f) the most perfect being exists if and only if the most perfect being exists and it is possible. However, (f) is trivial because the sentence ‘a exists’ entails ‘it is possible that a exists’. As far as the issue concerns Perzanowski’s simplification, the success of his main move based on (d) requires a very strong modal logic (the logic of strong rationalism). Although the scope of the term ‘logic’ is conventional to some extent, one can express some serious doubts as to whether logic should lead to such strong existential consequences. Personally, I would prefer to say that the proof of the Leibniz lemma and further steps of the entire argument proceed in the context of some formal theory involving modal concepts and relations between them and that this theory is not purely logical. Nevertheless, the fact that Anselm’s argument can be presented via a precise formal machinery shows the power of formalization applied to concepts of rational theology or theological ontology.9 Since, according to Leibniz, possibility is logically equivalent to consistency, (d) can be rendered as: (g) if the most perfect being is consistent, its existence is necessary. Let the letter B denote the predicate ‘is the most perfect being’. Assume that B is introduced by some consistent set K of sentences. By the Gödel-Malcev completeness theorem (a set X of sentences has a model if and only if this set is consistent), K has a model. This assumption meta-mathematically guarantees that K is non-empty. One can be even tempted to say: (h) if a set of sentences has a model, it possesses it necessarily (it is impossible that the set in question has no model).


See also M. Tkaczyk, Is the Ontological Proof of God’s Existence an Ontological Proof of God’s Existence?, “Logic and Logical Philosophy” 2007, no. 16, p. 289309 for embedding Anselm’s proof into first-order logic without modalities.

Theology and Logic


However, this last assertion requires additional constraints (see below). The way in which K is given as defining the item conceived as greater than any other being, is very important. First of all, the phrase ‘the conceived item’ means ‘the item described by a set of sentences’. Secondly, according to the common opinion, the most perfect being (MPB, for brevity) is introduced by the maximalization procedure. Meta-logically speaking, it consists in identification of MPB with a collection of perfections (omniscience, omnipotence, the greatest goodness, immutability, infinity, etc.); existence belongs to perfections. Assume that K0 is an initial consistent collection of perfection. It might be maintained that application of the Lindenbaum maximalization theorem (every consistent set of sentences has a maximally consistent extension) provides an argument for generating the adequate set K. However, the Lindenbaum extensions are not unique. More specifically, if X is a consistent set of sentences, it has more than one (in fact, there are infinitely many) maximally consistent extensions. Every such extension E has a model (a possible world) in which elements of E are true. Moreover, since these sentences cannot be false in this model, they must be true in it as well. Note that the relativisation to a specific model is crucial. Let us apply these observations to the set K0 and K. Clearly, the former set has several different maximally consistent extensions, including K as their part; in fact, K itself is not maximal, but it is a minor point. Meta-mathematical observations about K do not suffice for a demonstration that this set is true in every possible world. Note that we could consider K as a body of absolute necessities (necessary truths), that is, sentences true in every model. It is obvious now that necessity of truth, with respect to a specific model is not absolute, but just relative, because a sentence true in one model can be false in other possible worlds. Moreover, there is no a priori reason to refer to one and only one initial set K0 of perfections. One could ask, for instance, why the greatest goodness or immutability, but not changeability or goodness directed to people deserving it to some degree, should be regarded as prima facie perfections. Typical-


Jan Woleński

ly, listed perfections are recommended by Christian (or other similar) theology, but it is only a religious argument, but not a logical one. Thus, contemporary versions of Anselm’s ontological proof do not show that B is not empty in virtue of logical necessity. Furthermore, one cannot demonstrate independently the assumptions of a given monotheistic theology that the denotation of B is unique. If we apply a similar technique to any consistent collection of anti-perfections (that is, pointing out the lack of perfections in the ordinary or theological sense), it is easily provable by the dual logic and meta-mathematics (it takes falsity as the distinguished value) that there is a being such that nothing lesser can be conceived. We can interpret this being as the Absolute Evil (AE, for brevity). If we entirely omit the ordinary or religious meaning of perfection, AB is a maximal being, because it is constructed by a similar maximalization strategy as employed in the case of the denotation of B. There is no reason, at least logical, in order to maintain that AE is less real than MPB or Christian God. This leads to a Manichean theology with its perennial battle between forces of Goodness and Evil or even a radicalized Marcionic heresy in which the world presents itself as a emanation of the personified Evil, in particular, deceiving people in order to make them suffer more. The theist could presumably answer that the meta-mathematically phrased ontological proof is enough for demonstrating the existence of MPB, let us say a god of philosophers which has properties which are not entirely coherent with Christian theology. This being is necessary and thereby exists. However, this standpoint is not correct, even if we agree some perfections actually occur in our world. Lindenbaum’s maximalization procedure does not imply that there exist maximal perfections in the sense of Anselm and his followers. Although people know something, can do something or behave properly, these facts do not entail that there is a being (subject) which realizes these properties in the most maximal manner. Thus, there are models in which no MPB occurs. In fact, some maximally consistent extensions of K0 contain the sentence ‘there exists the MPB, according to a given specification of perfections’, but this

Theology and Logic


sentence is false in other extensions. Since we do not know which model represents our universe, we can only say that the gods of philosophers (and theologians as well) exist in some possible models.10 We can also make some general observations about the strategies of theism, fideism, atheism and agnosticism. In particular, we can say something about onus probandi in related proofs or disproofs. The onus probandi (the burden of proof) indicates which part in a discussion should prove or justify its thesis. In general, the onus probandi belongs to the party which asserts something positively. Clearly, the issue primarily concerns (***), because proving this sentence motivates the theist to take his or her position captured by (37). Hence, the atheist is not obliged to prove the negation of (***). Thus, the demand, frequently directed at atheists that they should prove God’s non-existence, is simply misguided for logical reasons. There is also a psychological argument in a similar direction. It applies to atheists and agnostics grouped together, let us call them disbelievers for a while. A religious attitude, independently of whether it is understood as knowledge or merely faith, has grounds in positive mental states. Of course, one can say that disbelievers lack such states and that the absence of a mental state is also a mental event. This is true, but there are serious doubts as to whether negative state of affairs can be causally efficient. However, the practice of religious disbelievers suggest that, if their atheism or agnosticism is not caused by growing in a non-religious environment, they always look for the weak points of theological arguments. Although I am skeptical about the scientific knowledge that God does not exist, I do not suggest that the atheist has no arguments for justifying his or her own standpoint. In general, the atheist asserts that the set {God} is empty. One argument consists in objecting to given proofs that this set is non-empty. This way is of limited force, See also J. Perzanowski, Reasons of Monodeism, [in:] J. Perzanowski, Art of Philosophy. A Selection of Jerzy Perzanowski’s Works, Ontos Verlag, Frankfurt am Main 2012, p. 41-44, where one can find a good example of model-theoretical thinking in theology; Perzanowski gives a set of definitions for monotheistic philosophical theology.



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just because the theist can always say that other proofs are possible. Other atheist arguments try to show that no empirical data suggest that the predicate ‘is God’ refers to a non-empty set. For example, many atheists claim that no empirical support can be given for assertions about beings devoid of empirically graspable arguments. Further analysis leads to a substantial discussion concerning reasons invoked by defenders of theism and atheism. Since the fideist does not appeal to knowledge (e.g. scientific knowledge) of God or His existence, he or she limits his or her arguments to personal ones. Thus, fideists consider the onus probandi as related to ‘I believe’, not to ‘I know’. Similarly, the agnostic stresses ‘I believe that God does not exist’ and claims that there are no sufficient reasons for believing that God exists. Fideists and agnostics, contrary to theist and atheists, are more interested in believing than in knowing, both related to God. They agree that knowledge as a cognitively rational enterprise does not apply to God, but they can disagree whether there is another kind of knowledge or what the word ‘knowledge’ means. Historically speaking, atheism and agnosticism usually point out the lack of scientific knowledge of God and insist that other types of knowledge are merely personal and subjective cognitive states and thereby cannot pretend to objectivity. Agnosticism is my own position. It is motivated by logical difficulties of all known proofs of God’s existence. They allow me to reject (***) and, consequently, to accept the statement ‘I do not believe that God exists’. Additionally, I argue that the language of theology and the language of science are semiotically incommensurable. Since any proof (deductive or empirical) assumes that the language in which it is embedded possesses semantic uniformity, prospects of proofs from the world to transcendence or in the reverse direction are not very great. On the other hand, logic helps in understanding the nature of the theological enterprise and its quality.

Jerzy Dadaczyński The Pontifical University of John Paul II Copernicus Center for Interdisciplinary Studies

What Kind of Logic Does Contemporary Theology Need?* 1. Introduction


his paper deals with one of the key aspects of the logic of contemporary theology.1 Due to the fact that one of the essential criteria for the classification of different types of logic is their intolerance or tolerance of contradictions, an important issue will be the question of the meta-theological acceptance or non-acceptance of the antinomies in theology. It should be strongly emphasized that this paper is not devoted to the issue of the antinomies in theology, their enumeration, classification or an explanation of their causes (sources).2 This paper is concerned with the meta-theological acceptance or non-acceptance of the antinomies which may occur in theology and thus have generated a selection of different sorts of the logics of theology. First of all, a controversy between K. Barth and H. Scholz, a logician (and theologian), will be outlined, concerning the ‘logics’ of theology in the context of the theological contradictions. Secondly, an This contribution was made possible through the support of a grant “The Limits of Scientific Explanation” from the John Templeton Foundation. 1 The Polish version of this paper was published in the volume of Polish Theologian Congress held in September 2010 in Poznań. 2 A contradiction may be generated by an investigated object, thinking (about this object) or the language in which one thinks. This classification of the sources of contradictions corresponds to the three modes of understanding of logic: nomological, inferential and reconstructive. *


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overview will be presented of how the problem of antinomies in these disciplines has been resolved in the history of theology and other sciences. Then, para-consistent logic will be presented as a logic which tolerates antinomies. Finally, the last part of this paper provides a set of arguments for classical logic to serve as the logic of theology. An example will be given of physics which ‘copes’ with antinomies while preserving classical logic. Some of the arguments for classical logic in theology will refer to the specificity and practice of theology.

2. Dialectical theology and its ‘derivatives’ – Scholz’s dispute with Barth about the logic of theology Barth’s theology, called dialectical theology after World War I, was a reaction to liberal historical Protestant theology. The Swiss scientist made revelation the absolute starting point of his theology, which was treated by him, in some sense, as non-historical. The revelation of God has, according to Barth, – and this is very important for the study presented here – a dialectical structure. This is mainly because it contains and, moreover, combines contradicting ‘elements’. These contradicting ‘elements’ are, for example, God and man, eternity and time. The unity of contradicting elements, which is generated by revelation, does not tolerate opposites. Therefore, this particular unity is ‘non-descriptive’ and perceptible only from the standpoint of opposites.3 For example, Barth argues that the Court (Gericht) is a negation (Verneinung) of man, by which – and only in this way – man will be accepted (bejaht) in Christ by God. In this context Barth concludes

The assumptions of the dialectical theology of K. Barth from the years 1918-1932 are reported here according to W. Pannenberg, Dialektische Teologie, [in:] Die Religion in Geschichte und Gegenwart. Handwörterbuch für Theologie und Religionswissenschaft, Hrsg. von Kurt Galling, Bd. 2, Mohr, Tuebingen 1958, p. 174ff. W. Pannenberg referred first of all to K. Barth, Der Roemerbrief, Baeschlin, Bern 1919 and K. Barth, Gesammelte Vortraege, Kaiser, Muenchen 1924.


What Kind of Logic Does Contemporary Theology Need?


that (Divine) ‘not’ is ‘yes’ and the non-existence (Nichtsein) of all things is their true existence (Sein). The structure of revelation – in the way Barth perceived it – implies a certain structure of theological statements. The dialectic of revelation implies, in this context, the dialectic of theological statements. In 1922 Barth described the dialectic of theology as a combination of dogmatic and critical speaking, taking into account their common assumption, which itself is an unnameable truth, located in the middle (centre), and gives meaning and significance to both the position and its negation. This middle (centre) is the unity of God with man in Jesus Christ. To try to capture the ‘non-descriptive’ essence of that unity, the theologian, who according to Barth is a dialectician, must correct each statement with the opposite statement. The Swiss scientist argued that the theologian should relate each statement to his negation and vice versa, and highlight ‘yes’ with ‘no’ and ‘no’ with ‘yes’, not lingering longer than just to say “yes” or “not”. It must be added that Barth protested this way against Luther who, in his opinion, made a fact from mystery. From the perspective of the study presented in this paper, it is not sufficient to conclude that Barth’s theology contains antinomies. Much more important is the assertion that the Swiss theologian, at the meta-theological level, fully accepted the antinomies of theology and thereby ‘suspended’ in (his) theology the validity of the principle of non-contradiction. According to Barth, the foundation of theology is revelation. According to him, revelation has a dialectical structure. Theology inherits the dialectical nature of the revelation and thus implies its (dialectical) method. Barth’s meta-theological position was shared, at least for some time, by the theologians who grouped around him in the early 1920s and created a background for dialectical theology. This position is also represented in modern Protestant meta-theological reflection. The main representatives are E. Schlink, G. Ebeling and G. Keil. The meta-theological complete acceptance of the antinomies of theology and the ‘suspension’ of the principle of non-contradiction


Jerzy Dadaczyński

refers not only to certain areas of Protestant theology. It appears that the leading meta-theological ideas represented in the 1920s environment of dialectical theology penetrated some groups representing Catholic theology in a very direct way. M. Schmaus, who until the 1960s was regarded as one of the most influential Catholic theologians, is the author of the following meta-theological statement: The analogy, however, must be completed by the dialectic, which is strictly associated with the analogy. It [the dialectic – J.D.] says the following: two statements on one and the same subject are indeed contradictory, but they are both necessary for the full understanding of a cognized object and cannot be reduced to a common denominator. The dialectical statement is important in all areas of theology as well as the analogous [statement – J.D.]. We will meet them on several occasions.4

The whole picture is completed by the statement that the Polish Catholic meta-theological reflection sometimes accepts antinomies in theology, too, and ‘suspends’ the rule of non-contradiction.5 “Die Analogie muss jedoch ergänzt werden durch die Dialektik, die sich auf das engste mit der Analogie verbindet. Sie besagt folgendes: Zwei Aussagen über einen und denselben Gegenstand stehen zwar im Gegensatz zueinander, sie sind jedoch beide zur vollständigen Erkenntnis des zu erkennenden Dinges notwendig, lassen sich aber nicht auf einen Nenner bringen. Die dialektische Aussage hat in allen Bereichen der Theologie ebenso Geltung wie die analoge. Wir werden ihr bei vielen Gelegenheiten begegnen.”, M. Schmaus, Der Glaube der Kirche, Bd. 1, Tbd. 2, St. Ottilien 1979, p. 109. For the sake of accuracy, it should be added that M. Schmaus completed his statement to the following passage: “Eine Parallele zu der Dialektik in der Theologie bildet das Prinzip der Komplementarität in der Naturwissenschaft, das erstmals von dem dänischen Atomphysiker Niels Bohr formuliert worden ist. Es drückt sich z.B. darin aus, dass das Atom zugleich Teilchen und Welle ist, ohne dass das eine mit dem anderen identifiziert werden kann. Es hängt jeweils von der Anfrage an das Atom ab, ob es sich als das eine oder als das andere zeigt”. The paper shows further one of the modern solutions of the logical problems of quantum physics which is not identical with the universal suspension (in quantum physics) of the principle of non-contradiction and the introduction of special quantum logic. 5 W. Hryniewicz, Bóg cierpiący? Rozważania nad chrześcijańskim pojęciem Boga, “Collectanea Theologica” 1981, no. 51 (2), p. 6-11. 4

What Kind of Logic Does Contemporary Theology Need?


The ‘suspension’ of the rule of non-contradiction is, in fact, the resignation of the validity of classical logic in (classical) theology. Therefore, Barth’s meta-theological resolution had to counter the reaction of the scientific community that generally accepted classical logic as the logic of scientific discourse. The main criticism of Barth’s meta-theological position was undertaken by his friend, the theologian, philosopher and, above all, logician, Scholz.6 The German logician addressed this issue in his paper Wie ist eine evangelische Theologie als Wissenschaft möglich? published in 1931 in the flagship journal of dialectical theology Zwischen den Zeiten.7 He formulated six postulates for evangelical theology which it must – in his opinion – satisfy in order to deserve the name of science (Wissenschaft). With reference to the second postulate (apart from the questions and definitions in science can contain only the sentences, that is expressions having a claim to truth) – that he considered minimal and indisputable – Scholz formulated the requirement for consistency of theology which inherited, according to him, the characteristics of minimalism and indisputability. Scholz argued that consistency is a fundamental requirement of any rational thought. He clearly stated that consistency is the condition for the scientific character of theology. Moreover, consistency is a prerequisite for the rationality of theology.8 The German logician based his requirement of the consistency of theology on two grounds: H. Schulz (1884-1956), the son of a Lutheran pastor, studied theology and philosophy under the direction of A. von Harnack. He earned his habilitation in philosophy of religion and systematic theology. Influenced by Principia Mathematica, he studied mathematics and physics in the 1920s. From 1928 he taught philosophy in Münster where he befriended Barth who also worked there. This chair was transformed in the 1930s into the chair of mathematical logic and basic researches. H. Scholz was a leading German logician of his time. 7 H. Scholz, Wie ist eine evangelische Theologie als Wissenschaft möglich?, “Zwischen den Zeiten“ 1931, no. 9, p. 8-53, also [in:] Theologie als Wissenschaft. Aufsätze und Thesen, Hrsg von G. Sauter, München 1971, p. 221-264 (pagination used in this paper). 8 H. Scholz, Wie ist eine evangelische Theologie als Wissenschaft möglich?, op. cit., p. 252. 6


Jerzy Dadaczyński

1. On the general methodology of sciences which traced its roots directly back to Aristotle. 2. On the principles of classical logic: from the law of this logic p ∧ ¬ p → q one can state that if a contradiction occurred in a (scientific) system, then this system includes any sentence, which means that this system becomes worthless from the logical (and epistemological) point of view. This possibility to join any proposition is sometimes called an ‘explosion’. Of course, one solution would be to say that theology (with contradictions) uses some special ‘theological’ or ‘Christian’ logic. Nowadays, one could suppose that it would be a logic which would tolerate contradictions in the sense that a conjunction of contradictory sentences would not lead to an ‘explosion’. Scholz took up this topic and stated that he did not know anything about the existence of such logic.9 In other words, he stated that what should be the logic of theology, like the logic of other sciences, is a classical logic. The German logician certainly ‘checked’ the collection of (formal) logic, which was known until the early 1930s. Apart from classical logic then, there was the intuitionistic logic and Łukasiewicz’s (Post’s) multivalued logic. These types of logic did not protect against the ‘explosion’ and, therefore, did not meet the ‘requirements’ of the special logic of theology. The fact that Scholz took these issues up in a journal published by dialectical theologians and that at that time he had befriended Barth in Münster indicates that he had dealt with the issue of the (non)contradictions of theology within the context of dialectical theology. He negated some meta-theological assumptions of Barth’s and, above all, he completely refuted his acceptance of contradictions in theology and his meta-theological suspending of validity of the non-contradiction principle in theology. The fact that Schulz discussed the issue of ‘Christian’ or ‘theological’ logic indicated that probably the postulates (of existence) of such logic were considered in the theological circles.


What Kind of Logic Does Contemporary Theology Need?


It should be noted that the German logician explicitly stated that in theology there were no contradictions (antinomies). Thus, one can only guess that he saw the only reason for the emergence of contradictions in dialectical theology in the dialectical method used. Therefore, Scholz’ fight to keep classical logic in theology was at the same time a fight for the method – a non-dialectical one – in theology.

3. Meta-scientific approaches to the contradictions in the various fields of science It is worth posing the question of how the case of scientific antinomies in the history of meta-sciences (meta-theology and other meta-sciences) was perceived at this point. Without negating the specificity of theology, one may try to find a meta-scientific model to deal with antinomies in sciences. Regarding theology, one must refer primarily to the Middle Ages. Before the formation of the classical theology of the Middle Ages and in the late Middle Ages, one can find two thinkers whose position on the issue under discussion is convergent with the dialectical position of today. It is striking that both can be classified as representatives of Christian Platonism. A theologian and philosopher, who fully accepted contradictions (antinomies) in theology at the meta-theological level, was Saint Peter Damian (1007-1072). According to him, the principle of non-contradiction applies neither to the reality of God, nor to revealed knowledge. What brought the medieval theologian to such conclusions is his attempt to resolve the paradox of God’s omnipotence. He insisted on preserving the limitless omnipotence of God and the resignation from validity of the principle of non-consistency in the mentioned area. Saint Peter Damian stated that God could create a stone which no one would be able to raise and indeed raise the stone.10 R. Poczobut, Spór o zasadę niesprzeczności, Towarzystwo Naukowe KUL, Lublin 2000, p. 30. 10


Jerzy Dadaczyński

Towards the end of the Middle Ages, Nicolas of Cusa (14011464) totally accepted and even promoted antinomies in theology on the meta-theological level. He explicitly claimed that God is coincidentia oppositorum. It is worth noting that Nicolas of Cusa identified God with infinity. In scientific literature it had not been noted previously that this identification may have led to defining God as coincidentia oppositorum, because from Aristotle (more precisely: since the formulation of the aporias of Zeno) until the 19th century infinity was regarded as a source of antinomies.11 It is also highly probable that Cusanus considered a project to create a new logic for theology. It was to be a logic which tolerated contradictions. It should be said, however, that in the classical period of medieval theology, which was dominated mainly by the thought of Aristotle, contradictions in theology were absolutely not tolerated on the meta-theological level. This obviously does not mean that medieval theology was free of antinomies but they constituted a fundamental issue for the meta-theological reflection. It can be evidenced by the fact that so much effort was devoted to the removal of these contradictions (for instance: antinomy of the omnipotence of God).12 In the history of mathematics, antinomies which appeared in its foundations were not strictly accepted. Antinomies of the set theory, on which the whole mathematics can be built, started a great ‘storm’ in the foundations of mathematics in the late 19th century. The criIf the hypothesis presented above were true, it could be argued that the antinomies which Cusanus perceived in theology were not timeless but relativized in terms of the paradigm of philosophy and mathematics of his time. 12 It is clear, that at least some antinomies in theology were generated by the philosophical and scientific context, more precisely, the Aristotelian context in which theology was cultivated in the Middle Ages. For instance, on the one hand, since its beginnings Christianity accepted the thesis of the temporal beginning of the world, and on the other the generally accepted Aristotelian philosophy maintained the thesis of eternity of the world. It can be argued that the antinomy ceased when the Aristotelian philosophical and scientific context was removed. That is an example showing that the antinomies, which are treated (were treated) as theological, may be something temporary, dependent on the (variable) paradigm of theology (culture). It seems that the last statement is a meta-theoretical point against the meta-theological acceptance of contradictions in theology. 11

What Kind of Logic Does Contemporary Theology Need?


sis was overcome by introducing an axiomatization (E. Zermelo) or typical ambiguity (B. Russell) of set theory.13 A very vigorous debate regarding the causes of the crisis led to the extraction of the principal directions in the philosophy of mathematics. When it comes to physics, it should be stated that the meta-theoretical reflection did not accept contradictions occurring in its history. Nevertheless, they did happen, like the paradox of the javelin in Aristotelian physics. According to the principles of Aristotelian physics, only such an object may be in motion that is currently affected by force. An experiment involving throwing the javelin is contrary to this principle. The javelin while flying is not affected by any force and yet, it is in motion.14 This contradiction which occurred in (Aristotelian) physics was not accepted at the level of meta-theoretical reflection, as indicated by the numerous (ineffective) attempts to remove it. The best example is the medieval theory of impetus. However, since there was no better physical theory – i.e. the theory which would explain at least the same set of physical issues and eliminate the mentioned contradiction – the theory of Aristotle was not rejected for centuries, despite the contradiction inherent in it. This situation changed in the 17th century when a new Newtonian paradigm of physics emerged. The new physics changed the old Aristotelian assumptions (principles) and, as a result, eliminated the antinomy of the javelin.15 In practice, therefore, despite the occurring contradictions, physical theory was retained. Still, contradictory statements were still subjected to special control, attempts were made to find solutions to For the sake of accuracy, it should be added that contemporary mathematics can be built based on para-consistent logic, e.g. on the logic which tolerates inconsistencies. In mathematics, the classical antinomies of Russell, Cantor and others can be reconstructed. These theories are non-explosive because they are the logic on which such mathematics is based. Based on these theories, no proposition can be proven. 14 Gravity, which in the Aristotelian paradigm of physics was not present, is not considered here. 15 Within this paradigm, a principle of dynamics which differs from the Aristotelian one is valid. It assumes that objects continue to move in a state of constant velocity unless acted upon by an external (net or resultant) force. 13


Jerzy Dadaczyński

contradictions ‘inside’ the obligatory paradigm of physics, and the contradictory propositions were not combined together (conjunctions of them were not created) to draw consequences. Therefore, antinomies within the existing paradigm of physics were in some sense ‘isolated’, under the control of a specific quarantine. In any case, at the level of meta-theoretical (meta-physical) reflection, antinomies in physics were not accepted. In this context, quantum physics requires a separate discussion. Its distinctness from other, historical and contemporary, theories of physics is that it allowed, and not infrequently, notoriously contradictory sentences. In particular, the quantum theory contains sentences which are mutually incompatible (complementary),16 which cannot be jointly stated as true. And so the sentences ‘the value of x-component of angular momentum is mħ’ and ‘the value of y-component of angular momentum is mħ’, referring to one object at a particular time and in a particular state, are not together true.17 What is relevant to the study presented in this paper is the fact that there has been no simple acceptance, at the level of meta-theoretical (meta-physical) reflection, of the coexistence of non-compliance (complementary)18 sentences in quantum physics. An intense debate has begun on this issue and two solutions, in fact, have been proposed: either to create a ‘special’ non-classical logic for quantum physics or retain classical logic as the basis of quantum physics, while treating complementary sentences in a special way.

Complementarity may be regarded as generalization of the concept of non-compliance (a form of contradiction), but one must not forget that the concept of complementarity is an empirical concept and the concept of non-compliance is a logical one. The complementarity of the above example sentences is not a logical issue but an empirical one, and is therefore an attribute of physical reality (Z. Hajduk, Filozofia nauk formalnych, Wydawnictwo Naukowe KUL, Lublin 2010, p. 69-73 (manuscript pagination)). 17 Z. Hajduk, Filozofia nauk formalnych, op. cit., p. 69-73. 18 See footnote 16. 16

What Kind of Logic Does Contemporary Theology Need?


It is important to indicate here the meta-philosophical approaches to antinomies in philosophy. As it turns out, since Hegel’s19 at least some representatives of the continental philosophy have not only accepted contradictions in philosophy, but also purposely constructed their philosophical systems on an inconsistent (dialectical) basis. Above all, Marx and the whole of Marxist philosophy, as well as Kierkegaard and Heidegger, should be mentioned here. In the conclusion of this overview it must be stated that, on the level of meta-scientific reflection, scientific contradictions were only accepted unconditionally in the meta-philosophy of leading representatives of continental philosophy, and – in the Middle Ages – in the meta-theology of some representatives of Christian Platonism. It should be noted straightaway, however, that the mainstream in medieval meta-theology firmly supported the validity of the principle of non-contradiction in theology. There was, and there is, also no simple acceptance, on the meta-scientific level, of contradictions occurring in other sciences. There is a striking coincidence of approving attitudes towards the contradictions within temporary (dialectical) meta-theology and meta-philosophy of important representatives of continental philosophy mentioned at the beginning of this paper. In fact, it is a temporal (the 19th and, moreover, the 20th century), geographical and cultural (the German language area) coincidence. If we add to this statement the evident dependence of theology – at least in the method – on its contemporary philosophy and cultural trends, then the hypothesis that the acceptance of contradictions in contemporary meta-theology is a derivative of their acceptance in continental metaphilosophy, appears to be highly probable.20

Earlier philosophers who rejected the principle of non-contradiction are not mentioned here deliberately. It was all about Heraclitus and some representatives of NeoPlatonism. 20 The general influence of post-Hegelian idealism on Protestant theology is indisputable, and was almost immediate. It will suffice to give the example of liberal theology of the Tübingen school of Baur in the first half of the 19th century. 19


Jerzy Dadaczyński

4. Paraconsistent logic as logic of (dialectical) theology? It was shown above that, between the 1920s and 1930s, when Scholz discussed the criteria of the scientificity of theology with Barth, no logic existed which would tolerate contradictions, i.e. in which contradictions would not lead to an explosion. However, since the discussion between Scholz and Barth, an essential progress in logic has been made. A number of formal, nonclassical, logical systems have been built and among them is the socalled paraconsistent logic, a logic which tolerates contradictions. The principle ex contradictione quodlibet is not an element of this theory and, therefore, a contradiction (antinomy) does not lead to every proposition being included in the system. In other words, paraconsistent logic may be considered as a natural candidate for the logic of dialectical theology and other types of theology in which the meta-theories accepted are theological antinomies. One may even ask whether Scholz, who rejected the existence of a specific theological logic because of the lack of appropriate ‘candidates’ (formal systems), would see a paraconsistent logic exactly as specific theological logic. It must be pointed out that the paraconsistent logic emerged outside those circles of philosophy which accepted the Hegelian dialectic or at least rejected the principle of non-contradiction, and – even more – outside of theological circles, which in meta-theological reflection accepted the antinomies in theology and preferred new paradigms for practicing theology. Paraconsistent logic, understood as formal systems, began to emerge in the circle of Latin American logicians (N.C.A. da Costa, F.G. Asenjo, A.I. Arruda, R. Chuaqui)21 in the 1960s and 1970s. Today, the most influential representative of this trend is the Australian

An ‘introduction’ to para-consistent logic was Jaskowski’s discussive logic (S. Jaśkowski, Rachunek zdań dla systemów dedukcyjnie sprzecznych, “Studia Societatis Scientiarum Toruniensis” 1948, no. 1). 21

What Kind of Logic Does Contemporary Theology Need?


G. Priest. He believes that para-consistent logic is the accurate formalization of Heraclitus’ and Hegel’s dialectic.22 Of course, what is important here is the question of how paraconsistent logic excludes the possibility that any sentence can be inferred from a conjunction of contradictory sentences (how to exclude the principle ex contradictione quodlibet?). To explain this issue it is necessary to investigate how the principle ex contradictione quodlibet is proven in classical logic. First, it is assumed that: (0.1)

p ∧ ∼p.

The reference to conjunction elimination (simplification) rule (CE) gives: (0.2)





From the line (0.2), under the disjunction introduction rule (DI), one can derive: (0.4)

p ∨ q,

where q is an entirely arbitrary proposition. Then, from (0.4) and (0.3), under the disjunction elimination rule (DE), one can derive: (0.5)


Then, from (0.1)-(0.5), under the rule of implication introduction (II), one can derive: G. Priest, In Contradiction. A Study of the Transconsistent, Martinus Nijhoff, The Hague 1987, p. 4, 6. 22

Jerzy Dadaczyński



(p ∧ ∼p) → q,

what completes the proof of ex contradictione quodlibet in classical logic. It must be noted that the removal of any rule used in the proof ((CE) or (DE) or (DI) or (II)) from the set of inference rules of classical logic makes it impossible to prove, in the way above, that the theorem ex contradictione quodlibet. Usually, the conjunction elimination (simplification) rule (CE) is treated as the most ‘natural’ and is not eliminated from the collection of the inference rules of logic. However, the rule of implication introduction (II) can be replaced by the cut rule.23 With these explanations the following conclusion may be given: By constructing paraconsistent logic, the impossibility of proving any proposition is achieved by removing the disjunction introduction rule or the disjunction elimination rule or the cut rule from the set of inference rules. It must be said that the presented proof is not the only proof of the thesis ex contradictione quodlibet in classical logic. For this reason, by constructing paraconsistent logic, other rules must be removed, too.

5. Classical logic as the logic of theology (I) In the previous paragraph, paraconsistent logic was presented as a possible candidate for the logic of theology. The next two paragraphs will present arguments, however, that classical logic should rather be the logic of theology.24 The first of these paragraphs presents an argument referring to the fact that classical logic is applied in nonThe cut rule (Schnittregel) may be formulated as follow: ∆├ A; A├ B ⇒ ∆├ B. Argument (LM3), formulated at the end of this paper, shows that, in fact, in theology on which meta-level contradictions are accepted nothing was done to adapt logic that does not contain the principle of non-contradiction. 23 24

What Kind of Logic Does Contemporary Theology Need?


theological sciences (it refers primarily to physics). The next paragraph includes arguments referring, inter alia, to the specificity and the ‘practice’ of theology. Previously in this paper, an analysis of the relationship between meta-scientific reflections and antinomies which (may) occur on the scientific level was conducted.Two main conclusions may be derived based on this analysis: 1. Within the reflection on the meta-level, there is no acceptance of antinomies occurring on the scientific level (with the exception of (meta)-theological trends and the continental meta-philosophy ‘associated’ with them, which has just been discussed). 2. Despite the lack of acceptance, it has not been decided (effectively) in any case to construct an appropriate, ‘specific’ for the scientific discipline, logic or to ‘fit’ in an adequate, already known non-classical logic. The last sentence needs clarification. It could be protested against by indicating the seemingly obvious example of quantum physics and the quantum logic built for it. It is currently stated that: 1. So far there has not been used any quantum logic to achieve correct results in the quantum physics. 2. Approximately 50 years of research in quantum logic have not produced significant results. This is explained by the fact that quantum physics uses only those mathematical theories which are built only on classical logic. It is well highlighted by the axiomatizations of quantum mechanics.25

Z. Hajduk, Filozofia nauk formalnych, op. cit., p. 72 (manuscript pagination). Z. Hajduk referred to: W. Stegmüller, Hauptströmungen, p. 208-220, 537-538 and S. Kiczuk, Język fizyki współczesnej, p. 57-75. 25


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In view of the above statements, a crucial question arises of how to deal with contradictions (more precisely: with non-compatible, complementary26 sentences) of quantum physics. It is stated now that a logic (Boolean logic, classical logic) should be capacious enough to allow the statement of mutually complementary propositions27and this is actually what happens in the sphere of everyday (natural) language. It is seen, for instance, in the case of such sentences as ‘he is writing now’ and ‘he is swimming now’ which cannot both be true for the same person and the same time. It is, however, a material and not a logical limitation. What should be done then, when complementary sentences are found is to stop from stating them altogether (jointly). It is believed that such an approach and not the search for a special (non-classical) logic for quantum physics is reasonable. It is to be concluded that the decision to ‘seek’ (in this case some ‘specific’ quantum logic) involves the confusion of the order of logic with the objective (physical, or more broadly: ontological) order.28 Generally, it should be stated then that the practice of meta-scientific reflection corresponds to the two statements formulated above. At this level, there is no acceptance of contradictions occurring at the scientific level and, at the same time, there is no agreement about how to build some specific logic for a specific scientific discipline. It is sometimes recommended, as in the case of quantum physics, that the contradictory (more precisely – in the case of quantum physics – non-compatible, complementary29) sentences should not be stated together (jointly). It is a kind of ‘isolation’ of pairs of this type of sentences. This reminds one also of the ‘isolation’ pairs (as mentioned above) of contradictory sentences in the old paradigms of physics. Such sentences were subjected to ‘quarantine’, were ‘observed’, and See footnote 16. Sometimes the task of quantum logic is referred to as exclusion of the mutual complementary propositions (Z. Hajduk, Filozofia nauk formalnych, op. cit., p. 71 (manuscript pagination)). 28 Ibidem. 29 See footnote 16. 26 27

What Kind of Logic Does Contemporary Theology Need?


were absolutely not combined in conjunction. They and their conjunctions were not used in the proofs (arguments) of physics. Efforts were made to remove the antinomies. But the non-removed antinomies frequently did not lead to the withdrawal or ‘collapse’ of old physical theories because of the lack of better candidates (theories). The old theories – with and despite the antinomies – were still accepted. It should also be noted that some antinomies were generated by the paradigmatic foundations (assumptions) of physics and ceased with the change of paradigm. It must be stated that the metaphysical practice appropriately represents the meta-scientific practice stressed in the two postulates mentioned above. At this level, there is no acceptance of the contradictions occurring at the scientific level and, at the same time, there is no agreement to build some specific logic for a specific scientific discipline or to ‘fit in’ an adequate, already known non-classical logic.30 Therefore, it seems that this practice should also dominate the meta-theological thinking. The reference to physics (and its history) indicates that the theological contradictory sentences should not be stated together (jointly), they should not be combined in conjunctions31 They and their conjunctions should not be used in the proofs (arguments) of theology. The latter postulate implies – as it seems – the requirement of the exclusion of the Hegelian dialectic from the set of methodological ‘tools’ applicable in theology. The reference to the history of physics 30 An exception is, for example, intuitionistic mathematics. It was built on philosophical foundations different from classical mathematics. Intuitionistic logic was later ‘dissected’ from this mathematics. Generally, it may be said that mathematics can be built on different logical foundations. This statement is related to the thesis of mathematical pluralism. However, if logic is adopted as a basis of mathematics, then in this mathematics the rules of inference permitted by this logic must be strictly followed. Mathematics (in plural) are primarily interesting as the object of meta-level studies. 31 ‘Locally’ then, with respect to pairs of contradictory sentences only, the rule of introducing conjunction would be suspended. On the one hand, it recalled the solution used in para-consistent logic, where, in order to eliminate the ‘explosiveness’ of the theory, some rules of inference are excluded. On the other hand, the prohibition of this very ‘natural’ rule would be only ‘local’.


Jerzy Dadaczyński

shows that, at least some, antinomies may be relativized to the paradigm of theology. This was the case with some antinomies in medieval theology: they disappeared with the resignation from (some of) their Aristotelian foundations.32 Moreover, it appears that most of the antinomies in theology of Barth’s type are a result of the adoption of methodological ‘tools’ from continental philosophy, firmly determining the paradigm of such theology.

6. Classical logic as the logic of theology (II) Previously, an important argument in favour of theology based on classical logic has been presented. While formulating the argument, in fact implicitly, a reference was made to the assumption that theology is one of the sciences, and therefore, theology must respect the meta-level solutions concerning the logic of science, undertaken in the reflection on the non-theological sciences. One could, however, refute this argument by saying that the specificity of theology as just science (theology is science, but science sui generis) does not allow this type of argument.33 Hence below some other arguments are given in favour of the logic of theology as the logic which does not tolerate contradictions. These arguments are divided into two groups: – ecclesio-theological arguments (ET) – logical and methodological arguments (LM). The first group of arguments seems to incorporate certain specificity of theology. In this group the following arguments can be distinguished: See footnote 12. Barth stated in his discussion with Scholz, that the term Wissenschaft used in philosophy and European culture did not mean such a theology as he understood it. Consequently, the criteria of scientificity do not necessarily have to be satisfied by theology (such as he understood it). Scholz replied, that the connotations of the term Wissenschaft were not an invention of positivism, but had a centuries-old tradition, partly developed by the Christian thought.

32 33

What Kind of Logic Does Contemporary Theology Need?


ET1 – from the proclamation ET2 – from the ecumenism ET3 – from the authority ET4 – from the catholicity (from the Magisterium). The second group contains following arguments: LM1 – from the rationality LM2 – from the meta-logic LM3 – from the factual non-elimination of ‘natural’ rules of inference. In what follows, the above arguments are discussed. ET1 (from the proclamation). It is worth recalling here that these ideas indicate servitude, submission, or even the identity of theology with the Christian teaching. If theology is expected to be at least in one of these relations to the proclamation – and it will probably not raise any resistance to adoption of at least the ancillary role of theology for proclamation – then theology must necessarily take into account the recipient of the proclamation. And the recipient – though often unintentionally – uses the classical logic. It is true that sometimes, for instance in the context of the problem of ‘mechanization’ of human thinking, it is indicated that occasionally a person may be irrational in the sense that they accept the contradictory sentences (judgments) in order to demonstrate that the human mind can be regarded as a contradictory Turing machine. But generally the recipient of Christian proclamation uses classical logic. Therefore, the inclusion of antinomies in the proclamation (theology) would be something which distracts the recipient from understanding the proclamation. One might at this point carry out the reasoning starting from the statement revelation needs to be understood by the recipient of the proclamation. However, such reasoning would be a highly speculative nature. ET2 (from the ecumenism). The undoubted aim of ecumenical activities is to unite Christians, including by theological agreement among the churches and communities of Christians. The current state


Jerzy Dadaczyński

of affairs is characterized by the fact that, on certain issues (also of purely theological-dogmatic nature such as the Eucharist) these communities accept contradictory judgments. If the different sorts of logic of the theology in which these judgments are formulated were the logic which tolerates contradictions, then there would be no lack of agreement in matters covered by these theses. This last statement is falsified by the current state of affairs. Thus, the appropriate theology does not tolerate the contradictions. ET3 (from the authority). At this point, the statement of John Paul II in his encyclical Fides et Ratio must be mentioned. It should be indicated that the statement is contained in the text by John Paul II which concerns the relationship between faith and reason and, therefore, is undoubtedly an important determinant in current attempts to build (construct) theology. In no. 32 the Pope states: “The unity of truth is a fundamental premise of human reason, expressed already in the principle of non-contradiction.”34 Hence the principle of noncontradiction should apply in all fields of activity of the human mind and, therefore, also in theological cognition. It should be noted that John Paul II is speaking here on the ‘basic postulate’ and does not indicate any exceptions. One could of course take the discussion in the context of this postulate formulated in the encyclical and indicate that man is indeed consistent, but in theology antinomies are generated (produced) by the theological reality (by the subject of theology). In other words, one could suggest that not epistemology but ontology is the source of theological contradictions. One could then counter-argue indicating that John Paul II pointed out the principle of non-consistency (of the cognition, of the thinking) as an ‘exponent’ of the principle of unity of truth. The concept of truth requires the reference to semantics, and therefore the reference to certain ontology. The unity of truth presupposes the unity of ontology and, therefore, its non-contradiction. It is worth indicating the convergence of the thesis of John Paul II with the previously mentioned statement of 34

John Paul II, Fides et ratio, no. 32.

What Kind of Logic Does Contemporary Theology Need?


Scholz about the non-contradictory nature of theology as a necessary requirement of its rationality. It should also be stated that John Paul II’s thesis, which can be treated as a meta-theological thesis, is contradictory with the meta-theological thesis of Schmaus. ET4 (from the catholicity/from the Magisterium). This argument relates strictly to Catholic theology. The Magisterium applies the classical logic which tolerates no contradictions. A contradiction between the official interpretation of faith and a theological statement gives the Magisterium the assumption of activities that are aimed at denying by the author their theological statement. If the theology of Catholic authors was to be built on the logic accepting contradictions, then the logical ‘basis’ of the Magisterium and those of theology would obviously be ‘incompatible’. It could happen then that in theology a thesis of the Magisterium and its negation would be both accepted. It seems that the Magisterium, guided by classical logic, would not accept such a situation. LM1 (from the rationality). As it has been indicated above, Scholz pointed out that the consistency of theology (as any other scientific, wissenschaftliche, theory) is a condition for its rationality. It is also currently stated that consistency is a requirement for rationality. Determining the reasons for this is an issue which goes beyond the scope of this study. LM2 (from the meta-logic). The classical logic is, in some way, highlighted among the many forms of constructed formal logic. Namely, in meta-logic just classical logic is used. In principle, one does not know how to answer the question of why it is so. However, this is a factual award of classical logic in the set of formal logic and may constitute a certain argument in favour of its use, also in theology. LM3 (from the factual non-elimination of ‘natural’ rules of inference). As shown above, in logic which tolerates contradictions their ‘non-explosiveness’ is achieved through the elimination of some, otherwise ‘natural’, rules of inference. However, in theology in which meta-theology accepts theological antinomies, no rules of inference are eliminated. In fact, no step has been taken to replace


Jerzy Dadaczyński

classical logic in such theology with the logic that tolerates contradictions. As a result, there is a logical and methodological chaos in that paradigm of theology, which may be aptly described as ‘touched by the spirit of Hegel’.

7. Conclusion The above study indicates the non-legitimacy of the meta-theological acceptance of contradictions occurring in theology. The most appropriate logic for theology appears to be classical logic. This, in turn, implies the above-described ‘isolation’ of any pair of contradictory theological sentences. Above all, these sentences should not be stated together (jointly), i.e. they and their conjunctions should not be used in theological proofs (arguments).35

The recognition of classical logic as the logic of theology and the taking of particular precautions with respect to possible contradictory sentences mentioned above determine the withdrawal from the deliberate ‘production’ of contradictions in theology. It may not be that a theology student who participates in their first year of study is restricted to the classical course of logic, and in the second or third year in the class of fundamental theology is ‘under attack’ by the rhetoric dominated by Barth’s ‘already and not yet’, which openly violates the principle of non-contradiction implanted in the student by the course of logic. In fact, it is, differently formulated, earlier demand for the resignation of Hegelian dialectic in the methodological ‘instrumentarium’ of theology. 35

Antonino Rotolo Erica Calardo University of Bologna

God’s Omniscience: A Formal Analysis in Normal and Non-normal Epistemic Logics* 1. Introduction


ne of the major problems which the logician willing to model knowledge and belief has to face is that of avoiding, or at least alleviating, the problem of omniscience. The efforts are usually focused on creating models for agents (either human or artificial) with bounded rationality and finite cognitive capabilities: such agents thus do not possess complete information about how the world is. Logical omniscience is often seen, therefore, as a problem to be solved and the solutions proposed so far are numerous.1 Nevertheless, if the issue to be addressed is that of defining divine omniscience, such a perspective should be reversed in order to push the concept of knowledge to its most extreme possibilities. The Christian thesis that God is omniscient is well-established, as the fact that God possesses the most perfect knowledge of all things follows

This contribution was made possible through the support of a grant “The Limits of Scientific Explanation” from the John Templeton Foundation. The paper was presented in May 2012 at a seminar at the Copernicus Center in Krakow, Poland. We would like to thank Bartosz Brożek and the other colleagues attending the seminar for their useful comments. 1 R. Fagin, J.Y. Halpern, Y. Moses, and M.Y. Vardi, Reasoning about Knowledge, MIT Press, Cambridge–London 1995; J.J. Meyer, Modal Epistemic and Doxastic Logic, [in:] D. Gabbay and F. Guenthner (eds), Handbook of Philosophical Logic, 2nd edition, Kluwer, Dordrecht 2001. *


Antonino Rotolo, Erica Calardo

from His infinite perfection.2 We shall not investigate the theological grounds of this idea. Rather, we will simply try to clarify the basic meaning of omniscience by pushing modal epistemic logic to its limits. From this perspective, an interesting question is that, even in the case of God’s knowledge, omniscience should have conceptual limits, otherwise the risk is to trivialize and make the notion of knowledge void. Perhaps, one may wonder whether this conclusion is problematic: after all, the transcendent nature of God exceeds our rational and conceptual resources. Since in God all things that ought to be are in fact the case (which trivializes the concept of normativity), we can likewise accept that divine knowledge about some A is equivalent to saying that A is true. This having been stated, one may think that omniscience is quite an easy property to get and hence to formalise. However, this is only partially true. If, on the one hand, it is quite easy to define logical omniscience in terms of knowledge of the logical truths,3 on the other, it turns out to be rather difficult to formally capture the insight of factual omniscience, which has to do with propositions having a different status.4 As we shall see, as soon as we try to capture the intuition behind divine knowledge, the whole scenario becomes foggy and slippery and all of a sudden even the strongest modal logics, those which are usually supposed to naturally define omniscient agents, turn out to be rather useless. A first informal definition may be that the agent that has complete or maximal knowledge is omniscient. What this informal definition means precisely is the research issue that we address here. Indeed, the aim of this paper is that of providing a formal account of such property. The Classical Propositional Calculus (henceforth CPC) being our foundation, we shall proceed by analyzing those modal schemata and rules which, on our view, are better suited to capture the intuitions L. Zagzebski, Omniscience, [in:] C. Meister and P. Copans (eds), The Routledge Companion to Philosophy of Religion, Routledge, London–New York 2007. 3 R. Fagin et al., Reasoning..., op. cit. 4 R. Girle, Modal Logic and Philosophy, Acumen, Teddington 2000. 2

God’s Omniscience: A Formal Analysis in Normal and Non-normal...


behind the property of omniscience. At first glance, an omniscient agent may be regarded as a being possessing complete knowledge about both those facts which are necessarily true and those which are just contingently so. Hence, it seems that there are two main types of omniscience. We shall call the first type logical and the second factual. Intuitively, this last account of divine omniscience looks the strongest candidate. The layout of the paper is as follows: Section 2 offers a brief outline of the modal logic language, the axiom schemata and inference rules, and the resulting modal systems that we discuss regarding the notion of divine omniscience. Section 3 focuses on a first family of epistemic logics expressing a very weak degree of omniscience: all logics are characterized in the most general semantic setting for modal logics, which, however, can hardly be conceptually re-framed in terms of well-known standard semantics for epistemic logics. A different path, which adopts a direct generalization of such standard semantics is depicted in Section 4. Section 5 considers stronger well-known epistemic logics, where we can have just a single accessibility relation (standard Kripke semantics) or, in the extreme case, where knowledge no longer requires the use of any accessibility relation. Section 6 goes beyond the threshold of logical omniscience and defines the idea of factual omniscience: this allows us to approximate God’s perfect knowledge, but the cost to pay is to trivialize epistemic logic. A short summary ends the paper.

2. Preliminaries As usual, we define the language of CPC as a containing countable set of propositional letters Prop := {p, q,...}, propositional constants ⊤, ⊥ (top and bottom), round brackets, boolean operations ¬, ∧, ∨, →, ≡ (negation, conjunction, disjunction, implication, double implication) and one modal unary operator □i for each agent i operating in the system (the index i being dropped whenever only one agent

Antonino Rotolo, Erica Calardo


is supposed to be present). An expression like □iA means that agent i knows/believes that A is true. Well-formed formulae (henceforth wffs) are defined as follows (where p is a propositional letter and A, B are meta-variables for well-formed formulae): p | ⊤ | ⊥ | ¬A | A ∧ B | A ∨ B | A → B | A ≡ B | □iA Let us recall some well-known inference rules and schemata we shall use for any □i:5 Inference Rules RE:= ⊢A ≡ B ⇒ ⊢ □iA ≡ □iB RM:= ⊢A → B ⇒ ⊢ □iA → □iB RN:= ⊢A ⇒ ⊢ □iA RR:= ⊢A ∧ B → C ⇒ ⊢ □iA ∧ □iB → □iC RK:= ⊢A1 ∧ ... An → B ⇒ ⊢ □iA1 ∧ ... ∧ □iAn → □iB n ≥ 0 Axiom Schemata efg:= M:= C:= K:= N:= Con:=

A ∧ ¬A → B □i(A ∧ B) → □iA ∧ □iB (□iA ∧ □iB) → □i(A ∧ B) □i(A → B) → (□iA → □iB) □i ⊤ ¬□i ⊥

D:= T:= 4:= B:= 5:= DEX:=

□iA → ¬□i¬A □iA → A □iA → □i□iA A → □i◇iA ◇iA → □i◇iA □iA ∧ □i¬A → □iB

There are different systems of propositional modal logics built to model various situations. In Figure 1, we consider some simple and well-known multi-modal systems which may be seen as a base for more complex ones. We use B.F. Chellas’ terminology and assume as usual that ◇i = def ¬□i¬ (cf. B.F. Chellas, Modal Logic, Cambridge University Press, Cambridge 1980). 5

God’s Omniscience: A Formal Analysis in Normal and Non-normal...

E, classical M, monotonic MN, N-monotonic R, regular K, normal KD, normal T, normal B, normal S4, normal S5, normal



Axioms E⊕M E⊕M⊕N E⊕M⊕C E⊕K⊕N K⊕D K⊕T E⊕K⊕N⊕B K⊕T⊕4 K⊕T⊕4⊕5

Figure 1: Modal systems

3. The starting point: minimal epistemic logics Any system of epistemic logic, if based on the standard modallogic paradigm,6 should assume some minimal formal properties. In particular, it is well-known that any modal logic should at least be closed under logical equivalence.7 This will be our starting point to formally analyse the notion of omniscience.

3.1. Principle of co-extensionality: the minimal epistemic logic E When dealing with CPC, one standard and well-known option is the adoption of a Fregean approach to semantics. In a given state of affairs, propositions are taken to be different names of the only two semantical objects populating the universe: Truth and Falsehood. A tautology is a proposition which is true only in virtue of its logical form: the truth values of its components do not influence the truth of the whole in the slightest. The set of tautologies can, hence be

6 7

J.J. Meyer, Modal Epistemic..., op. cit. B.F. Chellas, Modal Logic, op. cit.


Antonino Rotolo, Erica Calardo

described as the class of all the true names of Truth: those propositions whose truth is certain and unchangeable. A most famous result in formal logic states that all theorems of CPC are tautologies and vice versa. Two propositions that always share the same extension can be regarded as logically equivalent and, in a logical sense, identical. This can be expressed symbolically as A ≡ B: whichever truth value A is given, it would be identical to B’s and vice versa. A basic requirement the knowledge base of a divine (omniscient) being should meet is the principle that for any known sentence A, all its equivalents are also known. We are not yet including anything in God’s knowledge base. What we are stating is merely that if something is known, then all those facts which ‘look’ different but are actually the same (logically, extensionally) must also be known. This is a well-known modal principle and it can be compared to Leibniz’s Law, here applied to propositional logic. What this principle states is that if two propositions are logically equivalent, they are epistemically interchangeable. Following B.F. Chellas’8 terminology, this rule shall be henceforth referred to as RE: RE := A ≡ B / □A ≡ □B. It can be added to CPC to generate the minimal system of Classical Propositional Modal Logic E. Built on the foundation of propositional logic, the system E becomes our first step towards a logical definition of the concept of divine omniscience. When knowledge and belief are modelled in epistemic logics like E, which are much weaker than K (see above, Figure 1), then the epistemic logics can have a peculiar semantic reading, which is suitable to provide a fine-grained interpretation of logical omniscience. Modal logics weaker than K – which are called generically non-normal, in contrast with any normal logics that are stated to be as strong as, 8


God’s Omniscience: A Formal Analysis in Normal and Non-normal...


or stronger than, K9 – can be interpreted in semantic structures that consist of: • a set of possible worlds or states • a set of accessibility relations connecting pairs of worlds. The introduction of a plurality of worlds connected via a given accessibility relation R stems in epistemic logics from the need to represent agents’ relative ignorance (i.e. partial knowledge) about the world. Given a state w, all the R-associated worlds t are seen as epistemic alternatives to w itself10: when we have such a relation R which connects a world w with all alternatives where A is true, then we can say that □A is true in a world w – and □A is meant to say that an agent knows/believes that A is true. The plurality of worlds captures the notion of partial knowledge as follows. Suppose an agent i lives in Paris and does not know whether today it is raining in London (p:= ‘It is raining in London’). If i does not have access to any reliable source of information, he simply ignores all facts about the weather in London, hence he has at least two epistemic alternatives: for i from the perspective of Paris, (1) p is true, (2) p is false. However, as soon as the agent gains access to new pieces of information concerning the meteorological situation of London, the number of alternatives that he considers possible drops. If, for instance, he reads that it is currently raining in London, the epistemic alternatives he considers are only those which reflect the real situation, i.e. only those in which the proposition p is true. His knowledge base would then change accordingly. However, the plurality of worlds expresses only one aspect of agents’ relative ignorance. As we have said, we also assume to work with a plurality of accessibility relations. The resulting semantics is as follows:11 Ibidem. R. Fagin et al., Reasoning..., op. cit.; J.J. Meyer, Modal Epistemic..., op. cit. 11 Semantics for non-normal modal logics have a long and distinguished tradition which goes back to the work of D. Scott, Advice in Modal Logic, [in:] R. Hilpinen (ed.), Philosophical Problems in Logic, Reidel, Dordrecht 1970, p. 143-173; R. Montague, Universal Grammar, “Theoria” 1970, vol. 36, p. 373-398; K. Segerberg, An Essay 9



Antonino Rotolo, Erica Calardo

Definition 1 (Multi-relational semantics: frames, models and meaning of □).12 Multi-relational semantics is based on the following notions: • A multi-relational frame is a tuple ℱ := 〈W,ℛ〉 where W is a non-empty set and ℛ is a (possibly infinite) set of binary relations on W. • A multi-relational model is a tuple ℳ := 〈W,ℛ,V〉 where 〈W,ℛ〉 is a multi-relational frame and V is a function (assignment) V : Prop → 𝒫 (W). • Given a multi-relational model ℳ := 〈W,ℛ,V〉, a propositional letter p is true under V in a state w ∈ W iff w ∈ V (p). The truth conditions for all boolean operations are standard. The condition to evaluate □-formulae is as follows. For any w ∈ W: w ⊨v □A iff ∃Ri ∈ ℛ : ∀x(wRix ⇔ x ⊨V A). • Given a multi-relational frame ℱ := 〈W,ℛ〉, a model ℳ := 〈ℱ, V〉 and a formula A, we say that ℳ satisfies A iff for some world w, w ⊨V A; A is true in ℳ, ℳ ⊨V A, iff for all w, w ⊨V A; A is valid on ℱ, ℱ ⊨ A, iff for any model ℳ on ℱ, ℳ ⊨ A. Given a class of frames 𝔽, A is 𝔽-valid, 𝔽 ⊨ A, iff for any frame ℱ ∈ 𝔽, ℱ ⊨ A. • Abbreviations. For any formula A, Truth:13 ǁ A ǁV := {w | w ⊨V A} Validity: ǁ A ǁ := {w | w ⊨V A, for any valuation V}. in Classical Modal Logic, Filosofiska Studier, Uppsala 1971. Such modal logics are usually placed within the so-called neighborhood semantics, which is slightly different from the multi-relational one of Definition 3.1. Neighborhood semantics considers a set of collections of worlds related to w instead of connecting worlds via an accessibility relation. These collections are the neighborhoods of w. Formally, a frame is a pair 〈W, N〉 where W is a set of possible worlds and N is a function assigning to each w in W a set of subsets of W (the neighborhoods of w). A model is thus a triple 〈W, N, V〉, where 〈W, N〉 is a frame and V is a valuation function: □A is true at w iff the set of elements of W where A is true is one of the sets in N(w); i.e. iff it is a neighborhood of w. Despite the mathematical perspicuity of this semantics, neighborhood models are often considered far from intuitive when applied to provide philosophical insights of epistemic logics. 12 G. Governatori and A. Rotolo, On the Axiomatization of Elgesem’s Logic of Agency and Ability, “Journal of Philosophical Logic” 2005, vol. 34 (4), p. 403-431. 13 When clear from the context, we will omit the reference to V.

God’s Omniscience: A Formal Analysis in Normal and Non-normal...


Multi-relational semantics was originally proposed by Schotch and Jennings14 and Goble15. If we just confine our attention to multirelational modal frames, the question is how to philosophically interpret such a plurality of relations in epistemic logics. (Semantic structures are based on possibly infinite sets of accessibility relations.) In other words: what can we say about the intuitive reading of such multiplicity of criteria for selecting epistemic alternatives? Originally, multi-relational semantics was developed in the field of deontic logic. In deontic logic, Kripke’s accessibility relation selects for each world those states of affairs that are (morally, legally, etc.) ideal with respect to it: hence, if □A is true in a world w, this simply means that A is the case in all ideal alternatives to w. The interpretation of multi-relational models, as given for example in deontic logics, is thus that each accessibility relation corresponds to a particular ‘standard of value’ or a norm that selects those ideal worlds; however, it is not guaranteed that such worlds are still ideal according to different standards of value or norms, namely, according to different accessibility relations. From this perspective, different relations correspond to different deontic standards or that conflicting norms are obtained from otherwise consistent different systems of norms. If we import this intuition in the domain of epistemic logics, the multiplicity of relations may express the idea that there exist many epistemic standards and that the truth conditions for knowledge assertions can vary across contexts as a result of shifting epistemic standards. The idea of plurality of epistemic standards16 was defended within different philosophical theories of knowledge,17 none P.K. Schotch and R.E. Jennings, Non-Kripkean Deontic Logic, [in:] R. Hilpinen (ed.), New Studies in Deontic Logic, Reidel 1981, p. 149-162. 15 L. Goble, Multiplex Semantics for Deontic Logic, “Nordic Journal of Philosophical Logic” 2001, vol. 5 (2), p. 113-134; and by the same author: L. Goble, Preference Semantics for Deontic Logic, Part II: Multiplex Models, “Logique et Analyse” 2004, vol. 47, p. 113-134. 16 J.L. Pollock, Contemporary Theories of Knowledge, Rowman & Littlefield, Savage 1986, p. 190-193. 17 N. Malcolm, Knowledge and Belief, “Mind” 1952, vol. 61, p. 178-189; A. Goldman, Discrimination and Perceptual Knowledge, “The Journal of Philosophy” 1976, vol. 73, 14

Antonino Rotolo, Erica Calardo


of which should be necessarily assumed to confer a minimal philosophical meaning to epistemic multi-relational models. Let us just consider how Hector-Neri Castañeda18 illustrates what a plurality of epistemic standards means and how it may affect the truth conditions of knowledge assertions: Example 1 (Discovering America example adapted after H. Castañeda19). ‘What counts as knowing’ that Christopher Columbus discovered America on October 12, 1492 might change depending on whether we are considering (i) a television quiz show, (ii) a high school student’s essay, or (iii) a defence of the traditional dates of America’s culture from some famous Harvard historian. Hence, we have in this example three epistemic standards. The fact that □ (Columbus discovered America on October 12, 1492)


is true according to, for example, standard (i) does not entail that it is also true according to standard (iii), which is somehow more demanding. Hence, in general, we could tolerate epistemic expressions such as: □A ∧ □¬A


because different standards can lead to know that Columbus discovered America or to know that this was plainly false. Indeed, if we do not impose any special condition on multi-relational frames, then we have the modal system E. In this setting, it is easy to check that formula (2) is not contradictory (see Figure 2 below). A simple inspection of the figure shows that this is possible because an agent can know/believe that A is true and know/believe p. 771-791; R. Rorty, Philosophy and the Mirror of Nature, Princeton University Press, Princeton 1979. 18 H. Castañeda, The Theory of Questions, Epistemic Powers, and the Indexical Theory of Knowledge, “Midwest Studies in Philosophy” 1980, vol. 5, p. 217. 19 Ibidem.

God’s Omniscience: A Formal Analysis in Normal and Non-normal...


that it is false because the worlds selected by R1 (the epistemic alternative v selected by the standard R1) make A true, while the worlds selected by R2 (the epistemic alternative z selected by the standard R1) make A false. Hence, if we interpret relations as different epistemic standards, it is not required that the truth of (2) corresponds to a genuine cognitive dissonance,20 because there is no real epistemic conflict between □A and □¬A: each formula refers to a different standard. A true cognitive dissonance occurs rather when □(A ∧ ¬A) is true, because this sentence means that there is a logical conflict within a same standard. A v R1 R1



A 2A 2¬A

z ¬A

Figure 2: A simple model validating (2)

Let us assume now to formalize Example 1 above following the above semantic intuitions. Example 2 (Discovering America, cont’d). Let us denote ‘Columbus discovered America on October 12, 1492’ with A and represent standards as follows: (i) a television quiz show = R1 (ii) a high school student’s essay = R2 (iii) a defence of the traditional dates of America’s culture from some famous Harvard historian = R3 E. Aronson, The Theory of Cognitive Dissonance: A Current Perspective, [in:] L. Berkowitz (ed.), Advances in Experimental Social Psychology, vol. 4, Academic Press, New York 1969. 20

Antonino Rotolo, Erica Calardo


For formula □A


it is sufficient that A is true in all worlds selected by one standard, as the model in Figure 3 shows. A v R1 R1

¬A s R2


R2 , R3

A 2A 2¬A

z ¬A

Figure 3: A simple model illustrating Castañeda’s example

This analysis suggests that the agents’ ignorance is not only captured by having more alternatives for a given world, but also by having more standards. In fact, the standards (i), (ii), and (iii) of Example 1 and 2 represent different contexts as well as ‘perspectives’ of knowledge, which overall express the fact of a structural bounded epistemic capability with regard to the time when America was discovered. For an omniscient agent g, it would be odd to argue that from a certain perspective g knows that A is false, while from another perspective he knows that A is true, because an omniscient being is supposed to know precisely what is objectively true; hence, a multiplicity of epistemic standards reflects a certain degree of ignorance, at least insofar as the absence of ignorance is taken to correspond to omniscience. Indeed, reducing the number of relations lessens the structural degree of ignorance of agents and leads to a higher degree of agents’ omniscience.

God’s Omniscience: A Formal Analysis in Normal and Non-normal...


3.2. A Stairway to omniscience: an easy step after E As outlined in Figure 1, the first step in the path that leads from E to full divine logical omniscience is adding schema M, i.e.21 M := □(A ∧ B) → (□A ∧ □B). This schema seems relatively acceptable in epistemic logic. First of all, its validity is assumed in most non-normal modal systems – it is actually discarded only by the system E. Second, the schema looks conceptually harmless: if I know/believe both sentences together, at the same time, then it must be also true that I know/believe that America was discovered by Columbus on October 12, 1492 and that I know/believe that Betsy Ross reported in May of 1776 that she sewed the first American flag. Semantically, the following result holds for M:22 Proposition 1 (schema M). For any multi-relational frame ℱ, and any world w and relation Ri, let us Ri(w) denote the set of worlds accessible from w via the relation Ri. Let J and K be subsets of W in ℱ. Hence, the following holds: ℱ ⊨ □(A ∧ B) → □A ∧ □B iff ℱ is supplemented, i.e. for any valuation V, for any world w ∈ W, if there exists a relation Ri such that Ri(w) = J ∩ K, then there are two relations Rj and Rk such that Rj(w) = J and Rk(w) = K. In other words, if there is one epistemic standard according to which A and B are jointly true, there are two standards that validate respectively A and B. 21 A different but mathematically equivalent route can be taken by discussing stronger inference rules than RE, starting with RM. However, referring axiom schemata is much more philosophically perspicuous to discuss the notion of divine omniscience. Hence, we will mostly follow this second route. 22 G. Governatori and A. Rotolo, On the Axiomatization..., op. cit.

Antonino Rotolo, Erica Calardo


Consider the following example: Example 3 (Columbus and Betsy Ross). Let us denote ‘Columbus discovered America on October 12, 1492’ with P and ‘Ross reported in May of 1776 that she sewed the first American flag’ with Q. Again, let us suppose to work with the mentioned epistemic standards: (i) a television quiz show (ii) a high school student’s essay (iii) a defence of the traditional dates of America’s culture from some famous Harvard historian

= =

R1 R2



For formula □(P ∧ Q) → (□P ∧ □Q)


it is sufficient to have supplemented models (see Proposition 1) such as in Figure 4. ¬P Q v

P ¬Q s

R2 P ¬Q w R3

R3 R2

P Q z

R3 R1 Figure 4: A simple model ℳ illustrating Example 3

In the model ℳ represented in Figure 4, for any x ∈ W, we have that x ⊨ □(P ∧ Q) because there is a relation R1 such that R1(w) = {z} = ǁ P ǁ ∩ ǁQ ǁ. All epistemic alternatives that make both P and

God’s Omniscience: A Formal Analysis in Normal and Non-normal...


Q true are related to w via the perspective of the standard (i) a television quiz show; hence, it is true in w that the given agent knows/believes that P ∧ Q is the case. Also, the other two standards, (ii) a high school student’s essay and (iii) a defence of the traditional dates of America’s culture from some famous Harvard historian connect w respectively to precisely those worlds that make true the sentences P (via R3) and Q (via R2), hence: • □(P ∧ Q) is true in w (via standard (i), i.e. relation R1), □P is true in w (via standard (iii), i.e. relation R3) and □Q is true in w (via standard (ii), i.e. relation R2) • for any other world x ∈ {v, s, z}, we have that |= □(P ∧ Q); therefore • formula (4) is true in ℳ.

3.3. More and harder steps: conflicts, coherence and epistemic paradigms There is a further important schema that plays a central role in our quest for a logical definition of the concept of omniscience from the perspective of epistemic systems, namely, the schema C: C := (□A ∧ □B) → □(A ∧ B). Adding C to the formerly defined system M generates the system R, the smallest regular modal logic. This system shows very interesting properties. Let us focus on C. If there are two standards guaranteeing, respectively, that □P and □Q are true, then there is possibly a third standard that selects all the epistemic alternatives in which P ∧ Q is true, namely □(P ∧ Q) holds. In general, the result for C is the following: Proposition 2 (schema C). For any multi-relational frame ℱ, assuming the following holds:

Antonino Rotolo, Erica Calardo


ℱ ⊨ □A ∧ □B → □(A ∧ B) iff ℱ is closed under intersections, i.e. for any valuation V, for any world w ∈ W, if there are two relations Rj and Rk such that Rj(w) = J and Rk(w) = K, then there exists a relation Ri such that Ri(w) = J ∩ K. Example 4 (Columbus and Betsy Ross, cont’d). For formula (□P ∧ □Q) → □(P ∧ Q)


it is sufficient to have structures closed under intersections (see Proposition 2). Notice that the model in Figure 4 also validates (5). However, consider a subtle variation, as depicted in Figure 5. ¬P Q v

P ¬Q s

R1 , R2 P ¬Q w R3

R3 R2

P Q z

R1 , R3 Figure 5: A variation ℳ’ of the model ℳ of Figure 4 that validates (4) but falsifies (5)

The model ℳ’ in Figure 5 still validates (4). However, • □P is true in w (via standard (iii), i.e. relation R3) and □Q is true in w (via standard (ii), i.e. relation R2) • □(P ∧ Q) is false in w because ǁ P ǁ ∩ ǁQ ǁ = {z} but there is no accessibility relation Rj such that Rj(w) = {z}. • formula (5) is false in w and not valid in ℳ’. Here, we may have indeed two epistemic standards that individually support the agent’s knowledge/belief that ‘Columbus discovered America on October 12, 1492’ and ‘Ross reported in May of 1776 that she sewed the first American flag’ are true, but it is far from ob-

God’s Omniscience: A Formal Analysis in Normal and Non-normal...


vious that there is a standard that supports them jointly. On the other hand, the difficulty in saying that there is such a standard for P ∧ Q does not undermine the truth of (4), since, if there is no such relation, then the formula is trivially true in w (its antecedent is false). Notice that schema C plays a crucial role in enforcing cognitive dissonances and in making explicit epistemic conflicts. Indeed, let us take Example 4 and replace Q with ¬P. Hence, we can simply consider the following instance of C: (□P ∧ □ ¬P) → □(P ∧ ¬P)


Since (6) is true for example in w, then there is at least an epistemic standard (represented in the example and in Figure 4 by R1) that connects w to all epistemic alternatives that make P ∧ ¬P true. However, P ∧ ¬P ≡ ⊥, hence the standard refers to a contradiction, which makes void R1 and hence we should have that R1 (w) = Ø. In a different but related perspective, since the modal system R makes valid the inference rule RR, i.e. ⊢A ∧ B → C ⇒ ⊢ □A ∧ □B → □C23, then, if we have □P and □¬P, we obtain □X for any sentence X: (P ∧ ¬P) → X is in fact a tautology of CPC. Hence, suppose we know/believe that P and know/believe that ¬P. We could obtain □Q, □¬Q, □(Q ∧ ¬Q), □(Bologna is in tke UK), and so forth.

4. A different path: truth and logical omniscience We have discussed in Section 3 some very weak epistemic logics. In Section 3.3, however, we highlighted that combining schemata M and C results in the well-known modal system R, where a much stronger version of logical omniscience emerges: here, we can easily include any tautology and logical truth in God’s knowledge base as well as making explicit any cognitive dissonance. 23

B.F. Chellas, Modal Logic, op. cit., Chapter 2.

Antonino Rotolo, Erica Calardo


A different (and not equivalent) path can be taken to capture divine understanding by assuming M and state that God knows the Truth. This last statement is expressed by the axiom schema known in the alethic tradition as the Necessity of Truth: N := □⊤. As formerly observed, the propositional constant ⊤ is taken to mean the Truth and its truth value is, accordingly, always true. Notice that the schema N is enough to include any tautology and logical truth in God’s knowledge base. Knowing only one theorem, only one logical Truth would be sufficient to know all the classical theorems. Indeed, for any theorem A it holds that A ≡ ⊤ and it is enough to apply RE and MP to derive □A. Hence, it is sufficient to add the schema N to the system E to state that God knows all the truths of logic, i.e. all the theorems generated within the system. This intuition is usually captured by the rule RN: RN := A / □A This rule is derivable in the system NM, the smallest N-monotonic system, obtained by adding N to M: Lemma 1. The rule RN is derivable in the system NM. Proof. ⊢NM A ⊢NM A ≡ ⊤ ⊢NM □A ≡ □⊤ ⊢NM □⊤ ⊢NM □A

Assumption RE Axiom N MP

The most interesting feature about NM-systems is that it enables the switch from multi-relational strong semantics to weak

God’s Omniscience: A Formal Analysis in Normal and Non-normal...


frames. In fact, the logic NM is known to be sound and complete with respect to weak multi-relational frames, i.e. structures in which the truth condition for the □ operator is evaluated as follows: Definition 2 (Weak truth conditions for multi-relational frames). For any w ∈ W: w ⊨ □A iff ∃Ri∀x(wRix ⇒ x ⊨ A). The validity of the schemata M and N follows immediately by the adoption of this evaluation clause. Notice that the schema N states something rather strong. It says, in fact, the any agent operating within the system knows all the truths of logic, all the theorems. However, this type of omniscience still concerns the abstract truths of mathematics rather than contingent facts. This difference becomes rather more evident if looked at from a semantic perspective. What the schema claims, in fact, is that an agent knows all valid propositions, i.e. those formulae which are true everywhere, in all possible worlds of all possible frames and under all possible valuations. On the other hand, if a fact happens to be true in a specific state of a model, under a specific valuation (but it can still be false under other conditions), there is no way-yet-to infer that an agent knows it.

5. Normal modal logics and the nature of knowledge and knowers So far we have presented a semantic scenario designed to accommodate different epistemic perspectives and paradigms. However, this looks like a feature suited to depict the sort of knowledge possessed by humankind rather than divinity. In fact, given the laws of CPC, we are bound to accept that any proposition has one and, even more importantly, only one truth value: the law of excluded middle A ∨ ¬A is a classical tautology. Semantically, this is mirrored by the fact that the intersection of two complementary sets of epistemic alternatives is always empty. Hence, no genuine epistemic standard


Antonino Rotolo, Erica Calardo

(i.e. a relation which is not empty) can accommodate both A and ¬A. That known facts should be coherent is suggested, as we have already said, by the schema C. It states is that if an agent knows two distinct facts and such facts are contradictory, then he must also use a further epistemic standard which is trivial, i.e. a standard which makes him believe everything (semantically: an empty binary relation). On the other hand, if the two facts are indeed consistent with each other (semantically: the intersection of ǁ A ǁ and ǁ B ǁ is not empty), then, by C, the agent must possess another epistemic standard to accommodate both propositions. In general, for any couple of genuine epistemic paradigms, there must exist a third one which takes into account those facts that are common to both. This means that true knowledge is consistent and cannot handle contradictions, i.e. all non-trivial epistemic paradigms are coherent with each other. Intuition would suggest that this is equivalent to possessing only one epistemic standard and this is perfectly consistent with our idea of divine knowledge. Semantically, a multi-relational weak frame with only one binary relation is called a Kripke frame. Definition 3 (Kripke semantics). A Kripke frame is a multi-relational frame ℱ := 〈W,ℛ〉, such that the cardinality of ℛ is 1. Given a model ℳ := 〈W,ℛ,V〉, a world w and a wff A, w ⊨V □A iff for any v, if wRv then v ⊨V A. The theory generated by adding RE, M, N, C to CPC is precisely K, a system which is known to be sound and complete with respect to Kripke frames. Modal logics above K are called normal. It is often argued that from the epistemic perspective normal systems are too strong to model human agents: despite the manageability of such logics for AI applications and their low computational complexity, normal epistemic logics raise a number of difficulties if employed to philosophically clarify the nature of human knowledge and belief. One of the most well-known problems is that normal epistemic logics are

God’s Omniscience: A Formal Analysis in Normal and Non-normal...


affected by various forms of logical omniscience, which looks mostly unsuitable for modeling human epistemic capabilities.24 Nevertheless, if one’s goal is to model divine omniscience, normal logics seem no longer too strong, but rather not strong enough. In the following section, we shall further investigate this issue, by considering further axiom schemata within a normal-modal-logic setting.

5.1. Some well-known schemata Three well-known axiom schemata are widely discussed in the context of epistemic logics:25 they all contribute to characterising the concept of omniscience and can describe aspects of divine knowledge as well.

5.1.1. Knowledge vs belief: God’s infallibility A first schema is the following: T := □A → A, which can be called the Principle of Truthful Knowledge. It claims that an agent cannot be mistaken when he knows something: what is known is also true. The contrapositive of this schema is T* := A → ¬□¬ A and states that given a true fact, it is impossible to know the opposite. This schema is traditionally considered as defining knowledge versus belief: if the logic for □ does not contain T, then □ represents agents’ belief but not their knowledge.26 Indeed, it would be

R.Fagin et al., Reasoning..., op. cit.; J.J. Meyer, Modal Epistemic..., op. cit. R. Fagin et al., Reasoning..., op. cit.; J.J. Meyer, Modal Epistemic..., op. cit. 26 J. Hintikka, Knowledge and Belief: An Introduction to the Logic of the Two Notions, Cornell University Press, Ithaca–New York 1962. 24 25


Antonino Rotolo, Erica Calardo

quite odd to assume that anything believed is also true, whether the same is acceptable, if not desirable when modelling knowledge. If applied to modelling God’s omniscience, T has the following immediate positive effects: • If all beliefs of God are expressed with □, then we capture the idea that God is epistemically infallible: none of God’s beliefs can, in fact, be false when T is accepted. Although this schema is not directly linked to the idea of omniscience, it plays anyway an indirect but important role in characterising God’s knowledge.27 • The fact that God is epistemically infallible alleviates the problem described by the end of Section 3.3: if we have □P and □¬P, then we can derive □X for conclusions X that are factually false. Schema T guarantees that whatever X is known/believed by the agent is also true. • Since modelling God’s epistemic nature cannot admit the distinction between knowledge and belief, otherwise we should admit that God believes things that are not true, and so He is not infallible; then, if □(Bologna is in tke UK), then Bologna is indeed in the UK. • If this last conclusion, i.e. ‘Bologna is in the UK’, looks quite odd when we model human or non-divine agents, it should be seen as positive with divine knowledge: the mere fact that God knows that any X is true implies (or, perhaps, makes) such an X true. This analysis is, hence, in line with the idea that God believes no falsehoods, which was pointed out, for instance, by Plantinga28 and Gale29.

27 P. Weingartner, Omniscience: From a Logical Point of View, Ontos Verlag, Frankfurt–Paris–Ebikon–Lancaster–New Brunswick 2008, Chapter 1. 28 A. Plantinga, God, Freedom, and Evil, “Eerdmans” 1977, vol. 20. 29 R. Gale, On the Existence and Nature of God, Cambridge University Press, Cambridge 1991.

God’s Omniscience: A Formal Analysis in Normal and Non-normal...


5.1.2. God’s knowledge about His own knowledge (and ignorance) The other two schemata are the following: 4 := □A → □□A 5 := ¬□A → □¬□A The epistemic common interpretation of 4 is that it expresses the Principle of Positive Introspection. This states that an agent that knows (or believes, if T does not hold) something, is also aware of this fact. This form of positive awareness should apply also to God’s knowledge:30 if what God knows is true, then, if it is true that He knows some A, He should also know that He knows that. Schema 5 expresses the Principle of Negative Introspection. This states that an agent who ignores a fact is also aware of his ignorance. Of course, this axiom captures a form of omniscience as well, even though it could sound strange that God may not know that some A is the case, because this would mean that God does not know A. Indeed, this schema does not force God to be necessarily ignorant, but states that, if God were ignorant with respect to some A, then he would be aware of this fact. However, the schema does not logically exclude, too, that God could be ignorant with respect to any true A. This is not guaranteed by any system containing only T, 4, and 5. We will discuss this question in the subsequent sections.

5.2. The S5 system: the heaven of knowledge? The system resulting by adding T, 4 and 5 to the minimal normal logic K is usually called S5 and it has been widely studied throughout the years. Some modal schemata correspond to specific properties of 30

P. Weingartner, Omniscience..., op. cit., Chapter 12.

Antonino Rotolo, Erica Calardo


frames. For instance, we have that T characterises the class of reflexive frames, 4 isolates that of transitive ones and 5 that of Euclidean structures. The schema B :=A → □◇A, which is derivable in S5, characterises the class of symmetric frames. These observations lead to the well-known result stating that S5 is sound and complete with respect to the class of Kripke structures whose binary relation is an equivalence, i.e. R is: • Reflexive: ∀w(wRw) • Symmetric: ∀w∀v(wRw ⇒ vRw) • Transitive: ∀w∀v∀z(wRv & vRz ⇒ wRz). An interesting and well-known technical result states that the system S5 is also determined by the class of frames whose relation is universal31. This means simply that all worlds are connected to each other, and hence the accessibility relation no longer exists. Universal frames can be seen as epistemic islands, monads in which knowledge does not depend on the perspective of agents. Moreover, from the agents’ point of view, a plurality of epistemic alternatives is no longer a theoretic possibility, since all worlds are epistemically equivalent, thus making the notion of epistemic alternative void. Within a universal frame, only one epistemic paradigm is accepted and the information must be consistent. Semantically, this is mirrored by the fact that the axiom T imposes reflexivity, hence a type of seriality: this guarantees that agents can never have contradictory knowledge, i.e. the situation □A → □¬A is no longer acceptable. S5 frames cannot handle conflicts, nor cognitive relativism, since we have that □A → ¬□¬A holds, too. Hence, these are structures designed to accommodate perfect, logically omniscient epistemic agents. Nevertheless, although S5 expresses the strongest version of logical omniscience, it is still too weak to capture God’s omniscience. 31

B.F. Chellas, Modal Logic, op. cit., p. 97-98.

God’s Omniscience: A Formal Analysis in Normal and Non-normal...


The heaven of knowledge is not far away, but we still have to make a substantial step beyond normal logics in order to climb to the top of the stairway to divine knowledge.

6. Factual omniscience and divine knowledge So far, we have characterised the S5 epistemic agent as perfectly analytical, aware of all the consequences of his knowledge, logically omniscient, never mistaken, consistent, perfectly conscious of what he knows and ignores. But is it enough to characterise an omniscient being? Another type of omniscience, we have said, can be mentioned: the so-called factual omniscience. This is often characterised as maximal or complete knowledge. Formally speaking, a set of sentences is said to be maximal if, for any well-formed formula A, either A or its negation ¬A belongs to the set itself. Thus an epistemically maximal knowledge base should be such that, given a formula A, it contains either A or ¬A. Hence, an agent possessing maximal knowledge knows any true fact, i.e. for any proposition A either □A or □¬A belongs to the theory. CPC assumes that any proposition has a semantic univocal truth value, either true or false (mirrored syntactically by the law of the excluded middle). That an omniscient being may lack the knowledge of any true (respectively, false) fact seems to contradict the concept of omniscience itself. Therefore, it sounds not only reasonable, but actually necessary to add a further schema to our epistemic theory S5 to capture this intuition. Formally, such an axiom should look like an epistemic version of the law of the excluded middle, namely: FK := □A ∨ □¬A, the acronym FK standing for factual knowledge, as it seems to formalise factual omniscience, i.e. the knowledge of any true fact. By


Antonino Rotolo, Erica Calardo

adding FK to S5, the resulting system should be sufficiently strong to give a good account of God’s omniscience. Indeed, the idea of divine omniscience as expressed by FK was informally discussed, for instance, by Zagzebski.32 The trouble is that, upon a closer analysis, it may actually appear rather too strong. Indeed, a very troublesome schema is easily derivable by means of a rather trivial logical deduction: ⊢s5.FK □¬A ∨ □A ⊢s5.FK ¬□¬A → □A ⊢s5.FK A → ¬□¬A ⊢s5.FK A → □A

FK Tautology Axiom T* Law of concatenation

The conclusion A → □A is known as Anselm’s Principle (AP) as it is used in the modal version of the famous ontological argument expressed by Anselm.33 Alethically, it states that anything which happens to be true is also necessary; epistemically, that any true fact is known. A reader not familiar with modal logic may not notice the profound difference between the Anselm’s Principle and the rule RN := A/□A. However, these schemata state indeed two very different things. The RN rule is only concerned with logical truth and theorems: it states that whenever a formula happens to be a theorem or, semantically, valid, i.e. true under any assignment, then it is also known. Anselm’s Principle, on the other hand, states that any formula which is possibly true under some assignment is also known. This captures the inner difference between factual and logical omniscience: any God-like omniscient being should certainly possess both. The central point is, however, that along with the schema T := □A → A, Anselm’s Principle brings the system to be trivial as there is a complete collapse of modality: □A ≡ A. In this scenario, modalities turn out to be empty and the system is actually equivalent to CPC. L. Zagzebski, Omniscience, op. cit., p. 262. J. Ernst, Charles Hartshorne and the Ontological Argument, “Aporia” 2008, vol. 18 (1), p. 57-66.

32 33

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Hence, the trouble with FK (and with AP) is that it rather appears too strong, since the modal concept of knowledge vanishes. The system obtained by adding AP and T or, equivalently, D and A ≡ □A to K is usually called Triv34 and it is equivalent to CPC. Moreover, the schemata AP and FK we discussed above are perfectly equivalent under K plus T. However, these schemata express rather different properties. Where AP states that whatever is the case, it is also known, schema FK says something stronger: given any possible fact, the agent knows either the fact itself or its negation. Their difference is mirrored semantically. Lemma 2 (Characterisation of AP in Kripke semantics). For any Kripke frame ℱ := 〈W,ℛ〉, ℱ ⊨ A → □A if and only if for any world w ∈ W, if R(w) ≠ Ø, then R(w) = {w}. Lemma 3 (Characterisation of FK in Kripke semantics). For any Kripke frame ℱ := 〈W,ℛ〉, ℱ ⊨ □A ∨ □¬A if and only if for any world w ∈ W, R(w) ≠ Ø, then |R(w)| ≤ 1. These conditions show that FK entails the validity of AP, whereas the converse does not hold, unless the frame is reflexive (and hence validates T). Therefore, it is not true, in general, that the two principles are logically equivalent, as maintained by Wierenga:35 this happens only if we assume the omniscient being to be epistemically infallible.

G. Hughes and J. Cresswell, A New Introduction to Modal Logic, Routledge, London–New York 1996, p. 64-65. 35 E. Wierenga, Omniscience, [in:] E.N. Zalta (ed.), The Stanford Encyclopedia of Philosophy, Spring 2010. 34


Antonino Rotolo, Erica Calardo

7. Summary In this paper, we have analysed the concept of divine omniscience within the setting of propositional epistemic logics. We considered different aspects that semantically model the limits for an agent to be fully omniscient: • a plurality of possible worlds or states, a standard idea in epistemic logics that captures the fact that, when an agent does not possess full knowledge in a certain situation w, then we conceptually refer to all the epistemic alternatives to w; • a plurality of accessibility relations connecting pairs of worlds, which express different epistemic standards; the fact that, for the same agent, we may have that □A and □¬A are both true entails that there are at least two standards and that the agent does not possess any true and objective knowledge in regard to A. This semantic picture was the general framework within which we discussed the epistemic meaning of several axiom schemata and inference rules. The analysis proceeded from the weakest epistemic systems to the strongest ones by adding different schemata step by step. The result depicted an ascending path to divine omniscience that first gave up the assumption that we may have different epistemic standards and, subsequently, removed any accessibility relation connecting worlds. This said, however, we argued that the obtained logic system is still under the threshold of divinity, as it does not yet model the concept of complete or maximal knowledge. This final concept may be obtained by adopting schemata that, in fact, trivialise the notion of knowledge: epistemic logics collapse into the classical propositional calculus.

Kazimierz Trzęsicki University of Białystok

Problems of Omniscience and Infallibility. A Temporal-Logical Approach* Rarely affirm, seldom deny, always distinguish. Thomas Aquinas, Summa Theologiae

1. Introduction


often ask how the sacred realm can be described in a coherent and acceptable way from the point of view of the divine message, as well as rational knowledge. It is a fundamental question of theology. Christian theology has to comply with the Holy Scripture and its tradition, i.e with the Bible or – as it is the case in the Roman Catholic Church – with its interpretation within the context of the sacred tradition and the community of the Church. In this aspect, there is a substantial difference between theology and philosophy. In particular, in the philosophy of the divine reality and the philosophy of God, no authority is unquestionable. The interpretation of the divine message can be undertaken in accordance with the intellect and the power of reason in line with its principles or according to its content. There are four combinatorial cases. Interpretation can be provided in accordance with the intellect: 1. only to the principles of interpretation, 2. only to the content of interpretation,

* This contribution was made possible through the support of a grant “The Limits of Scientific Explanation” from the John Templeton Foundation.


Kazimierz Trzęsicki

3. to the principles and to the content of interpretation, 4. to neither the principles nor the content of interpretation. We need to add what we understand by rationality or accordance with the intellect. For example, is the interpretation under authority an interpretation in accordance with the intellect? In a narrow sense, any interpretation of the divine message should be recognized as not being in accordance with the intellect, since any system of knowledge about the divine realm is based on an authority (individual, collective or mythical, etc.), deciding the questions of interpretation. If ‘to be rational’ means the same as ‘to be logical’, then the largest notion of rational interpretation, when not any interpretation under authority is questionable or – in other words – some questions can be decided rationally by an authority, is such that the authority decides about interpretation only in questions for which any answer ‘yes’ or ‘no’ has the same logico-methodological status. For example, the thesis that angels1 exist has the same logico-methodological status as the thesis that such beings do not exist. The same is true about the thesis that angels are intelligent and the thesis that they are not intelligent. Similarly, the question of the angelic hierarchy is outside natural knowledge. But the thesis that an angel is blue and red at the same time and in the same aspect, i.e. that an angel is a self-contradictory object, is illogical. This thesis has a different logico-methodological status than the thesis that no angel is blue and red at the same time and in the same aspect. Considering rationality, we may discuss the question of prayer, magic or witchcraft. Many examples of the effectiveness of such actions are known. We may ask if, for example, a statistical method may be applied to the investigation of the effectiveness of prayer. But the thesis concerning the effectiveness of prayer should not be examined by scientific method. The thesis would not be falsified if statistically the health of people who pray would be the same

Of course, if angels are conceived as beings and are cognizable only based on the sacred message.


Problems of Omniscience and Infallibility. A Temporal-Logical Approach


as that of people who do not pray and even if the health of praying people would be worse than the health of people who do not pray.2 The effects of prayer exist in a realm that is outside of human cognitive power. Thus, there are reasons to distinguish two notions of rationality: 1. negative – a thesis is rational if and only if the thesis is not incompatible with principles of logic or methodology, 2. positive – a thesis is rational if and only if the thesis is in accordance or is compatible with the principles of logic or methodology. The question arises as to how these conditions should be understood. For the idea of rationality discussed here, the notion of logic is fundamental. We have to answer the question as to what is acceptable from the logical angle and what is not acceptable. The answer to this question will be given in the first part of the paper. In the second part, in the light of the largest notion of logicality, the questions of omniscience and of infallibility will be discussed as an example. Solving theological problems can be done in accordance with human intellect or against it, irrationalistically: Credo quia absurdum. The rationalistic view that human reason, or understanding, is the sole source and final test of all truth results in the rejection of genuine theology. Theology can be anti-irrationalistic and it can be developed in the spirit of anti-irrationalism, like logistic anti-irrationalism.3 This situation is possible because sick people are more disposed to pray in hope that due to prayers they will get better. A similar situation is in the case of magic: communities practicing magic usually are poor and in bad health. 3 The term was introduced by Kazimierz Ajdukiewicz. For characterization of antiirrationalism see: K. Ajdukiewicz, Zagadnienia i kierunki filozofii (teoria poznania, metafizyka), Czytelnik, Kraków 1949, p. 37-77. This work was translated into English as Problems and Theories of Philosophy, and also into Spanish; K. Ajdukiewicz, Logistyczny antyirracjonalizm w Polsce, “Przegląd Filozoficzny” 1934, no. 4, p. 399-408. English translation [in:] K. Ajdukiewicz, Logistic Anti-Irrationalism in Poland, [in:] W. Krajewski, (ed.), Polish Philosophers of Science and Nature in the 20th Century, “Poznań Studies in the Philosophy of the Sciences and the Humanities”, vol. 74, Ro2


Kazimierz Trzęsicki

2. Notion of a theory A theory ॎ consists of: • P – the set of all propositions of the language of the theory, or shortly the language of the theory • L – rules of reasoning (logic) • T – the set of theorems of the theory. The set T of theorems can be conceived: • as the set of these and only these propositions that are actually – it depends on the methodology of the considered theory – well-grounded, justified or proved, • as the set of actual theorems and all their logical consequences, i.e. the set of propositions that is closed under the rules of logic. In our consideration in any case it should be clear how T is understood. An abstractly conceived theory is such that T = L(T), i.e. the set of all and only theses is closed under the rules of reasoning: any proposition that according to L can be obtained from T belongs to T. Our considerations are limited to a classical case: it is supposed that a proposition is or a proposition is not an element of T. We do not consider the possibility that a proposition is in a certain degree or probability an element of T. When we speak about language, logic and theorems, we will mean language P, logic L, and the set of theorems T of the theory ॎ, i.e.: ॎ = 〈P, L, T〉.

dopi, Amsterdam 2001, p. 241-249. Anti-irrationalism demands that every rationally accepted proposition be intersubjectively communicable and testable (stanford.edu/entries/lvov-warsaw/). Philosophical activity must begin with a very careful linguistic analysis of investigated problems and their meaning. Philosophy is a science, like any other. K. Ajdukiewicz defends the standpoint of moderate empiricism.

Problems of Omniscience and Infallibility. A Temporal-Logical Approach


L can be defined: • syntactically (⊢), • semantically (⊨). Σ ⊢ φ means that the proposition φ is inferable from the set Σ of propositions according to the syntactical rules ⊢ of logic L. Σ ⊨ φ means that the proposition φ is justified according to semantic principles ⊨ of logic L by propositions of Σ. { φ : Σ ⊢ φ} is the set of all and only propositions that could be inferred from the set Σ according to the rules ⊢. { φ : Σ ⊨ φ} is the set of all and only propositions that could be justified by propositions of the set Σ. Let R-theory be a rational theory, i.e. a theory that does not contradict the principles of intellect, a theory that does not deny the principles of logic or methodology. A theory ॎ = 〈P, ⊢, T〉 is R-theory only if: 1. T ≠ ∅, i.e. the set of theses is not empty, 2. {φ : T ⊢ φ} ≠ P, i.e. the set of all propositions that are inferable from T is not equal to the set P of all sentences, i.e. the theory is not overfull, 3. T ⊆ { φ : T ⊢ φ}, i.e. it is excluded that any thesis could be rejected by the rules of inference ⊢. These conditions characterize R-theory syntactically. The condition 1 presupposes that the theory says something. The condition is important because if the set of theses were empty, the conditions 2 and 3 would be satisfied. This condition presupposes non-emptiness of the set P of sentences.


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The condition 2 is weaker than the condition that for any φ: φ, as well as not-φ are not both inferable from T. Our condition is especially important in the case of, e.g. para-consistent logic.4 The condition 3 does not exclude rules of rejection. It is supposed only that an application of a rule to the theses does not result in the rejection of a thesis. Otherwise it would be a contradiction: according to some rules a sentence would be a thesis and according to other rules it would not be a thesis. This condition expresses the meta-theoretical syntactic principle of non-contradiction: for any proposition φ, it is excluded that the proposition is a thesis and is not a thesis of ॎ. It does not mean that it is excluded – at least by this condition – that for some φ, both the propositions φ as well as not-φ are theses. Classically conceived satisfaction is a relation between a proposition and a domain (in that the proposition is satisfied). A proposition that is satisfied in a domain of the theory ॎ – according to Tarski’s definition of truth5 – is true in the domain of the theory ॎ. Let ै be a model (reality) and let ⊨ be a 2-place relation such that ै ⊨ φ holds if and only if the proposition φ is satisfied in ै. ै ⊨ Σ means that all the propositions of Σ are satisfied in ै. A theory ॎ of reality ै is R-theory only if: 4 . { φ: ै ⊨ φ} ≠ ∅, i.e. at least one proposition is satisfied, 5 . { φ: ै ⊨ φ} ≠ P, i.e. not all propositions of P are satisfied in ै, 6. (a) ै ⊨ T, i.e. theorems (elements of T) are satisfied in ै, The first formal para-consistent logic to have been developed was discussive (or discursive) logic by the Polish logician Jaśkowski (cf. S. Jaśkowski, Rachunek zdań dla systemów dedukcyjnych sprzecznych, “Studia Societatis Scientiarum Torunensis” 1948, Sec. A, I, p. 57-77. English translation [in:] S. Jaśkowski, Propositional Calculus for Contradictory Deductive Systems, “Studia Logica” 1969, no. 24, p. 143-157. 5 A. Tarski, Pojęcie prawdy w językach nauk dedukcyjnych, Prace Towarzystwa Naukowego Warszawskiego, Warsaw 1933. See also A. Tarski, Der Wahrheitsbegriff in der formalisierten Sprachen, “Studia Philosophica” 1936 (for 1935), no. 41, p. 261-405. English translation: The Concept of Truth in Formalized Languages, [in:] Logic, Semantics, Metamatematics: Papers from 1923 to 1938, Clarendon Press, Oxford 1956. 4

Problems of Omniscience and Infallibility. A Temporal-Logical Approach


(b) T ⊆ { φ: T ⊨ φ} , i.e. semantic rules of reasoning do not result in the rejection of satisfied propositions. The condition 4 is important because if the set of satisfied propositions was empty, the other conditions would be satisfied. The condition presupposes non-emptiness of the set P of propositions. The condition 5 says that the language P should be sufficiently rich in order to express some non-satisfied propositions. This condition expresses semantically what the condition 2 expresses syntactically. The case of condition 6 is a similar one. Semantic characteristics should be completed by the provision that not only the rules of reasoning do not result in the rejection of satisfied proposition but that also elements of T are satisfied. The condition 6 does not exclude the possibility of revision of the theory. It rejects only the possibility that the rules of reasoning result in the rejection of a satisfied proposition. If a proposition is satisfied, then it is not possible that this proposition is not satisfied. This condition expresses the meta-theoretical principle of semantic non-contradiction. For R-theory it is not supposed that the rules of reasoning are such that if they are applied, then: • if premises are theses, then the conclusion is a thesis, • if premises are satisfied, then the conclusion is satisfied, too. This means that it is admitted that some logical rules of L, if applied to theses, may not result in a thesis. The syntactical and semantic conditions are comparable: if ⊢ is equivalent to ⊨, then they say the same. In any case, the condition of meta-theoretical (syntactical, semantic) non-contradiction is a basic requirement. The discussed requirement of non-contradiction is meta-theoretical. Therefore, it concerns statements about members of T, i.e. it says something about φ ∈ T but not about the members of T, so not about


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φ. It is excluded that a statement is and is not a member of T. The meta-theoretical principle of non-contradiction does not exclude it that for a certain statement φ, φ as well as non-φ are members of T. The discussed requirements for R-theory seem to be minimal. The question posed is if there are other conditions that R-theory has to fulfil. Unambiguous language. Language has to be meaningful as interpreted in the domain of a theory, i.e. it has to have a model.6 The unambiguity of language means that this model is exactly one (up to isomorphism). The demand of ambiguity seems to be natural. Let us bring up the Löwenheim-Skolem-Tarski theorem.7 The theorem – let us omit the technical details – states that any theory if it has an infinite model, then it has models of any cardinality which is the same or greater than the cardinality of the language. From the theorem it follows that any (sufficiently rich) language has more than one interpretation, more than one meaning. The theory that is formulated in such a language is not unambiguous. If the requirement of unambiguity is not fulfilled by theories that are developed according to the most rigorous logical conditions, then should such a requirement be imposed on other theories? The unambiguity of the language does not have to be accepted as a condition that R-theory has to fulfil. It is also expected that the language of a theory should be such that all truths about the domain of a theory could be put into words 6 A model of a language P of a theory ॎ is not the same as a model of ॎ. A theory has a model only if the language of the theory has a model but not conversely. 7 L. Löwenheim, Über Möglichkeiten im Relativkalkül, “Mathematischen Annalen” 1915, no. 76, p. 447-470. English translation [in:] J. v. Heijenoort, op. cit., p. 232251; T.A. Skolem, Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theorem über dichte Mengen, I, Skrifter utgitt av Videnskapsselskapet i Kristiania, Mat. Naturv. Kl. 4, 1920; T. Skolem, Einige bemerkungen zu axiomatischen begründung der mengenlehre, Mathematikerkongressen i Helsingfors 4-7 July 1922, Den femte skandinaviska matematikerkongressen, Redogoerelse 1923, p. 217-232; T. Skolem, Über einige grundlagenfragen der mathematik, Skrifter ut-gitt av det Norske Videnskaps-Akademi i Oslo: I. Matematisk-naturvidenskabelig Klasse 7, 1929, p. 1-9.

Problems of Omniscience and Infallibility. A Temporal-Logical Approach


of this language, i.e. the language has to be sufficiently expressive. From the Löwenheim-Skolem-Tarski theorem – if we suppose that there are truths about models of greater cardinality that are not the truths about smaller cardinality – it follows that no sufficiently rich language (that has infinite models) is satisfactorily expressive to pronounce all the truths of the investigated domain. The demand that any truth should be expressive in a language of R-theory is thus too strong. The most rigorous mathematical theories do not fulfil such a condition. Decidability. The expectation that R-theory is decidable seems to be a demand of our intellect. A question whether φ is a member of T is decidable if there is an effective procedure that results in a solution to a problem in a finite number of steps. A set T is decidable if there is a procedure that for any proposition φ of the language P, after a finite number of operations, gives ‘yes’ or ‘no’ answer to the question whether φ is a member of T. Is there such a procedure for any theory? The idea of decidability has been stated by David Hilbert who believed that Wir müssen wissen – wir werden wissen!8 Procedures that should not leave any doubt as effective procedures of deciding9 were first described independently by Alonzo Church10 and Alan Turing11. Based on his procedure, Church and also This is the inscription on D. Hilbert’s grave. It is shown that to date all described procedures that could be intuitively accepted as effective are mutually equivalent. 10 A. Church, An Unsolvable Problem of Elementary Number Theory, “Bulletin of the American Mathematical Society” 1935, no. 41, p. 332-333. Preliminary Report (abstract), supplied on 22 March 1935. Also by the same author: An Unsolvable Problem of Elementary Number Ttheory, “American Journal of Mathematics” 1936, no. 58, p. 345363. Presented to the American Mathematical Society on 19 April 19 1935; A Note on the Entscheidungsproblem, “Journal of Symbolic Logic” (received on 15 April 1936), no. 1, p. 40-41, p. 356-361. Correction ibidem, 1936, p. 101-102; Reviews of Turing (1936-1937) and Post (1936), “Journal of Symbolic Logic” 1937, no. 2 (1), p. 42-43. 11 A.M. Turing, On Computable Numbers, with an Application to the Entscheidungsproblem, Proceedings of the London Mathematical Society, vol. 42 (Series 2), 19361937, p. 230-265. Reprint [in:] A.M. Turing, On Computable Numbers, with an Application to the Entscheidungsproblem, 1965, p. 116-151. 8 9


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Turing proved the insolvability of certain problems. It is proven that the first order predicate calculus is not decidable. The first-order predicate calculus is the basic tool of all developed theories. If the predicate calculus is not decidable, the demand that R-theory should be decidable is too rigorous. Completeness. It seems rational to expect that for any theory there is a consistent set of theorems (axioms) that can be listed by an effective procedure from which all the other true statements could be extracted by the rules of logic L. It transpired, as a result of Gödel’s12 first incompleteness theorem, that in the case of any sufficiently powerful and consistent theory (that includes the theory of the natural numbers), it is not possible. For any such system, there will be a statement that can be shown to be true, but that does not emerge from the axioms of the theory. The demand that R-theory should be such that, starting with a decidable consistent set of statements we would be able to infer from them all other true statements, is thus too rigorous for R-theory. Any theory of a sacred realm is based on a limited number of truths revealed in a sacred message.13 By analogy to Gödel’s theorem, we may suspect that from these revealed truths not all truths about the sacred realm are inferable. Hence, there are such questions for which an answer could be given only as hypothetical, as one of possible answers. This conclusion may be used to justify the presence of mystery about the sacred realm.

K. Gödel, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, “Monatshefte für Mathematik und Physik” 1931, no. 38, p. 173-198. Translated in: S. Feferman (ed.), Kurt Gödel: Collected Works, vol. 1, Oxford University Press, Oxford 1986, p. 144-195. Online version: transl. by B. Meltzer, 1962, net/ygg/etext/godel/; PDF version: transl. by M. Hirzel, http://nago.cs.colorado.edu/ ~hirzel/papers/canon00-goedel.pdf. 13 Usually, there are problems with the interpretation of a sacred message and the list of truths included in it is disputable. 12

Problems of Omniscience and Infallibility. A Temporal-Logical Approach


Proof of consistency. The set of all theorems should not be equal to the set of all propositions P. In the case of classical negation, this condition is equivalent to the requirement that in T there are no two propositions such that one of them is the (classical) negation of the other. If it were a case by the law of Duns Scotus (if φ and not-φ, then ψ), any proposition would be inferable. It seems that a requirement of a proof of consistency of a theory is reasonable. Of course, the proof of consistency should not evoke any suspicion of fallibility. In any case, the proof should be accomplished by rules of logic that are certain and without any doubt regarding certainties. It was proven by Gödel in his second incompleteness theorem, using the methods of first-order predicate calculus, that for any sufficiently powerful and consistent theory (that includes the theory of the natural numbers) the proof of consistency does not exist.14 Stated more colloquially, any theory that is interesting enough to formulate its own consistency can, with the help of ⊢, prove its own consistency if and only if it is inconsistent. The hope of Hilbert that consistency could be established by finitistic means, i.e. – roughly speaking – by reliable logical means that concern only a finite number of calculations, has been lost. Mathematicians who want to develop mathematics as a deductive theory do not doubt the consistency of Zermelo-Fraenkel set theory, in particular. The belief is justified by the fact that after the inconsistencies found in Cantor set theory were solved, and in spite of intensive development of the theory and its applications in other fields of mathematics, no new contradictions have occurred. Is it reasonable to expect that a theory is R-theory, though there is no proof of its consistency, if there is no such proof in the case of mathematics which provides tools for all the science and technology?

Gerhard Gentzen proved the consistency and completeness of arithmetic using the transfinite induction. In his opinion, the transfinite induction is a credible logical means (cf. G. Gentzen, Die Widerspruchsfreicheit der reinen Zahlentheorie, “Mathematischen Annalen” 1936, vol. 112, p. 493-565). 14


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Are there any other unquestionable conditions than those listed above that R-theory should fulfil? The attempt to add some other conditions has failed: neither the unambiguousness of language, nor decidability, nor completeness nor even the proof of consistency are demands that are fulfilled in mathematics. Thus, we have rational reasons to accept as sufficient the conditions that have been listed above as necessary. The issues of R-theory are interesting in themselves. In our notion of R-theory, the role played by logic and methodology is restricted to demonstration that there is no contradiction if there are any reasons to suspect its occurrence. Below, the issue of God’s omniscience, God’s infallibility and human free will shall be discussed. We will argue that there are solutions of the issues that fulfil the conditions of R-theory.

3. God’s omniscience and human free will I have declared a spiritual war upon all coercion that restricts man’s free creative activity. Prof. Jan Łukasiewicz, March 7, 1918

The problem of God and time is one of the most complicated and subtle questions of theology (and philosophy). The ease of making an error has to be taken into account and erroneous steps in reasoning may have unpredictable consequences. We do not have an adequate notion of God. To date, neither physics nor philosophy has had an undisputable answer to the question of what is time, what is its nature and structure. Can we discuss the question of omniscience, infallibility and human free will, if the notions of God and time are fundamental for this consideration? What cognitive value will such a consideration have? As humans we want to have an answer. Let us take an attempt to resolve the problem in such a way that will fulfil the conditions of R-theory. It seems that the proposition that God is omniscient is not consistent with the proposition that any or some human actions are free. It

Problems of Omniscience and Infallibility. A Temporal-Logical Approach


seems that God’s omniscience excludes human free will. The universe is completely determined throughout eternity and, as a result, a man’s fate is decided in advance. Our belief that we are able to have conscious choice of different actions is only pure illusion. For Łukasiewicz:15 The determinist looks at the events taking place in the world as if they were a film drama produced in some cinematographic studio in the universe. We are in the middle of the performance and do not know its ending, although each of us is not only a spectator but also an actor in the drama. But the ending is there, it exists from the beginning of the performance, for the whole picture is completed from eternity.

This ancient problem, which Aristotle formulated considering the logical value of a proposition that tomorrow there will be a seabattle, went through the various speculations of logicians and philosophers of the Middle Ages. In contemporary logic, inspired by Łukasiewcz’16 investigations of many-valued logics, this was a problem deliberated by Arthur Norman Prior17 when conceiving temporal logic. For Łukasiewicz, any proposition about the contingent event has a logical value, though it is not necessarily one of the two: true or false.

4. Omniscience and infallibility Let us suppose that the considered theory is based on classical logic. Let Greek small letters φ, ψ, ..., possibly with indices, denote 15 J. Łukasiewicz, Selected Works, Polish Scientific Publishers, North-Holland Publishing Company, Warsaw–Amsterdam 1970, p. 113. 16 J. Łukasiewicz, O determinizmie. Z zagadnień logiki i filozofii. Pisma wybrane, Państwowe Wydawnictwo Naukowe, Warszawa 1961, p. 114-126. English translation [in:] J. Łukasiewicz, On Determinism, op. cit., p. 19-39 and Łukasiewicz, Selected Works, op. cit., p. 110-128. 17 A.N. Prior, Three-Valued Logic and Future Contingents, “Philosophical Quarterly” 1953, no. 3, p. 317-26.

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(these are meta-language variables) sentences of the language of the theory. Let AP (Atomic Propositions) be the set of propositional letters. The elements of AP are non-compound propositions. The following symbols of logical connectives will be used: ¬ – negation, ∨ – disjunction, ∧ – conjunction, ⇒ – implication, ⇔ – equivalence. Compound propositions are defined in the usual way. Let ‘B(φ)’ mean: B knows that φ. The letter ‘B’ will also be used to denoted a being about which we ascertain that it knows. ‘B is omniscient’ means: 1. φ ⇒ B(φ), i.e. if φ, then B(φ). ‘B is infallible’ means: 2. B(φ) ⇒ φ, i.e. if B(φ) , then φ. Anybody who for any question knows the answer ‘yes’ is omniscient. Anybody who for any question knows the answer ‘no’ is infallible. Thus, there is no problem with being omniscient or with being infallible. The true problem is with being omniscient and as well infallible. Usually, it is admitted that only God is omniscient and infallible. These formulations of omniscience and infallibility omit temporal aspects of facts. We can take into account this aspect introducing the parameter of time. Any fact takes place at definite instant of time. For our purposes, the introduction of temporal operators will be sufficient. They are able to express everything what is substantial for the questions of omniscience and infallibility. Tense operators are a particular type of temporal operators. Let us introduce symbols of tense-operators:18 Tense Logic was introduced by Arthur Prior as a result of an interest – from the position of science – in the relationship between tense and modality. The philosophical and 18

Problems of Omniscience and Infallibility. A Temporal-Logical Approach


• F – It will at some time be the case that... • G – It will always be the case that... • P – It has at some time been the case that... • H – It has always been the case that... F and P are weak tense operators. G and H are strong tense operators. Using tense operators the temporal aspects of omniscience and infallibility can be expressed. ‘B is omniscient’ can be expressed as: 4. φ ⇒ HBFφ. Such a formulation can be justified by sacred texts. By the thesis about infallibility of B, we have that: 5. BFφ ⇒ Fφ. If we accept the rule: H(φ ⇒ ψ) Hφ ⇒ Hψ’ then we have: HBFφ ⇒ HFφ. Finally, we obtain:

φ ⇒ HFφ. The proven proposition expresses the thesis of pre-determinism PREDET:19 if it is such that φ then it always has been such that there will be φ. theological questions were important as the source of inspiration (cf. A.N. Prior, Time and Modality, Oxford University Press, Oxford 1957; by the same author: Past, Present and Future, Oxford University Press, Oxford 1967; Papers on Time and Tense, Oxford University Press, Oxford 1968). 19 We discern the thesis of pre-determinism (PRE-DET) and the thesis of post-determinism (POST-DET). According to thesis of pre-determinism, any fact that was, is or will be, is determined in any instant earlier than the fact has occurred. Usually, the thesis of pre-determinism is understood as the thesis of determinism.

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May we maintain that in classical logic from omniscience and infallibility the thesis of (pre-)determinism is provable? Is it proven that in classical logic omniscience and infallibility are not compatible with free will? In R-theory a contradiction is not acceptable. Thus, there are at least two solutions. We may consider some changes in logic. Such a solution was proposed by Łukasiewicz. It gives the philosophical ground of his many-valued logics. Łukasiewicz’ investigations have paved the way for the elaboration of many-valued logics. In Łukasiewicz’ manyvalued logic the law of excluded middle is no longer valid. He wrote:20 I maintain that there are propositions which are neither true nor false but indeterminate. All sentences about future facts which are not yet decided belong to this category. Such sentences are neither true at the present moment, for they have no real correlate, nor are they false, for their denials too have no real correlate. If we make use of philosophical terminology which is not particularly clear, we could say that ontologically there corresponds to these sentences neither being nor non-being but possibility.

Another way of solving this problem was considered by A.N. Prior. He reflected on the concept of tense operators. The avoidance of undesirable consequences is possible if the meanings of tense operators are changed. Let time ॎ be a pair 〈T, ⊳〉, where T is a non-empty set (of points of time, instants) and let ⊳(⊆ T × T) be a binary relation earlier-later. The idea of time as a set of points is prima facie natural and common. There are other concepts of time, e.g. time as an interval, i.e. instants (of time) are intervals. ै (= 〈T, ⊳, V〉) or 〈ॎ, V〉), where V (: T → 2AP) is a valuation, is a Kripke model. According to the thesis of post-determinism, any fact that was, is or will be, is determined in any instant later than the fact occurred. In our formal language, POST-DET, the thesis of post-determinism is expressed as follows: φ ⇒ GPφ. 20 J. Łukasiewicz, Selected Works, op. cit., p. 126.

Problems of Omniscience and Infallibility. A Temporal-Logical Approach


The model ै (= 〈T, ⊳, V〉) is based on time 〈T, ⊳〉, or – what signifies the same – on time ॎ. ै, t ⊨ φ means that the proposition φ in model ै is satisfied at instant t. Meanings of the expressions of the language P can be defined using ⊨. Let us define only the meanings of tense operators as the other connectives are conceived classically. Definition 1. G. ै, t ⊨ Gφ F. ै, t ⊨ Fφ H. ै, t ⊨ Hφ P. ै, t ⊨ Pφ

iff21 for any t1, t ⊳ t1 : ै, t1 ⊨ φ. iff there exists t1, t ⊳ t1 such that ै, t1 ⊨ φ. iff for any t1, t1 ⊳ t : ै, t1 ⊨ φ. iff there exists t1, t1 ⊳ t such that ै, t1 ⊨ φ.

It can be shown that ⊨ φ ⇒ HFφ. It means that in any model the proposition φ ⇒ HFφ is satisfied. If the proposition is understood as expressing the thesis of pre-determinism, then our conceiving of tense operators is deterministic. The language is ontologically committed. Thus, there are two possibilities: 1. either we resign with interpretation of φ ⇒ HFφ as expressing the thesis of pre-determinism, 2. or the meaning of tense operators will be changed in such a way that the interesting us issue will not be decided by meanings of tense operators. Let us consider the second possibility. Thus, we have to define indeterministically conceived tense operators. In particular, the notion should be such that the proposition ‘it will be such that φ’ is true only if it will be φ independent of the possible courses of events, independent of possible futures. In other words, it is (now) true that at t occurs φ, only if all causal possibilities include the occurrence of φ at t; it is (now) true that at t occurs ¬φ, only if no causal possibilities include the occurrence of φ at t. φ is undetermined, contingent only 21

‘Iff ‘ is a short for ‘if and only if’.


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if neither φ is true nor ¬φ is true. Of course, the language can be extended in different ways: 1. the deterministic tense operators may remain and the language will be supplemented with new indeterministic tense operators 2. some operators will be introduced which together with deterministic tense operators will form tense operators with indeterministic meaning. The notion of maximal linearly ordered set is useful in our considerations. Definition 2. Let π(t) will be a subset of T, i.e. π(t) ⊆ T such that: 1. t ∈ π, 2. if t1, t2 ∈ π, then t1 = t2 or t1 ⊳ t2 or t2 ⊳ t1 – linearity, 3. if t ∈ T and for any t1 ∈ π such that t1 ≠ t: t ⊳ t1 or t1 ⊳ t, then t ∈ π – maximality. Let Π(t) = {π(t): π(t) ⊆ T}. Let Π = {π(t): t ∈ T}. Tense operators are defined as below. Definition 3. G. ै, t ⊨ gφ F. ै, t ⊨ fφ H. ै, t ⊨ hφ P. ै, t ⊨ pφ

iff for any t1, t ⊳ t1 : ै, t1 ⊨ φ. iff for any π ∈ Π(t) there exists t1 ∈ π, t ⊳ t1: ै, t1 ⊨ φ. iff for any t1, t1 ⊳ t : ै, t1 ⊨ φ. iff for any π ∈ Π(t) there exists t1 ∈ π, t1 ⊳ t: ै, t1 ⊨ φ.

The tense operators are defined in such a way that if time is linear (non-branching), then there is no difference in meaning between deterministic and indeterministic tense operators. The new operators, unlike former discussed deterministic tense operators, are not ontologically committed. The thesis of determinism depends on the nature of time but is not a thesis of logic. The aim of logic is not to decide whether the thesis of determinism is true or not. Logic aims to prepare tools that are ontologically neutral. Philosophical theses should depend on

Problems of Omniscience and Infallibility. A Temporal-Logical Approach


the domain being the subject matter of the theory but not be decided by logical tools. If we accept that ‘if φ, then it has been that it would be φ’ expresses the thesis of (pre)determinism, then the proposition should not be a thesis of logic. The new notions of tense operators fulfil the discussed condition. Their meanings are neutral with respect to the thesis of determinism (as expressed in the tense operators language). Let us repeat after Łukasiewicz:22 [...], I should like to state only one thing, namely that determinism is not a view better justified than indeterminism.

The time of an indeterministic world could be modelled as a branching structure. In the case of time as determined by courses of events, any possible course of events determines a branch of (possible) time. In a deterministic world, there is only one possible course of events and the time is linear. In an indeterministic world, there are various possible courses of events. In such a world, there are possible time forks. Real time is one of many branches that happen to become realized. In the case of time conceived as independent of the course of events, only one branch of time is ‘chosen’ by the sequence of (real) events. Nevertheless, let us remark that in fact we are not speaking about determinism but about inevitability. An event φ is determined if the time of its occurrence is definite. An event is inevitable if earlier or later the event occurs. Let us try to define temporal operators that would better render our intuition. It seems that it is reasonable to admit that on any branch time flows similarly. Hence, there is a map that for any two branches defines one-to-one correspondence between elements of these branches, preserving earlier-later relation between these elements. Such a map is called isomorphism. We admit that this map is one and the same for any two branches.


J. Łukasiewicz, Selected Works, op. cit., p. 127.

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Isomorphism divides set T for equivalence classes. Let [t] be a class of all and only elements of T that are isomorphic counterparts of t. The equivalence classes can be ordered. Let [⊳] be a binary relation in the class {[t]: t ∈ T} such that: • [t][⊳][t1 ] iff t ⊳ t1. Let us remark that relation [⊳] is a linear order in class {[t]: t ∈ T}. Now the tense operators are defined as bellow. Definition 4. G. ै, t ⊨ Gφ F. ै, t ⊨ Fφ ै, t2 ⊨ φ. H. ै, t ⊨ Hφ P. ै, t ⊨ Pφ ै, t2 ⊨ φ.

iff for any t1, t ⊳ t1 : ै, t1 ⊨ φ. iff there exists t1 such that t ⊳ t1 and for all t2, t2 ∈ [t1]: iff for any t1, t1 ⊳ t : ै, t1 ⊨ φ. iff there exists t1 such that t1 ⊳ t and for all t2, t2 ∈ [t1]:

The definitions could be formulated equivalently as follows. Definition 5. G. ै, t ⊨ Gφ iff for any t1, [t] [⊳] [t1] : ै, t1 ⊨ φ. F. ै, t ⊨ Fφ iff there exists [t1] such that [t] [⊳] [t1] and for all t2, t2 ∈ [t1]: ै, t2 ⊨ φ. H. ै, t ⊨ Hφ iff for any t1, [t1] [⊳] [t] : ै, t1 ⊨ φ. P. ै, t ⊨ Pφ iff there exists [t1] such that [t1] [⊳] [t] and for all t2, t2 ∈ [t1]: ै, t2 ⊨ φ. This definition is stronger than the former one: a proposition φ, if satisfied according to the new definition, is satisfied according to the former but generally not conversely.23 For example, now the proposiIt does not mean that the set of true propositions, i.e. propositions satisfied in all models, in the case of the first definition differs from the set of propositions satisfied in the case of the second definition. 23

Problems of Omniscience and Infallibility. A Temporal-Logical Approach


tion ‘it will be such that φ’ is satisfied only if there is a point such that on any branch φ is satisfied at its isomorphic counterpart. According to the former definition, ‘it will be such that φ’ is satisfied only if for any branch there is a point in which p is satisfied. Thus, the new definition better renders our conception of determinism. If we are stating that tomorrow there will be a sea battle, we are talking about tomorrow but not about any day in the future. Nevertheless, we have to remark that in the case of the new definition, we have supposed certain property, namely that all the branches are isomorphic. In both considered languages, the thesis of (pre)determinism is not true (is not satisfied in all models). It means that there is a model such that an instance of:

φ → hfφ is not satisfied and there is a model such that an instance of:

φ → HFφ is not satisfied. Let us remark that in the case of deterministic tense language the formula: Fφ ∨ F¬φ is true (is satisfied in all models). In the case of the new tense languages, there is a model such that an instance of the formula: fφ ∨ f¬φ is not satisfied and there is model such that an instance of the formula: Fφ ∨ F¬φ is not satisfied.

Kazimierz Trzęsicki


Unfortunately, the former proof of the thesis of determinism from the formula expressing omniscience and the formula expressing infallibility can be repeated in any of the languages. In any case, the only needed rule holds: h(φ ⇒ ψ) hφ ⇒ hψ and H (φ ⇒ ψ) Hφ ⇒ H ψ respectively. Thus, once again we ask if there is any possibility to show that the existence of a being that is omniscient and infallible is compatible with human free will. As we see from the thesis of omniscience, the thesis of infallibility can be drawn from the thesis of determinism,24 hence the thesis of determinism is independent of how the tense operators are conceived. This also entails that not even a temporal mind endowed with infinite memory, infinite computational skill, and complete knowledge of the present data, could predict the set of those facts that are not decided. Hence, the existence of an omniscient and infallible being whose nature is temporal, such as Laplace’s demon of supercomputer, contradicts human free will. To solve our problem, we have to revise the notion of B. It was assumed that omniscience is expressed by the formula:

φ ⇒ HBF φ. The formula expresses usual formulation of omniscience: if φ, then it has always been that B knew that it would be φ.


Determinism entails the refusal of free will (but not vice versa).

Problems of Omniscience and Infallibility. A Temporal-Logical Approach


This formulation of omniscience presupposes that the knowledge of B has a temporal aspect, i.e. that there are temporary definite states of knowledge of B.25 Why would we not then admit that knowledge of B could be atemporal? Usually B is conceived as an atemporal being. We have to answer a very old question. Is God outside of time or subject to time? Has time always existed, or is it created? Every description of God’s action contains a temporal reference. We have no other way of speaking, first of all because we experience God’s actions within a temporal frame of reference. God is an eternal being. Before the mountains were brought forth, or ever you had formed the earth and the world, from everlasting to everlasting you are God (Psalm, 90:2).

Time involves an extension of moments, but God is not subject to the passing of time. He simply is: Ego sum qui sum. For Boethius26 “[...], eternity is the possession of endless life whole and perfect at a single moment”. The view that God experiences time as an eternal ‘now’ is transmitted from the pre-Socratic philosophers through Plato, Aristotle and Plotinus, to Christian thinking by Boethius and continued by Thomas Aquinas. But how does the ‘eternal now’ have to be understood? In Book XI of Augustine’s Confessions27 we read: […] yet I say boldly that I know, that if nothing passed away, time past were not; and if nothing were coming, a time to come were not; and if nothing were, time present were not. It does not mean that the states are changing. The states may not differ from one another. 26 Boethius discusses the problem in reconciling genuine human freedom with God’s foreknowledge in Divine Foreknowledge and Freedom of the Will (Proses III–VI). (Cf. Boethius, The Consolation of Philosophy of Boethius, [email protected], http:// ebooks.adelaide.edu.au/b/boethius/consolation (2007), transl. into English Prose and Verse by H.R. James, Book IV, Prose 6). 27 Augustine of Hippo, The Confessions, eBooksAdelaide, http: //ebooks. adelaide.edu. au/a/augustine/a92c/ (2007), transl. by Edward Bouverie Pusey. 25

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If so, God’s present is not a temporal present. God is not within time, but the origin of time: In the beginning God created the heavens and the earth (Genesis, 1:1)

involved in the creation of time as well as space and matter. Augustine maintains:28 At no time then hadst Thou not made any thing, because time itself Thou madest. And no times are coeternal with Thee, because Thou abidest; but if they abode, they should not be times.

God is not in time and has no temporal properties, so God does not have any knowledge that is temporarily definite. God has placed us in this world of time. The whole of temporal reality, all events in the entire span of time are being before God’s mind in an atemporal present. Whenever God acts in creation, those acts take place in a specific moment of our time. But it is not true to say that these acts also take place in a specific moment of God’s time. God’s being and knowledge are infinite and eternal. With Him there are no ‘distinctions of time’. Consequently, God does not actually have foreknowledge, He just has knowledge of the atemporal ‘now’. In Boethius’s Consolations we read: Since, then, every mode of judgment comprehends its objects conformably to its own nature, and since God abides forever in an eternal present, His knowledge, also transcending all movement of time, dwells in the simplicity of its own changeless present, and, embracing the whole infinite sweep of the past and of the future, contemplates all that falls within its simple cognition as if it were now taking place. And therefore, if thou wilt carefully consider that immediate presentment whereby it discriminates all things, thou wilt more rightly deem 28

Ibidem, Book XI.

Problems of Omniscience and Infallibility. A Temporal-Logical Approach


it not foreknowledge as of something future, but knowledge of a moment that never passes.

God’s foreknowledge of future events is not some knowledge of events in the future, but knowledge of a never changing atemporal present. When we say about the knowledge of B, the atemporal ‘is’ should be used. Such an ‘is’ is used in e.g. ‘two plus two is four’. It does not have any sense to say ‘two plus two was four’ or ‘two plus two will be four’. It does not have any sense to say ‘the proposition two plus two is four has always been true’. Mathematical objects are atemporal. The mathematical knowledge acquired by humans is temporal. The knowledge as such is the subject of history of mathematical thought. Mathematical objects do not have any history. In the case of atemporarily conceived knowledge the formulas: • φ ⇒ Bφ, • Bφ ⇒ φ are the proper way to express omniscience and infallibility, respectively. Now we have to consider the type of language in that the knowledge could be expressed. Let us ask now about the consequences of the above defined omniscience and infallibility in the case of temporal facts. By tense facts let us call that temporal facts that refer to time by tense operators. In the case of tense facts proposition may change its truth value with changing of its tense. Let us consider the following argumentation. 1. B knows tense facts only if B acts within a temporal frame of reference. 2. If B does not act within a temporal frame of reference, then B does not know tense facts. 3. If B does not know tense facts, then B is not omniscient. 4. B is omniscient.

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W: B knows tense facts and acts within a temporal frame of reference. Ad 1. Let us say that tomorrow there will be a sea battle. In order to know that tomorrow there will be a sea battle, the knowledge about the reference of ‘tomorrow’ is indispensable. To know when tomorrow is, the experience of (transient) today is necessary. Ad 2. For an atemporal being, the present is unchangeable, there is no before and no after. For such a being, there is no tomorrow. What tomorrow is, is knowable only for a being that is in time. Ad 3. If for B present is eternal and unchangeable, then B is not able to know what tomorrow is. Ad 4. B knows, what tomorrow is, hence B knows the changeable and transient present. The world was not created in time. It was created together with time. The knowledge of B is atemporal but it does not mean that there is no time distinction of acts of B concerning the world. If there are cause-effects chains, there are time differences between causes and effects. To describe a cause and its effect, time references are necessary. Hence, B has knowledge that is described in temporal language. For this reason, to express omniscience and infallibility, we may use tense language. Such a language is acceptable. To express the omniscience, i.e. knowledge about a future fact φ, we use the formula: Fφ ⇒ BFφ. Our language is indeterministic. Therefore, for undetermined φ, neither Fφ nor F¬φ is true. Thus, from the fact that BFφ the thesis of determinism is not inferable (on the basis of the law of excluded middle). Neither it could be true that tomorrow there will be a sea battle nor it is true that tomorrow there will not be a sea battle. If the sea battle is not determined today, then neither is it true that tomorrow

Problems of Omniscience and Infallibility. A Temporal-Logical Approach


there will be a sea battle, nor is it true that tomorrow there will not be a sea battle. Now, we shall show that, in the case of indeterministic tense operators, the thesis of determinism does not follow from the thesis of omniscience and infallibility. To do that, we have to construct a model in which both the theses of omniscience and infallibility are satisfied and the thesis of determinism is not. First of all, let us remark that from the thesis of omniscience and the thesis of infallibility it follows that:

φ ⇒ Bφ , it means that the operator B is redundant. The thesis of post-determinism is usually accepted: everything that has been is unchangeable and eternal. For example, Augustine wrote:29 Sententia quippe qua dicimus aliquid fuisse, ideo vera est, quia illud de quo dicimus, iam non est. Hanc sententiam Deus falsam facere non potest, quia non est contrarius veritati. The proposition asserting anything to be past is true when the thing no longer exists. God cannot make such a proposition false, because He cannot contradict the truth.30 Augustinus of Hippo, Contra faustum manichaeum libri triginta tres, Citrà Nouva Editrice, (2010), edizione completa latino, Book 26.5. 30 An English Translation of the above (2012). In S.B. Bagnoregis, Commentaria in Quatuor Libros Sententiarum. Commentaries on the Four Books of Sentences Magistri Petri Lombardi, Episc. Parisiensis, Ad Claras Aquas, http://www.franciscan-archive. org/bonaventura/ (1882), p. 751 we read: “Item, Augustinus in vigesimo sexto libro contra Faustum: ‘Hanc sententiam, qua dicimus praeteritum fuisse, Deus falsam facere non potest’. Et rationem reddit: ‘Si enim hoc faceret, hoc esset facere, ut ea quae vera sunt, eo ipso quod vera sunt, essent falsa, et ita esset contrarius veritati’”. “Likewise, (St.) Augustine (says) in the twenty-sixth book Against Faustus: 7 ‘This sentence, by which we say that (something) past was, God cannot make false”. And he gives a reason: “For if He would do this, this doing would be, such that those which are true, by this very (fact) that they are true, would be false, and thus (this) would be con29


Kazimierz Trzęsicki

According to Thomas Aquinas (Qu. 25, art. 4): “Praeterita autem non fuisse, contradictionem implicat” (For the past not to have been implies a contradiction). Yes, the logical God is not able to change the past but what is not possible for logical God, can be possible for the God: God is Love for Whom nothing is impossible (1 John, 4:8; 1 Cor., 13:8; Luke, 1:37). Oscar Wilde31 believed that: Of course the sinner must repent. But why? Simply because otherwise he would be unable to realise what he had done. The moment of repentance is the moment of initiation. More than that: it is the means by which one alters one’s past. The Greeks thought that impossible. They often say in their Gnomic aphorisms, ‘Even the Gods cannot alter the past.’ Christ showed that the commonest sinner could do it, that it was the one thing he could do. Christ, had he been asked, would have said – I feel quite certain about it – that the moment the prodigal son fell on his knees and wept, he made his having wasted his substance with harlots, his swine – herding and hungering for the husks they ate, beautiful and holy moments in his life. It is difficult for most people to grasp the idea. I dare say one has to go to prison to understand it. If so, it may be worthwhile going to prison.

An unchangeable past is questioned by Łukasiewicz. He points out a sort of symmetry between the past and the future: we should not treat the past differently from the future.32 He rejects the Latin saying facta infecta fieri non possunt that is, what once has happened cannot be undone. He observes:33 trary to the truth’.” (http://www.franciscan-archive.org/bonaventura/opera/bon01750. html). 31 O. Wilde, De Profundis, http://upword.com/wilde/de_profundis.html (1998). HTML edition by R. van Valkenburg, [email protected], from e-text scanned and proofed by David Price. 32 J. Łukasiewicz, Selected Works, op. cit., p. 127. 33 J. Łukasiewicz, On Determinism, [in:] L. Borkowski (ed.), Selected Works, Polish Scientific Publishers/North-Holland Publishing Company, Amsterdam 1970, p. 127-128. The article in question is a revised version of the address that Łukasiewicz delivered as a Rector during the inauguration of the academic year 1922-1923 at the Warsaw

Problems of Omniscience and Infallibility. A Temporal-Logical Approach


If the only part of the future that is now real is causally determined by the present instant, and if causal chains commencing in the future belong to the realm of possibility, then only those parts of the past are at present real which still continue to act by their effect today. Facts whose effects have disappeared altogether, and which even an omniscient mind could not infer from those now occurring, belong to the realm of possibility. One cannot say about them that they took place, but only that they were possible. It is well that it should be so. There are hard moments of suffering and still harder ones of guilt in everyone’s life. We should be glad to be able to erase them not only from our memory but also from existence. We may believe that when all effects of those fateful moments are exhausted, even should that happen after our death, then their causes too will be effaced from the world of actuality and pass into the realm of possibility. Time calms our cares and brings us forgiveness.

We admit, roughly speaking, that the future exists only in the present data and the past exists only in the present data; whereas such data do not suffice to assign the truth value to φ, φ remains undecidable and belongs to the realm of possibility. Time 〈T, ⊳〉 is forwards linear (linear future) iff for any t1, t2, t3: if t1 ⊳ t2 and t1 ⊳ t3, then t3 ⊳ t2 or t2 = t3 or t2 ⊳ t3. Time 〈T, ⊳〉 is backwards linear (linear past) iff for any t1, t2, t3: if t2 ⊳ t1 and t3 ⊳ t1, then t3 ⊳ t2 or t2 = t3 or t2 ⊳ t3. Time that fulfils both the conditions (is forwards and backwards linear) will be called linear.34 There are four possible combinations of the theses pre- and postdeterminism: University. Later on, Łukasiewicz revised the address giving it the form of an article, without changing the essential claims and arguments. First published in Polish in 1946 and also in Łukasiewicz, op. cit., p. 114-126). It is worth noting that when the author delivered the talk at issue the theories and discoveries in the field of atomic physics that undermined determinism where still unknown. 34 This condition is fulfilled by parallel time, i.e. T is divided into classes of linearly ordered subsets.

Kazimierz Trzęsicki


1. POST-DET ∧ PRE-DET – both the theses hold: post- and predeterminism 2. –POST-DET ∧ PRE-DET – the thesis of post-determinism is rejected and the thesis of post-determinism holds 3. POST-DET ∧ ¬PRE-DET – the thesis of post-determinism holds but the thesis of pre-determinism is rejected 4. –POST-DET ∧ ¬PRE-DET – both the theses are rejected: postand pre-determinism. Ad 1. In this case, time is linear in both directions, in the past and in the future. The tense operators are semantically equivalent to the deterministic counterparts. Both the formulas:

φ ⇒ GPφ, φ ⇒ HFφ, are true (satisfied in any model). Ad 2. In this case, time is forwards linear but can branch in the past. Ad 3. In this case, time is backwards linear but can branch in the future. Ad 4. In this case, time can branch in the future as well as in the past. We only show that: 1. the thesis of post-determinism φ ⇒ GPφ is satisfied in any model, if time 〈T, ⊳〉 is backwards linear 2. there is a model in that an instant φ ⇒ HFφ of the thesis of predeterminism is not satisfied, if time 〈T, ⊳〉 is branching in the future. Theorem 1. Let L-LIN be class of times 〈T, ⊳〉 such that for any t, t1, t2: if t1, t2 ⊳ t, then t1 ⊳ t2 or t1 = t2 or t2 ⊳ t1 (backwards linearity). L-LIN ⊨ φ ⇒ GPφ.

Problems of Omniscience and Infallibility. A Temporal-Logical Approach


Proof. We have to prove that if time 〈T, ⊳〉 does not branch in the past, then for any ै, t : ै, t ⊨ φ ⇒ GPφ. Let time 〈T, ⊳〉 be backwards linear. Let ै, t ⊨ φ. From backwards linearity, we have that for any t1 such that t ⊳ t1 : ै, t1 ⊨ Pφ. Thus ै, t ⊨ GPφ. Theorem 2. Let R-LIN be class of times 〈T, ⊳〉 such that for any t, t1, t2: if t ⊳ t1, t2, then t1 ⊳ t2 or t1 = t2 or t2 ⊳ t1 (forwards linearity). R-LIN |= φ ⇒ HFφ. Proof. We have to show that for an instant p ⇒ HFp of the formula φ ⇒ HFφ, there is a model ै based on forwards linear time such that for some t: ै, t |= ⇒ HFp.

Let b, b1 be linearly ordered mutually disjoint non-empty sets such that: • there is t0 ∈ b such that for any t1 ∈ b1, t0 ⊳ t1 • for any t1 ∈ b if t0 < t1, then for any t2 ∈ b1 neither t1 ⊳ t2 nor t2 ⊳ t1 • there are t1, t2 ∈ b such that t1 ⊳ t0 and t0 ⊳ t2. Let T = b ∪ b1. T is branching in the future: at t0 is forking into two branches. Let for any t ∈ b : ै, t ⊨ p and for any t ∈ b1 : ै, t |= p. Let t1 ∈ b, t0 < t1. We have ै, t1 ⊨ p and ै, t0 |= Fp. Hence ै, t1 |= p ⇒ HFp. This entails that not even an omniscient and infallible being could at present predict the set of those facts that are not decided by present data. It has been shown that atemporal omniscience and infallible B knows temporarily facts but only those facts that are knowable at

Kazimierz Trzęsicki


the moment. If the truth value of φ is (now) not determined or – in other words – φ is contingent, then even for B φ is not determined. It would be a type of contradiction if B knows that φ is not determined and B knows the truth value of φ, what means that p is determined for B. It is true that neither φ nor not φ, then B – as omniscient – knows that neither φ nor not φ. To solve the problem of determinacy and free will, we need no third truth value. The semantic principle of bivalence is not rejected: in a model ै, it is true that φ or not-φ: ै ⊨ φ ∨ ¬φ.

For indefinite propositions, it could be that in a model ै neither φ nor not-φ is satisfied: neither ै ⊨ φ nor ै ⊨ ¬φ.

5. Summary Formal theories, i.e. theories with recursive language, recursive set of axioms, recursive set of rules of definition and rules of proof can be a pattern of methodological correctness. If any such theory does not satisfy some requirement, then such a requirement should not be absolutely obligatorily for other theories. Theology considering the divine realm undertakes questions and surpasses common sense that is shaped by experiences of material and human realm. For this reason, it can be that some solutions are accepted as correct but should not be accepted as such or some solutions are rejected as incorrect but should not be rejected as such. Logical fallacy is common even in such disciplines as formal theories and mathematics. The application of logic as a tool is crucial for the possibility of grasping the meaning of the theological language. Where there is direct insight into reality, notions are corrected by reality. Where, as

Problems of Omniscience and Infallibility. A Temporal-Logical Approach


it is in the case in theology and in abstract branches of mathematics, there is no such possibility, reasoning and argumentation are governed only by notions. Our intuition alone fails to give us an answer. In such cases, logic in fullness reveals its role as a tool of correctness of reasoning and argumentation. In the case of refined questions such as omniscience, infallibility and human free will and determinism, due to the tools created by temporal logic, some meaning distinctions of different formulation of the problem have been grasped and their logical consequences have been drawn. The age-old question concerning the compatibility of human free will and the existence of an omniscient and infallible being has been the subject of critical examination. It has been shown that the answer ‘yes’ is logically possible. Therefore, it has been proven that human beings enjoy free will even if there is an omniscient and infallible being.

Bartosz Brożek Jagiellonian University Copernicus Center for Interdisciplinary Studies

Adam Olszewski The Pontifical University of John Paul II Copernicus Center for Interdisciplinary Studies

Miracles: A Logical Perspective*


here is an extensive bibliography devoted to the problem of miracles.1 It is not our goal to analyze those works, particularly as they offer diverse definitions of miracles depending on the accepted conceptual scheme, such as Aristotelian ontology. Our goal is less ambitious: by adopting two explications of the notion of a miracle, we shall try to answer the question of how the knowledge concerning miraculous events may be embedded into one’s belief system. In particular, we shall be interested in uncovering the logical mechanisms at work in this process.

1. Introduction Simplifying considerably, one can argue that miracles are events which are contrary to the common course of nature (communum cursus naturae). Medieval philosophers posited that one should distinThis contribution was made possible through the support of a grant “The Limits of Scientific Explanation” from the John Templeton Foundation. 1 Cf. e.g. J.T. Driscoll, Miracle, [in:] The Catholic Encyclopedia, vol. 10, Robert Appleton Company, New York 1911, http://www.newadvent.org/cathen/10338a.htm (30 March 2010). *


Bartosz Brożek, Adam Olszewski

guish between two orders: so-called ordo naturalis or natural order, which is fully penetrable by human reason, and ordo supranaturalis vel miraculis, or the supernatural order, which can be known only through revelation. From this perspective, miracles – belonging to the supernatural order – are phenomena which cannot be accounted for with the use of natural reason.2 This understanding of miracles leads us to an important controversy. One should ask whether miracles are contrary to the principles governing the universe, or whether they strike us as contrary to the physical laws, but it is just a result of our ignorance. The answer to this question hangs together with how one understands the status of ordo naturalis and ordo supranaturalis: are they ontological concepts or only epistemological ones. According to the former interpretation, ordo naturalis is a set of laws governing the universe, and as a result, a miracle constitutes a violation of those laws. On the latter reading, the distinction between the two ordines coincides with the distinction between what can potentially be known through natural reason (ordo naturalis) and what reason cannot grasp (ordo supranaturalis). Thus, a miracle would be an event which complies with the laws governing the universe, but such laws which human reason cannot comprehend naturally. The declarations of the Doctors of the Church pertaining to this problem are ambiguous. Augustine claims that miracles are not contrary to nature, but only contrary to what we know about nature. He says: So great an author as Varro would certainly not have called this a portent had it not seemed to be contrary to nature. For we say that all portents are contrary to nature; but they are not so. For how is that contrary to nature which happens by the will of God, since the will of so mighty a Creator is certainly the nature of each created thing?

Cf. B. Brożek, The Double Truth Controversy, Copernicus Center Press, Kraków, 2010, Chapter 1.4. 2

Miracles: A Logical Perspective


A portent, therefore, happens not contrary to nature, but contrary to what we know as nature.3

On the other hand, Aquinas observes in Summa Theologica that miracles lie outside of the natural order or “are done outside the order of nature”.4 The declaration of Augustine may be interpreted as a confirmation of the epistemological theory of miracles: a miracle is an event which is contrary to our knowledge of the universe (or, more precisely, contrary to what we can potentially know about it). The stance of Aquinas seems to support the ontological interpretation: a miracle is an occurrence contrary to the laws governing the universe. It is not our goal to answer the question of which of these interpretations is correct. We assume that both options are interesting from a logical point of view. Before we proceed with further analysis, however, one more distinction is needed. It seems that one can speak of two kinds of miracles. The first category involves the unrepeatable miracles, that is such that constitute unique events, e.g. when Lazarus was raised from the grave. The second category comprises the repeatable miracles, an example being the Eucharistic transubstantiation. Such repeatable miracles can be captured by a general ‘law’ (e.g. each time wine and bread are consecrated during a holy Mass, they become the blood and body of Christ). The question we venture to answer is: what is the formal structure of theological knowledge if it allows the existence of miracles? In order to carry out our analysis, let us formulate the following two examples: Example 1: a repeatable miracle. According to Aristotle, there are two kinds of entities: substances, which exist simpliciter (per se), and accidents, which exist secundum quid or in substances. It is not possible for an accident to exist per se. Thus, it is impossible to explain

3 4

Augustine, The City of God, Book XXI, Chapter 8. Thomas Aquinas, Summa Theologica, I:110:4.


Bartosz Brożek, Adam Olszewski

what happens during transubstantiation. The teaching of the Catholic Church is that after transubstantiation the accidents of bread and wine exist – miraculously – without an underlying substance. Let us formalize this example in the first order logic.5 Let ACC stand for ‘is an accident’, DEP – ‘is dependent in its existence’, SUB – ‘is a substance’, IND – ‘is independent in its existence’, and HOS – ‘is an accident of bread after transubstantiation.’ Thus, we get: (1) ∀x (ACC(x) → DEP(x)) premise (natural reason) (2) ∀x (SUB(x) → IND(x)) premise (natural reason) (3) ∀x (IND(x) ≡ ¬DEP(x)) definition (4) ∀x (HOS(x) → IND(x)) premise (faith) (5) ∀x (HOS(x) → ACC(x)) premise (natural reason) Example 2: an unrepeatable miracle. It is impossible to raise a dead man from his grave. However, Lazarus was raised from his grave in a miraculous way. Let us assume that MOR stands for ‘is dead’, and RES – ‘to be resurrected’, while l is an individual constant (a proper name) for Lazarus. We get: (i) ∀x (MOR(x) → ¬RES(x)) premise (natural reason) (ii) MOR(l) (iii) RES(l)

2. Epistemological understanding of miracles Let us assume, first of all, that miracles should be understood epistemologically and look from this perspective at the repeatable We formalize this example in the first order logic, which may seem incorrect since we use individual variables to refer to both accidents and substances. It would be more natural to use the second order logic. At the price of gross simplification, we have decided to utilize the first order logic, which is sufficient for our goals.


Miracles: A Logical Perspective


miracles as depicted in Example 1. Let h stand for the accident of some particular bread after transubstantiation: (6) HOS(h)


Based on the premises (1)-(6), we may conclude that: (7) IND(h) (8) ACC(h) (9) DEP(h) (10) ¬DEP(h)

from (4) and (6) from (5) and (6) from (1) and (8) from (4) and (7)

Thus, we obtain a contradiction. A way to avoid it is to take into account a piece of information which comes from revelation, namely that the accidents of bread after transubstantiation exist independently in a miraculous way. This knowledge must be incorporated into the general law based on natural reason, according to which accidents have no independent existence. Thus, we must modify the premise (1) in the following way: (1)∗ ∀x ((ACC(x) ∧ ¬HOS(x))→ DEP(x)) Under those new circumstances, it is impossible to derive (9) as it is not true that ¬HOSh. Let us note that such a reformulation of the premise (1) complies with the epistemological understanding of miracles. A miracle – in our case, the independent existence of the accidents of bread after transubstantiation – is not something contrary to the laws of nature, but rather incompatible with what we know about those laws. To put it differently: the revealed truth pertaining to the independent existence of the accidents of bread after transubstantiation constitutes a piece of information about the laws governing the world, and so it must be incorporated into the general law expressed in the premise (1).

Bartosz Brożek, Adam Olszewski


It may be added that in a similar way – through the incorporation of exceptions – the knowledge based on the natural reason alone is expanded. Let us consider a general law expressed in the following way: (P) ∀x (R(x) → S(x)) Let us assume further that we have observed an exception to this rule such that for any x, which is R but also T, x is not S. In such a case, one needs to revise the law (P): (P*) ∀x ((R(x) ∧ ¬Τ(x)) → S(x)) This type of revision is an example of the concretization of a law of physics, i.e. dispensing with some idealizing assumption of that law. The situation is different in the case of unrepeatable miracles. In our example: (i) ∀x (MOR(x) → ¬RES(x)) (ii) MOR(l) (iii) RES(l)

premise (natural reason)

The contradiction is easy to spot since the premises (i) and (ii), through the elimination of the universal quantifier and modus ponens, yield: (iv) ¬RES(l) An attempt to take advantage of the strategy of incorporating exceptions into the general law results, in this case, in the following modification of the premise (i): (i)* ∀x ((MOR(x) ∧ ¬x=l) → ¬RES(x)) This modification is unfortunate as it requires for each and every occurrence of an unrepeatable miracle which contradicts the law ex-

Miracles: A Logical Perspective


pressed in (i) the introduction of a clause ‘∧¬x=n’, where n is an individual constant, i.e. a proper name of an object which does not submit to the law (i). This manoeuvre leads to fatal consequences: our knowledge is no longer universal, but becomes ‘casuistic’. Such laws as the one expressed in (i)* are not algorithmically compressible.6 Moreover, the exceptions in the form ‘∧¬x=n’ are purely redundant in any system of knowledge: they are always formulated ex post factum and serve no role in the prediction of future events. The situation changes little when one – instead of modifying (i) by introducing clauses in the form of ‘∧¬x=N’ – opts for the following solution: (i)** ∀x ((MOR(x) ∧ ¬MIR(x)) → ¬RES(x)) where MIR stands for ‘is subject to a miraculous event’. In order to make inferences on the basis of (i)** one would need to confirm first that the given object x is not subject to a miraculous event, while this may be stated only ex post. The above-presented analysis shows that – on the epistemological interpretation of miracles – only repeatable miracles may be reasonably incorporated into one’s web of beliefs. Unrepeatable miracles, on the other hand, are anomalies, which – introduced into our knowledge – constitute redundant information. It must be noted, however, that since the mechanism of incorporating ‘repeatable miracles’ into our knowledge is – from the logical point of view – the same as the mechanism of concretizing the laws of physics, the distinction between ‘natural’ and ‘supernatural’ exceptions is always based on an extra-logical criterion. Indeed, as our example clearly shows, repeatable miracles are always non-empirical (they cannot be observed). If they were observable, they would be indistinguishable from other – natural – phenomena, which are in conflict with some general law.

Cf. M. Heller, Czy świat jest matematyczny?, [in:] Filozofia i Wszechświat, Universitas, Kraków 2006, p. 51.


Bartosz Brożek, Adam Olszewski


3. Ontological understanding of miracles In the case of the ontological interpretation of miracles, the solution outlined above, i.e. that exceptions constituted by miraculous events should be incorporated into the formulation of general law, is unacceptable, since our goal is for our knowledge to mimic the structure of the universe (it should describe the laws governing the reality). Thus, in our Example 1, the premise (1) should be preserved in the form: (1) ∀x (ACC(x) → DEP(x)) The problem here is that – together with the premises (2)-(6) – the premise (1) yields a contradiction. The only way out of this trouble is to abandon classical logic and admit that the existence of ontologically understood miracles forces one to express the laws depicting the structure of the universe in the so-called defeasible logic.7 Such defeasible logic (let us call it DL) operates at two levels. On the first level, from the given set of premises one constructs arguments; on the second level, the arguments are compared in order to decide which of them prevails. The conclusion of the prevailing argument is also the conclusion of the entire set of premises. The language of DL is the language of the first order logic extended by the addition of a new connective, the so-called defeasible implication ⇒. For the defeasible implication, there is a special inference rule of defeasible modus ponens: A⇒B A ––––– B

Cf. H. Prakken, Logical Tools for Modelling Legal Argument, Kluwer, Dordrecht 1997; B. Brożek, Defeasibility of Legal Reasoning, Zakamycze, Kraków 2004.


Miracles: A Logical Perspective


The difference between the material and defeasible implications is visible only at the second level of DL. The language of DL serves to construct arguments. Let us have a look at the following example: (a) A ⇒ B (b) C ⇒ ¬B (c) A This set of premises may serve us to construct the following argument: (ARG1) (a) A ⇒ B (c) A ––––––––– (d) B Let us extend our set of premises by adding the following sentence: (e) C Now, we can construct another argument: (ARG2) (b) C ⇒ ¬B (e) C –––––––––– (f) ¬B Given two such arguments, we can move onto the second level of DL, where we decide which of the sentences – B or ¬B – should be the conclusion of our extended set of premises. On the second level of DL, two concepts play a key role: attack and defeat. Simplifying considerably, one can say that an argument

Bartosz Brożek, Adam Olszewski


ARGA attacks an argument ARGB, when the conclusions of both arguments are contradictory. This is the case in our example, where B and ¬B are contradictory, and thus (ARG1) attacks (ARG2). When two arguments attack each other, it must be decided which of them prevails, or defeats, the other. There are many possible ways of comparing arguments. The easiest and most flexible is to assume that there is an externally (extra-logically) given ordering of defeasible implications. One checks what the defeasible implications that served to construct the two competing arguments are and declares that the one which prevails is that whose defeasible implication is higher in the ordering. In our example, the first argument is constructed with the use of the implication A⇒B, and the second: C⇒¬B. Let us assume that the second implication is higher in the ordering. If so, it is the second argument that prevails in the comparison with the first one, and it is its conclusion, ¬B, which becomes the conclusion of our extended set of premises.8 Let us assume further that the arguments which are constructed based on no defeasible implications always prevail in comparison with arguments constructed with the use of some defeasible implications. Let us now apply this simple idea to our Example 1, replacing all material implications which appear in the premises expressing the laws of natural reason with their defeasible counterparts: (1) ACC(x) => DEP(x) (2) SUB(x) => IND(x) (3) ∀x (IND(x) ≡ ¬DEP(x)) (4) ∀x (HOS(x) → IND(x)) (5) HOS(x) => ACC(x) (6) HOSh Those premises may serve to construct the following two arguments: 8

Thus, DL is a non-monotonic logic.

Miracles: A Logical Perspective


(ARG3) (6) HOS(h) (4) ∀x (HOS(x) → IND(x)) (7) IND(h) from (4) and (6) (3) ∀x (IND(x) ≡ ¬DEP(x)) (10) ¬DEP(h) from (3) and (7) (ARG4) (6) HOS(h) (5) HOS(x) => ACC(x) (8) ACC(h) from (5) and (6) (1) ACC(x) => DEP(x) (9) DEP(h) from (1) and (8) The arguments attack each other, since their conclusions are contradictory. In this particular case, it is (ARG3) that prevails as it takes advantage of no defeasible implication, when (ARG4) uses two such implications. The logical conclusion of our entire set of premises is then: (10) ¬DEP(h) Example 2 may be formalized in a similar way: (i) ∀x (MOR(x) => ¬RES(x)) (ii) MOR(l) (iii) RES(l)

premise (natural reason)

On the basis of these premises, one can construct two arguments: (ARG5) (iii) RES(l)

Bartosz Brożek, Adam Olszewski


(ARG6) (i) MOR(x) => ¬RES(x) (ii) MOR(l) (iv) ¬RES(l) Again, we have a conflict of arguments. In this case, it is argument (ARG5) that prevails since it takes advantage of no defeasible implication. Thus, the logical conclusion of the entire set of premises is: (iii) RES(l) This formalization is satisfactory for two reasons. Firstly, it enables one to preserve the form of the laws of nature: one does not need to introduce any exceptions into the formulation of a law of nature, which is in compliance with the ontological understanding of miracles. Secondly, within the defeasible logic one can identify a logical (formal) difference between our knowledge of miracles and our scientific knowledge. In the former case, a miraculous event ‘blocks’ the application of a general law of natural reason; in the latter – when a phenomenon is observed which is incompatible with some general law – one can revise the law by introducing an exception or through some more complex modifications. An important drawback of the present solution is that the defeasible logic described above does not have a traditionally understood semantics. Instead, it is based on the so-called argument-based semantics, in which the concepts of attack and defeat play the crucial role. However, there is no (logical) interpretation in the traditional sense here, and defeasible implications are ascribed no truth values. Such an approach is uncontroversial in the context of the medieval debates pertaining to miracles (or to natural and supernatural orders). Medieval philosophers deemed the laws discovered by the natural reason probabiles. This concept of probabilitas should not be understood, however, in the contemporary sense of ‘probability’,

Miracles: A Logical Perspective


but rather as expressing that something ‘can be proved’, ‘is justified’ or ‘is with arguments in its favour’.9 With such an understanding of the epistemic status of the laws discovered by the natural reason, the defeasible logic equipped with the argument-based semantics is a perfect tool for rendering the medieval intuitions. From today’s perspective, however, a different solution is needed. One option is to take advantage of the intentional semantics for defeasible logic.10 An inspiration for this type of formal structure is the semantics for counterfactual conditionals. Such a counterfactual conditional as P → Q is true in an actual world if and only if in a subset of possible worlds Z, which are ‘the most similar to the actual world’, Q is true in all those worlds belonging to Z in which P is true. Defeasible implications may be treated similarly. A defeasible implication, say P => S, is true if in a subset of possible worlds N, which are ‘the most normal worlds’, S is true in each world belonging to N in which P is true. Thus, one of our defeasible implications, (i) MOR(x) => ¬RES(x), is true if for every x, ¬RES(x) is true in every most normal world in which MOR(x) is true. Let us observe that the actual world does not have to be one of 'the most normal worlds'. In order to make inferences on the basis of a defeasible implication such as (i) MOR(x) => ¬RES(x), we assume that it is a most normal world. However, if a miraculous event occurs in the actual world, our assumption is undermined and we are no longer entitled to reason with the use of (i). In such a semantics, considerable controversy is connected to the way of determining the set N (of the most normal worlds). This, however, is an extra-logical problem. To put it simply: N is a set of those worlds in which the laws of the natural order are always fulfilled. The most normal worlds are the worlds in which there are no miracles.

Cf. B. Brożek, The Double Truth Controversy, op. cit., Chapter I. Cf. H. Prakken, G. Vreeswijk, Logical Systems for Defeasible Argumentation, [in:] D. Gabbay (ed.), Handbook of Philosophical Logic, 2nd edition, vol. IV, Kluwer, Dordrecht 2002, p. 219-318. 9


Bartosz Brożek, Adam Olszewski


It turns out that the ontologically understood miracles are easier to model logically than miracles understood epistemologically. The use of defeasible logic makes it possible to account for both repeatable and unrepeatable miracles. This is achieved at the price of abandoning the classical logic, which is an interesting consequence of accepting the ontological interpretation of miraculous events.

4. Summary The problem which we have analysed above constitutes an aspect of a broader controversy, namely the question pertaining to the possibility of God’s intervention in the course of events. Contemporary theology – and especially its strands inspired by the development of the natural sciences – makes a number of attempts designed to show that God’s intervention in the world is possible without violating the laws of physics. Some see a place for this in the probabilistic character of the laws of quantum mechanics. Others regard such proposals as a subtler version of the ‘God of the gaps’ argument and opt for an alternative solution. Józef Życiński observes: Instead of God hidden in Heisenberg’s uncertainty, or expressed in the so-called physical chaos, we propose a model, in which the role of God immanent in cosmic history is contained in laws of nature as well as in what we metaphorically call the ‘boundary conditions.’ The expression denotes theologically conceived boundary conditions in which non-physical (i.e. biological, psychic, spiritual) factors are also taken into consideration in a system considered ‘from God’s point of view’ (again metaphor).11

It must be stressed that both formal explications of miracles we have presented are incompatible with the understanding of God’s immaJ. Życiński, The Laws of Nature and the Immanence of God in the Evolving Universe, “Studies in Science and Theology” 1997, vol. 5, p. 15. 11

Miracles: A Logical Perspective


nence in the world as outlined by Życiński. On both interpretations of miracles – epistemological and ontological – miracles remain phenomena which cannot be explained by science. This is incompatible with methodological reductionism, which constitutes the backbone of modern science: there are no such observable facts which a priori may be deemed inexplicable by science. Such aprioricity is directly related to miracles understood ontologically, where the natural order is ontologically different from the supernatural order. There is no such difference within the epistemological interpretation of miracles. Thus, one may postulate a reformulation of the epistemological concept such that it would comply with the reductionist strategy of science. However, this would lead to the rejection of such revisions of the laws of nature which introduce redundant information, as e.g. replacing (i)* ∀x(MOR(x) → ¬RES(x)) with (i)* ∀x((MOR(x) ∧ ¬x=l) → ¬RES(x)). On this new account, miracles would be ‘naturalized’, and the goal of science would remain to formulate general laws which explain – through ‘natural causes’ – why Lazarus was raised from the grave or why in Cana the water was turned into wine. The question is, however, whether we are still speaking of miracles in this case.

Damian Wąsek The Pontifical University of John Paul II

The Role of Dialectics in Peter Abelard’s Concept of Theology* 1. Introduction


hat does theology need logic for? Does a discipline based on clearly defined rules of human reasoning have anything in common with a science based on the authority of God, which is far beyond our understanding and which calls for the acceptance of mysteries that cannot be fully understood? Is there any point in mixing two seemingly opposing disciplines? Such and similar questions were asked by philosophising theologians and theologising logicians in the course of the history of science. Answers were given mostly in the medieval period, as it was the period abundant in philosophers competent enough in logic and theology to provide reliable conclusions. For this paper, I have decided to explore the thought of Peter Abelard,1 as it seems, on the one hand, an underestimated and unex* This contribution was made possible through the support of a grant “The Limits of Scientific Explanation” from the John Templeton Foundation. 1 P. Abelard (1079-1142) – one of the most prominent philosophers and theologians of the Middle Ages. He was born in Le Pallet near Nantes. He studied philosophy under Roscelin of Compiègne, Guillaume de Champeaux and other scholars of the day. He taught philosophy and theology mostly in Paris, Melun, Corbeil and Mont Ste. Geneviève. His teaching was condemned by the Council of Soissons (1121) and the Council of Sens (1140). He is best known for his unhappy love affair with Heloise. Reconciled with the Church, he died in Chalon at the age of 63. He described his experiences in an autobiography entitled: Epistola 1: Historia calamitatum sive ad amicum suum consolatoria, J. Monfrin (ed.), Vrin, Paris 1978, p. 62-109. Other biographies of Abe-


Damian Wąsek

plored page in the history of the abovementioned disciplines2 and, on the other hand, it brings a new perspective on the mutual relations between them and benefits that may result from their combination. The aim of this study is to present, on the basis of Abelard’s works, the multidimensionality of the term ‘dialectics’, and in consequence, the diversity of its possible applications in the creation of the theological system, as well as dangers resulting from an overemphasis on the role of dialectics in the field of faith. The problem can be best expressed as a question: To what extent and in what dimensions – according to Abelard – can dialectics help in pursuing theology? In order to find an answer to the above question, I will outline the theological-philosophical context of Abelard’s thought, define the notion of dialectics itself and its theological implications in his concept and point to certain limitations of dialectics as a tool to explore the truths of faith. The research presented in this paper is based on a critical edition of Peter Abelard’s works, though some of his writings have not undergone critical analysis yet. The theological treatises most vital for my research were edited and published by E.M. Buytaert and C.J. Mews in three volumes, as a part of Corpus Christianorum Continuatio Medievalis. Among these works, the most important was Theologia, written around year 1220, which is nowadays known un-

lard worth mentioning include: R. Pernoud, Héloïse et Abélard, A. Michel, Paris 1970; G. Ballanti, Pietro Abelardo. La rinascita scolastica del XII secolo, La Nuova Italia, Firenze 1995; M.T. Clanchy, Abelard: A Medieval Life, Blackwell Publishers, Oxford 2000; É. Gilson, Héloïse et Abélard, Vrin, Paris 2000; S.P. Bonanni, Abelardo, San Paolo, Milano 2003; A. Pamparana, Abelardo. Ragione e passione, Ancora, Milano 2007. 2 The issues presented in this paper are further analyzed and elaborated on in my dissertation Koncepcja teologii Piotra Abelarda, UNUM, Kraków 2010. Other studies making reference to this topic include D.F. Blackwell’s Non-Ontological Constructs. The effects of Abaelard’s Logical and Ethical Theories on His Theology: A study on Meaning and Verification, Peter Lang, Bern–Frankfurt am Main–New York–Paris 1988. However, the author mentions only briefly the issues that are of interest to us, as he focuses mainly on the logical methods used by Abelard and on the argument over universals. More information on the role of dialectics in theology can be found in Antonio Crocco’s Abelardo. L’ “altro versante” del Medioevo, Liguori, Napoli 1979.

The Role of Dialectics in Peter Abelard’s Concept of Theology


der the title Theologia Summi Boni.3 It this text, Abelard made an attempt at explaining the mystery of the Holy Trinity with reference to the thought of Christian authors, the evidence provided by pagan philosophers and the arguments of reason. Abelard extended and modified the text many times and two of its later versions, that survived up to now, are entitled Theologia Christiana4 and Theologia Scholarium5. Dialectica6 is one of the most significant logical works of Abelard, in which he presents his views on the main philosophical issues, comments on the existing solutions to the problem of universals and puts forward the idea of general concepts. All quotations from Abelard’s writings used in this paper come from readily accessible English translations or, in the case of untranslated sources, from the author. Despite some necessary changes with respect to the original texts, all effort was made to render the thought of Abelard correctly. As a method I will employ the analysis of Abelard’s writings that include comments on the application of dialectics in the field of theology. However, disputes over the possible application of secular disciplines in the field of doctrina sacra in the 11th and 12th century will have to be presented from a synthetic perspective.

2. Theological-philosophical background In the 12th century, the nature of theology and its status in the catalogue of sciences were widely disputed. The vital question asked at that time was as follows: ‘To what extent can secular sciences be used in the exploration and presentation of the truths of faith?’ AnP. Abelard, Theologia Summi Boni, [in:] E.M. Buytaert, C.J. Mews (eds), Petri Abaelardi opera theologica, III, CCCM 13, Brepols, Turnholt 1987, p. 39-201. 4 P. Abelard, Theologia Christiana, [in:] E.M. Buytaert, C.J. Mews (eds), Petri Abaelardi opera theologica, II, CCCM 12, Brepols, Turnholt 1969, p. 1-371. 5 P. Abelard, Theologia Scholarium, [in:] E.M. Buytaert, C.J. Mews (eds), Petri Abaelardi opera theologica, III, CCCM 13, Brepols, Turnholt 1987, p. 203-549. 6 P. Abelard, Dialectica, L.M. De Rijk (ed.), Van Gorcum, Assen 1970. 3


Damian Wąsek

swers were formulated along two opposing lines. A group of thinkers described as antidialecticians provided many justifications for a ban on any attempts at combining the human and divine order, and pointed to the key role of the conclusions suggested by the Church Fathers. The other group, called dialecticians, created a plane on which an encounter of sacrum and profanum could be possible. Abelard, who belonged to the dialecticians, had to deal with the arguments presented by the 11th century thinkers and embraced also by his contemporaries. From among the most representative antidialecticians of the 11th century, I have chosen Peter Damian (1007-1072), prior of the isolated hermitage of Fonte Avellana.7 According to him, the main concern of a monk should be the pursuit of sanctity and not that of knowledge. A monk should strive after a mystical union with Christ and not try to explain His teaching. Peter Damian warned his subordinates against the pursuit of rational explanations for the truths of faith, drawing a parallel between practicing dialectics and committing the sin of impurity. Only holy simplicity (sancta simplicitas), complete trust in all that has been given to us in the Sacred Scripture, the writings of the Church Fathers and liturgical traditions allows us to call ourselves the members of the Church.8 Damian claimed that logic, and other secular disciplines of that kind, can be of any importance only in the context of research on the nature of the material world in which conclusions can be verified. According to him, using human reasoning in the field of theology will reduce God, who exceeds the material reality, to a limited being. These claims, however, should not be treated as an absolute condemnation of all achievements obtained by secular sciences. The biography of Peter Damian and an overall description of his views are presented by Z. Kadłubek in Rajska radość. Św. Piotr Damiani, Gnome, Katowice 2005 and Św. Piotr Damiani, Wydawnictwo WAM, Kraków 2006. He is also described by M. Kowalewska in Powszechna Encyklopedia Filozofii, vol. II, Polskie Towarzystwo Tomasza z Akwinu, Lublin 2001, p. 395-397. 8 See G. d’Onofrio Historia teologii. Część II: Epoka średniowieczna, Wydawnictwo M, Kraków 2005, p. 150-151. 7

The Role of Dialectics in Peter Abelard’s Concept of Theology


Some basic logical tools should be used in order to better clarify some truths of faith but never to verify their veracity.9 In this context, Peter Damian wrote: However, in the techniques of humanities10 be used in the study of revelation, they must not arrogantly usurp the rights of the mistress, but should humbly assume a certain ancillary role, as a maidservant to her lady, so as not to be led astray in assuming the lead, nor to lose the enlightenment of deepest virtue, nor to abandon the right road to truth by attending only to the superficial meaning of words.11

According to Peter Damian, all knowledge should serve as a path to a stronger communion with God and to moral perfection, as only such high aims justify its usefulness. Among those presenting an opposite view, namely the dialecticians of the 11th century, the most prominent figure was Anselm of Canterbury (1033/1034-1109). He based his concept of theology on Augustine’s definition of Truth, which called for a certain correspondence (rectitudo) between the thing that is being studied (res), the inner thought that reflects this thing (intellectus) and the word that expresses this thought in the spoken or written language (vox).12 Liberal arts, and dialectics in particular, were supposed to ensure the cohesion and correctness of that intellectual process. According to See T.J. Holopainen, Dialectic and Theology in the Eleventh Century, E.J. Brill, Leiden 1996, p. 6-44. 10 The term ars humana used by Damian refers most probably to dialectics. 11 “Quae tamen artis humanae peritia, siquando tractandis saris eloquiis adhibetur, non debet ius magisterii sibimet arroganter arripere, sed velut ancilla dominae quodam famulatus obsequio subservire, ne, si praecedit, oberret et, dum exteriorum verborum sequitur consequentias, intimae virtutis lumen et rectum veritatis tramitem perdat”, P. Damian, De divina omnipotentia, [in:] Peter Damian, Lettre sur la toute-puissance divine, “Sources Chretiennes” 1972, vol. 191, p. 416 (On Divine Omnipotence [in:] The Fathers of the Church: Medieval Continuation. The Letters of Peter Damian 91120, transl. O.J. Blum, The Catholic University of America Press, Washington 1998, p. 356). 12 See C.É. Viola, Anselmo d’Aosta. Fede e ricerca dell’ intelligenza, Jaca Book, Milano 2000, p. 86-96. 9


Damian Wąsek

Anselm, information concerning what should be accepted on faith and only then elucidated by reason was included in the revelation and patriarchal texts and was to serve as the criteria for veracity and the source of knowledge about anything that we want to describe by means of reasoning. Anselm’s method, called sola ratio, does not aim, therefore, at expounding the faith on the basis of ‘pure’ intellect, but advocates the use of reason for a better understanding of previously accepted truths. The prologue to Monologion seems to be the key text for understanding this method: Some of the brethren have often eagerly entreated me to write down some of the things I have told them in our frequent discussions about how one ought to meditate on the divine essence, and about certain other things related to such meditation, as a sort of pattern for meditating on these things. Having more regard to their own wishes than to the ease of the task or my ability to perform it, they prescribed the following form for me in writing this meditation: absolutely nothing in it would be established by the authority of Scripture; rather whatever the conclusion of each individual investigation might assert, the necessity of reason would concisely prove, and the clarity of truth would manifestly show, that it is the case, by means of a plain style, unsophisticated arguments, and straightforward disputation.13

“Quidam fratres saepe me studioseque precati sunt, ut quaedam, quae illis de meditanda divinitatis essentia et quibusdam aliis huiusmodi meditationi cohaerentibus usitato sermone colloquendo protuleram, sub quodam eis meditationis exemplo describerem. Cuius scilicet scribendas meditationis magis secundum suam voluntatem quam secundum rei facilitatem aut meam possibilitatem hanc mihi formam praestituerunt: quatenus auctoritate scripturse penitus nihil in ea persuaderetur, sed quidquid per singulas investigationes finis assereret, id ita esse piano stilo et vulgaribus argumentis simplicique disputatione et rationis necessitas breviter cogeret et veritatis claritas patenter ostenderet”; Anselm of Canterbury, Monologion, Prologus, [in:] Anselm of Canterbury, Opera omnia, F.S. Schmitt (ed.), Friedrich Frommann Verlag (Günther Holzboog), Stuttgart–Bad Cannstat 1968, english edition: Anselm of Canterbury Monologion and Proslogion with the Replies of Gaunilo and Anselm, transl. Thomas Williams, Hackett Publishing Company, Indianapolis 1996, p. 3. 13

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Thus, a common ground for discussion between dialecticians and antidialecticians, proposed by the archbishop of Canterbury, seems free from objections raised by the former against deprecating the power of reasoning and the fears of the latter that the revelation would be reduced to human knowledge. It served, therefore, as an incentive for the 12th century thinkers and an argument against their opponents: as Anselm managed to create a system based on dialectics and acceptable within the boundaries of orthodoxy, attempts at using dialectics in the field of faith can be justified. The above argument failed to convince the most prominent antidialectician of Abelard’s time, Bernard of Clairvaux (1090/10911153).14 In Sermones super Cantica canticorum,15 he characterised faith as disinterested and pure, striving after God whom we cannot comprehend and with whom we cannot be united without love. Therefore, the main task for a man is to love God, and all human knowledge and aspirations of reason should be reduced to piety, which is a bridge to truth. All that we are trying to understand with the help of liberal arts is of no consequence in the context of salvation which is the ultimate goal of man. Hence, there is no point in discussing issues 14 In 1112 he entered the Abbey of Citeaux, and two years later founded the monastery of Clairvaux, where he served as an abbot till the end of his life, though he was repeatedly offered to take episcopal orders. He was highly regarded not only within the Church but also by the monarchs of the time. He served as a spiritual director and teacher of Pope Eugene III, cardinals, bishops and state leaders. He was often called ‘a pope with no crown’, and he mentioned this fact in a letter to Pope Eugene III (“Aiunt non vos esse Papam, se me: et undique ad me confluent, qui habent negotia”; Bernard of Clairvaux, Epistola CCXXXIX, PL 182, c. 431). His opinions were taken into consideration not only in religious but also social and political matters. He perceived fighting against the enemies of the Church as his main task, and understood it as actions against external enemies, but also against Christian thinkers, who according to Bernard, may have been a threat to the purity of the doctrine. He was also very much involved in the moral revival of the clergy, which can be seen in his works: De moribus et officio episcoporum (PL 182, cc. 809-834A), De laude novae militia (PL 182, cc. 917-940B) and De consideratione (PL 182, cc. 727-808A); as well as in his interventions concerning episcopal nominations, as he wanted to make sure that those in charge of dioceses were ready to introduce reforms; see P.P. Gilbert, Wprowadzenie do teologii średniowiecza, Wydawnictwo WAM, Kraków 1997, p. 99. 15 PL 183, cc. 785A-1198A.


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that cannot lead man to real joy, and what is more, that can harm the soul by dragging it away from real knowledge (vera scientia) or trying to replace what was left to us in the Sacred Scripture, the teaching of the Church Fathers and the tradition of the Church.16 Examining faith with the use of ratio loses its raison d’être, as any beneficial knowledge in the field of faith, which could be obtained with the use of reason, can already be found in patristic texts. Moreover, not a single bit of the revealed knowledge should raise any doubts, and therefore there is no place for logic, which is a tool used for solving questionable issues, in the field of theology.17 Such an approach to the relationship between fides and ratio equated theological understanding only with a mystical contemplation of truth within the Church by means of sacraments. Any reflection on God undertaken outside these boundaries would be an empty discourse.18 Among those who disagreed with Bernard’s view was Gilbert of Poitiers (1076-1154),19 who defined theology as the highest form of knowledge which tries, with the use of ratio, to explain the unfathomable divine mysteries that in themselves are impossible to reach and to express. Gilbert stressed the necessity for trust in the revelation, as reason cannot reach fully reliable conclusions, but he also pointed to the fact that the inadequacy of human research tools in reference to the truths of faith does not allow thinkers to resign from making attempts at justifying the Christian doctrine in a rational way.20 A theoloSee G. d’Onofrio, Historia teologii..., op. cit., p. 246. See E. Gilson, La teologia mistica di San Bernardo, Jaca Book, Milano 1995, p.106. 18 See J. Leclercq, Esperienza spirituale e teologia. Alla scuola dei monaci medievali, Jaca Book, Milano 1990, p. 189-191. 19 He was born in 1076. He was educated at Laon, where he learned about Augustinian realism. He started his career as a master at Chartres, where he studied the Platonising naturalism of late antiquity ncyclopaedists and Boethius. He served as the chancellor of Chartres for twelve years. Then he moved to Paris in order to take the chair of logic and theology at the cathedral school, where he further improved his skills thanks to contacts with the most prominent logicians of his time. He died in 1154 as the bishop of his hometown of Poitiers. See S. Swieżawski, Dzieje europejskiej filozofii klasycznej, PWN, Warsaw–Wrocław 2000, p. 488. 20 See M.D. Chenu, La teologia Nel dodicesimo secolo, Jaca Book, Milano 1986, p. 429. 16 17

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gian should be lector, interpres, which means that he should not only present in a mechanical way the auctores’ views, but also comment on them, explore, revise and bring out the intention of the author. In order to do so, he must rely on rational knowledge, mostly on grammar and logic, being aware at the same time of the possibility of making a mistake and being corrected by the authorities of the Church.21 Abelard created his writings at the time when monastic theology dominated. The views presented by Anselm of Canterbury and Gilbert of Poitiers were important but also exceptional in the light of the theology of their time. In this context Abelard’s achievements seem to open a new chapter in the reflection on faith and reason, and on dialectics, as its tool, in particular.

3. The notion of dialectics and its theological implications in Abelard’s thought Dialectics is the basic term used by Abelard to describe the activity of human intellect. There is no coherent definition of that notion, as Abelard himself did not say explicitly what he had in mind while using the term. He provided two synonyms: logica end ratio, but from the linguistic point of view, even if we take them into consideration, still not much can be explained. A fragment of Abelard’s logical treatise Dialectica carries some referents: Dialectic, to which all judgement of truth and falsehood is subject, holds the leadership of all philosophy and the governance of all teaching.22 See G. d’Onofrio, Historia..., op. cit., p. 211. “Haec autem est dialectica, cui quidem omnis veritatis seu falsitatis discretio ita subiecta est, ut omnis philosophiae principatum, dux universae doctrinae, atque regimen possiedat”; P. Abelard, Dialectica, transl. C.J. Mews, Peter Abelard on Dialectic, Rhetoric, and the Principles of Argument, [in:] Rhetoric and Renewal in the Latin West 1100-1540: Essays in Honour of John O. Ward, C.J. Mews, C.J. Nederman, and R.M. Thomson (eds), Brepols, Turnholt 2003, p. 43. 21 22


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An analysis of the above sentence allows us to distinguish three meanings of the term. In the first meaning, dialectics is a tool for separating truth from falsehood. Therefore, on the logical-formal level, it enables us to determine the value of a statement by using the rules of logic. That definition, however, can be extended so as to include every type of discourse. Dialectics would make it possible then to judge the veracity of argumentation and suggested claims. In the second meaning, dialectics is a guide to all fields of knowledge. It serves as a tool to verify the scientific nature of theories that we are presented with and guards the scientific value of arguments. Understood in such a way, it contains all verification rules and methodological principles necessary for determining the value of every type of discourse. Thus, Abelard equated dialectics with classical logic.23 It would serve then as a critical tool for any scientific research and a general methodology for all sciences.24 Referring to that function at a different place Abelard quoted St. Augustine: In On Order the most honourable doctor Augustine praised dialectic is such words: ’They call dialectic the discipline of disciplines. It teaches to teach, it teaches to learn. In it reasoning itself makes itself evident and reveals what it is, what it intends. It alone knows; it not only intends to make people knowledgeable but also can do so’.25 “Logicam vero idem dicimus quod dialecticam et indifferenter utroque nomine in designatione utimur eiusdem scientiae”; P. Abelard, Logica ‘Nostrorum petitioni sociorum’, Peter Abelards Philosophische Schriften, Geyer B. (ed.), BGPTM, Münster 1933, p. 506. 24 See M.B. Brocchieri, La logica di Abelardo, La Nuova Italia, Firenze 1969, p. 17, 30. 25 “De cuius laude excellentissimus doctor Augustinus in libro De ordine his verbis scribit: ‘Disciplinam disciplinarum, quam dialecticam vocant. Haec docet docere, haec docet discere. ln hac seipsa ratio demonstrat atque aperit quid sit, quid velit; scit scire, sola scientes facere non solum vult, sed ctiam potest’.”; P. Abelard, Theologia Summi Boni, II, 5, op. cit., quote from On Order, [in:] Letters of Peter Abelard, Beyond the Personal, transl. J.M. Ziółkowski, The Catholic University of America Press, Washington 2008, p. 180; and similarly: “Unde et artem ipsam disputandi secundo De Ordine libro caeteris praeferens disciplinis, et tamquam ipsa sola sciat vel scientes faciat, eam commendans ait: ‘Disciplinam disciplinarum, quam dialecticen vocant. Haec docet do23

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In the third meaning, dialectics is the queen of the sciences that holds primacy over the whole of philosophy. The claim made by Abelard was very brave and controversial. Even St. Augustine, who in many cases served as an example for the theologian, valued dialectics above artes liberales but not above philosophy for which liberal arts, and dialectics in particular, served as preparatory disciplines. A. Crocco proposed a thesis that Abelard’s overevaluation of dialectics could have been caused by the fact that he equated this discipline with reasoning in general (ratio), which made it possible to perceive it as the foundation and queen of all philosophical investigations.26 Such an honourable position of dialectics could have resulted also from Abelard’s passion for this discipline which he expressed in A Letter to a Friend: Of all the areas of philosophy, my primary interest lay in the weapons of dialectical reasoning, so I traded all my arms for these and gave up the trophies of war for the noisy clash of argument.27

In the latter part of Dialectica, Abelard presents yet another meaning of dialectics. He treats it as a tool of linguistic analysis that monitors the right choice of terms, determines their scope of meaning

cere, haec docet discere. In hac se ipsa ratio demonstrat, quid sit, quid velit, scit sola. Scientes facere non solum vult, sed etiam potest’.”; P. Abelard, Dialogus, R. Thomas (ed.), Frommann, Stuttgart 1970, 1458-1464, (“In the second book of On Order [St. Augustine] values the discipline of disputation above all other disciplines, and as if it were the only knowledge or created the knowledgeable, recommends it so: ‘hey call dialectic the discipline of disciplines. It teaches to teach, it teaches to learn. In it reasoning itself makes itself evident’”). 26 See A. Crocco, Abelardo..., op. cit., p. 92ff. 27 “Et quoniam dialecticarum rationum armaturam omnibus philosophiae documentis praetuli, his armis alia commutaui et tropheis bellorum conflictus praetuli disputationum. Proinde diversas disputando perambulans prouincias, ubicumque huius artis vigere studium audieram, peripateticorum aemulator factus sum”; P. Abelard, Epistola 1: Historia calamitatum sive..., transl. W. Levitan [in:] Abelard and Heloise. The Letters and Other Writings, Hackett Publishing Company, Indianapolis–Cambridge 2007, p. 2.

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and functions in the structure of a sentence.28 As Crocco pointed out, in this case, logic-dialectic is presented as scientia sermocinalis, the philosophy of language, whose function is to analyse the logical basis of terms and names by verifying the semantic link between voces, or terms of discourse, and the reality of referents.29 The theological implications of this kind of dialectics pervade Abelard’s thought. According to him, separating truth from falsehood was important, above all, from the apologetic point of view. In Letter 13: Invectiva in quemdam ignarum dialectices, he wrote: For we are not equipped to rebut the attacks of heretics or any infidels whatsoever, unless we are able to unravel their disputations and rebut their sophisms with true reasonings, so that falsity may yield to truth.30

Thus, a true defence of faith should consist in revealing the falsehood in those doctrines that attack Christianity. The revelation in itself is not sufficient and, therefore, we need tools provided by dialectics.31 As a methodological tool we use it on the inner “Hoc autem logicae disciplinae proprium relinquitur, ut scilicet vocum impositiones pensando quantum unaquaque proponatur oratione sive dictione discutiat. Physicae vero proprium est inquirere utrum rei natura consentiat enuntiationi, utrum ita sese, ut dicitur, rerum proprietas habeat vel non. Est autem alterius consideratio alteri necessaria. Ut enim logicae discipulis appareat quid in singulis intelligendum sit vocabulis, prius rerum proprietas est investiganda. Sed cum ab his rerum natura non pro se sed pro vocum impositione requiritur, tota eorum intentio referenda est ad logicam”; P. Abelard, Dialectica, p. 286ff. 29 See A. Crocco, Abelardo..., op. cit., p. 95ff. 30 “Non enim haereticorum vel quorumlibet infidelium infestationes refellere sufficimus, nisi disputationes eorum dissolvere possimus et eorum sophismata veris refellere rationibus, ut cedat falsitas veritati”; P. Abelard, Epistola 13: Invectiva in quemdam ignarum dialectices, [in:] Peter Abelard: Letters IX-XIV, E.R. Smits (ed.), Rijksuniversiteit, Groningen 1983, p. 274, transl. J.M. Ziółkowski [in:] Letters of Peter Abelard, Beyond the Personal, The Catholic University of America Press, Washington 2008, p. 183. 31 “Fallacia membra sua adversus veritatem iam effrenis armat atque unum iam superest, ut qui non possumus factis, pugnemus verbis”; P. Abelard, Theologia Christiana, III, 3, op. cit. (“Nowadays falsehood impudently arms itself against truth and there remains only one solution, to fight on words if we cannot do so in actions...”). 28

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field in order to better understand the truths of faith by studying the Sacred Scripture32. Paré, Brunet and Tremplay found another advantage of transferring this characteristic of dialectics to the field of theology. Abelard did not want theology to be a compilation of revealed and patristic texts, but a scientific explanation of the truths of faith, that would be credible both for believers and those considering the adoption of faith. Therefore, strict scientific canons must be followed in the study of theology and conclusions need to be formulated according to certain requirements, while the scientific nature of the discipline is guaranteed by its submission to scientific methodology, i.e. to dialectics.33 While defining theology as the queen of the sciences, Abelard made reference to Christ who used not only quotations from the Bible but also the power of rational arguments to expose slanders spread by Jews with whom he held discussions.34 A Christian, following this example, should try to refute accusations in the same way, as the better he become Abelard quoted St. Augustine: “‘Restant ea quae non ad corporis sensus sed ad rationem pertinent, ubi disciplina regnat disputationis et numeri. Sed disputationis disciplina ad omnia genera quaestionum quae in sanctis litteris sunt penetranda, plurimum valet’”; P. Abelard, Theologia Christiana, II, 117, op. cit. (“The disciplines that have reason and not senses as their subject remain, and among them the art of discussion and arithmetic hold primacy. The art of discussion, though, is of great use in explaining all kinds of issues that are to be examined in the Sacred Scripture”). 33 See G. Paré, A. Burnet, P. Tremplay, La Reinassance du XIIe siècle. Les écoles et l’enseignement, Paris–Ottawa 1933, p. 296. 34 “Quis denique ipsum etiam Dominum Iesum Christum crebris disputationibus Iudaeos ignoret conuicisse et tam scripto quam ratione calumnias eorum repressisse […]? […] Cum autem miraculorum iam signa defecerint, una nobis contra quoslibet contradicentes superest pugna, ut quod factis non possumus, verbis conuincamus, praesertim cum apud discretos vim maiorem rationes quam miracula teneant, quae utrum illusio diabolica fatiat, ambigi facile potest”; P. Abelard, Epistola 13: Invectiva in Quemdam..., op. cit., p. 276 (“Who could not know that even Lord Jesus himself refuted the Jews in repeated disputations and crushed their slanders in writing as well as in reasoning [...]? Since, however, miraculous signs have now run short, one means of combat remains to us against any people who contradict us: that we may overcome through words, because we cannot do so through deeds, especially since among people of discernment reasoning carries greater force than miracles; it is easy to be uncertain whether or not a diabolic illusion produces miracles”), transl. J.M. Ziółkowski, Letters of Peter Abelard..., op. cit., p. 186. 32


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in repulsing the attacks against the Church (and the best way to repulse them is to use dialectics) the better follower of Christ he will become.35 Dreyer elaborates on this point, seeing in it the most original aspect of Abelard’s thought, and points to a regularity of etymological nature: Abelard started from the description of Christ as Logos and presented it as the source of the term ‘logic’. Thus, logic would be closely related to divine wisdom and would be the most suitable discipline for the study of the truths of faith. Such a relation would also make it possible to justify pagans who studied liberal arts and to perceive them as equal to Christians and as true philosophers.36 “Cum ergo verbum Patris, Dominus Iesus Christus, ‘logos’ Graece dicatur, sicut et ‘sophia’ patris appellatur, plurimum ad eum pertinere videtur ea scientia quae nomine quoque illi sit coniuncta et per derinationem quandam a ‘logos’ logica sit appellata et sicut a Christo christiani, ita a ‘logos’ logica proprie dici videatur. Cuius etiam amatores tanto verius appellantur philosophi quanto veriores sint illius sophiae superioris amatores. Quae profecto summi patris summa sophia cum nostram indueret naturam ut nos verae sapientiae illustraret lumine et nos ab amore mundi in amorem converteret sui, profecto nos pariter christianos et veros effecit philosophos”; P. Abelard, Epistola 13: Invectiva in Quemdam..., op. cit., p. 274ff. (“Therefore, since the Word of the Father, Lord Jesus Christ, is called logos in Greek, just as it is named the sophia of the Father, this knowledge seems to relate very much to him which is connected with him also by name and which is by a certain derivation from logos called logic; and just as Christians seem properly to be so called from Christ, so is logic from logos. In addition, lovers of logic are all the more truly called philosophers as they are truer lovers of that higher sophia. Indeed, when that highest wisdom of the highest Father assumed our nature so that it might illuminate us with the light of true understanding and turn us from love of the world toward love of itself, it made us at once Christians and true philosophers.”), transl. J.M. Ziółkowski, Letters of Peter Abelard..., op. cit., p. 185. 36 “Quae duo, de hoc videlicet amore et doctrina eius quibus tamphilosophi quam summi efficerentur logici, hymnus ille Pentecostes ‘Beata nobis gaudia’ diligenter distinguit, cum dicitur: Verbis ut essent proflui Et caritate feruidi. Haec enim duo maxime ille superni spiritus adventus in igneis linguis revelatus eis contulit ut per amorem philosophos et per rationum virtutem summos efficeret logicos”, P. Abelard, Epistola 13: Invectiva in Quemdam..., op. cit., p. 275ff; see M. Dreyer, Razionalità scientifica e teologia nei secoli XI e XII, Jaca Book, Milano 2001, p. 65ff, (“That hymn of Pentecost ‘Beata nobis gaudia’ (Blissful Joys to Us) carefully differentiates those two – which is to say, about this love and its teaching by which philosophers as well as highest logicians are produced – when it is said: ‘So that they may be fluent 35

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Dialectics as a linguistic tool played an important role especially in searching for a solution to the disagreement over the nature of universals. Abelard shifted the main focus from metaphysics to linguistics. As a result, he formulated a view that came to be known as conceptualism. Solving the problem of universals, though not directly related to theology, had an influence on the formulation of theological claims especially those concerning the nature of the Holy Trinity which, when formulated in the spirit of nominalism, were rejected as promoting tritheism. Drawing of the current state of research, it is hard to provide a final interpretation of Abelard’s thought in this matter. An interpretation provided by De Rijk seems most popular. He claims that Abelard applies realism when referring to theological issues, whereas in the field of philosophy he relies on nominalism, of which conceptualism is a variety.37 Thus, Abelard treated dialectics as a multidimensional discipline. The term comprised practically all manifestations of human intellectual activities. Thus faith, seen by Abelard as a rational reflection on the revelation, needed dialectics as the main tool. It was not, however, an unconditioned permission for ratio to enter the field of fides.

4. Limitations of dialectics in the field of theology Where do errors, such as that committed by Berengarius of Tours, come from? Why do some philosophers, while applying dialectics, arrive at wrong conclusions? Answers to these questions do not lie in in words and aflame with charity’. For that arrival of the heavenly Spirit, having been revealed in fiery tongues, conferred upon them these two things especially, that through love it should produce philosophers and through the power of reason the highest logicians.”) transl. J.M. Ziółkowski, Letters of Peter Abelard..., op. cit., p. 185-186. 37 See L.M. de Rijk, The Semantical Impact of Abailard`s Solution of the Problem of Universals, [in:] Thomas R. (ed.), Petrus Abaelardus (1079-1142). Person, Werk und Wirkung, Paulinus, Trier 1980, p. 139-149. Blackwell is of the opposite opinion, as he suggests combining logical nominalism and philosophical realism into one conception. D.F. Blackwell, Non-Ontological..., op. cit., p. 177-179.


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the nature of philosophical sciences but in the way they are pursued. According to Abelard, no science could in its nature contradict the revelation. Each scientific discipline is supposed to lead to wisdom and God is the Supreme Wisdom. Therefore, if any science distanced anyone from God, it would also distance him from wisdom and by doing so, it would not fulfil the basic criterion of scientificity.38 Accordingly, those who make theological mistakes are bad scholars. In the field of dialectics, Abelard called them pseudodialecticians and described them in the following words: Worse still than all enemies of Christ – heathens, Jews and pagans – are professional dialecticians. They penetrate the mystery of the Holy Trinity and use more difficult arguments, thereby displaying the impudence of the sophists, of whom Plato said mockingly that they find their happiness in a flood of words and an infinite series of proofs. We know what they murmur in silence, believing in the power of their arguments, when they have no opportunity to shout. In any case, I would like to stress that they do not follow the art of dialectics but use it in a perverse way. I do not mean to condemn dialectical knowledge, or any other of the liberal arts for that matter, but I condemn the perversity of the sophists, since even the holy Fathers recommend this art, as I have mentioned before, and regard it higher than others. Therefore dialectics serves philosophy as a razor-sharp sword at its defence, which can be used by a tyrant to

“Sed neque ullam scientiam malam esse concedimus, etiam illam quae de malo est; quae iusto homini deesse non potest, non ut malum agat sed ut a malo praecognito sibi provideat [...] Si qua autem scientia mala esset, utique et malum esset quaedam cognoscere ac iam absolui a malitia Deus non posset, qui omnia nouit. In ipso enim solo omnium plenitudo est scientiarum cuius donum est omnis scientia”, P. Abelard, Theologia Christiana, III, 6, op. cit. (“In general, I do not consider any knowledge as evil, even the knowledge about the nature of evil which a just man should possess, not in order to do evil but to avoid evil after recognizing it [...]. If any knowledge was evil, then knowing some things would be certainly bad, but in such a case, God would never be able to release Himself from evil, as He knows everything because He is the height of all knowledge and all knowledge is given by Him”). 38

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kill with fury, but which can be of equally great help and harm for those who use it for what it was intended.39

What would be the characteristics of pseudodialecticians? The worst vice of a pseudoscholar is arrogance that does not allow him to see the limitations of the science that he pursues. He trusts in his knowledge to such an extent that he does not perceive mysteries, questions without answers or gaps in the discipline that he studies.40 In formulating theories, he is driven more by the desire to be original and to win renown than to seek truth.41 Moreover, arrogance is what makes it impossible for a pseudosage to accept God as the source of all knowledge and to see himself only as its promoter. Accepting “Supra universos autem inimicos Christi, tam haereticos quam Iudaeos sive gentiles, subtilius fidem sanctae Trinitatis perquirunt et acutius arguendo contendunt professores dialecticae seu importunitas sophistarum, quos verborum agmine atque sermonum inundatione beatos esse P1ato irridendo iudicat. Hi argumentorum exercitio confisi, quid murmurent scimus, ubi facultas aperte garriendi non datur hi, inquam, non utentes arte sed abutentes. Neque enirn scientiam dialecticae aut cuiuslibet liberalis artis sed fallaciam sophisticae condemnamus, praesertim cum sanctorum quoque patrum iudicio haec ars maxime, ut supra meminimus, commendetur et caeteris praeferatur. Tenet itaque haec philosophia, acutissimi gladii instar quo tyrannus ad perniciem princeps utitur, ad defensionem ac pro intentione utentium, sicut plurimum prodesse ita et plurimum nocere potest”; P. Abelard, Theologia Christiana, III, 4ff, op. cit. 40 “Quorum tanta est arrogantia, ut nihil esse opinentur quod eorum ratiunculis comprehendi atque edisseri non queat, contemptisque universis auctoritatibus, solis sibi credere gloriantur. Qui enim id solum recipiunt quod eis ratio sua persuadet, profecto sibi solis acquiescunt, quasi eos habeant oculos qui in nullis caligare noverint”; P. Abelard, Theologia Christiana, III, 20, op. cit. (“They are so provocative in their impudence that they do not even want to allow for any issue to be beyond their understanding and inexplicable with the use of their feeble brains. They reject any commonly acknowledged authority and are proud of the fact that they rely solely on their own reason which tells them what can be accepted as true. Therefore, it is clear that they rely only on themselves, as if they had such good eyesight that they could see in complete darkness”). 41 “Non enim ignorantia haereticum facit sed superbia, cum quis videlicet, ex novitate aliqua nomen sibi comparare desiderans aliquod inusitatum proferre gloriatur, quod adversus omnes importune defendere nititur, ut superior omnibus videatur, aut ne confutata sententia sua inferior caeteris habeatur”; P. Abelard, Theologia Christiana, III, 17, op. cit. (“Not ignorance but arrogance is what makes a heretic. It can be seen for example when someone wants to win renown by being original and boasts about some unusual theories that he is ready to impudently defend in front of others, so as to present himself as superior to them, but to avoid being regarded as inferior to others if his theories were proved wrong”). 39


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such a position of God is tantamount to seeing the need for a life of prayer42 and concern for a pure heart.43 According to Crocco, a comparison and explanation of two verbs: intelligere and comprehendere, used by Abelard to describe the activity of intellect in the field of faith, can help in the right understanding of the value and simultaneous insufficiency of reason in theology.44 The first verb: intelligere cannot be reduced in Abelard’s writings to define empirical knowledge or cognition that focuses on the actual experimental reality. He used this term in reference to faith “O utinam vel per somnium et isti qui se philosophos profitentur a praesumptione sua compescerentur, ut veri et summi numinis incomprehensibilem maiestatem esse non negarent, nisi eam aperte discuti audirent! Sed fortasse inquiunt: quid eam veritatem dici attinet ab aliquo, quam ipse se non valet explicare ut intelligi possit? Multum equidem, respondeo. Cum enim auditur de Deo quod non intelligitur, excitat auditorem ad inquisitionem, quod non fieret nisi audiretur. Inquisitio vero facile intelligentiam parit, si devotio adsit cui se Deus revelare dignetur. Et saepe quod divina inspiratione dicitur vel divina operatione geritur, non ad opus eorum fit per quos agitur sed ad commodum aliorum extenditur, sicut de miraculis supra meminimus. Aliis itaque dicendi gratia data est, aliis intelligendi reservatur, quousque opus sit secundum divinae consilium providentiae; et aliquorum sincera devotio, tum orationibus, tum bonis operibus id promeruerit”; P. Abelard, Theologia Christiana, III, 47-49, op. cit. (“May those scholars who perceive themselves as philosophers be prompted, even by a dream, to temper their arrogance and to start promoting the view that the majesty of the Highest God is incomprehensible and they should not dare to discuss it publicly. Some may ask though: What is the benefit of someone talking about a truth that he himself cannot explain and the others cannot understand? To that I say: It’s a great benefit. When someone hears that it is very difficult to understand God, he is encouraged to undertake intensive research, which would not happen if he had not heard about the nature of God first. But research results in cognition when accompanied by piety. To such a person God benignly reveals Himself, and it is often the case that what someone says out of God’s inspiration or does with God’s help does not serve for the benefit of the one who acts but of the others, as it is the case with miracles. Some are given the grace of talking, others of understanding but in both cases it is limited by the plans of Providence and depends on one’s true piety proved by prayers or good deeds”). 43 “Qui etiam ea maxime quae ad notitiam Dei attinent nonnisi eo revelante percipi posse, quem soli mundicordes conspicere valent, valida similitudine conuincit”; P. Abelard, Theologia Christiana, III, 30, op. cit. (“Especially those issues that are related to the cognition of God can be understood only on the basis of His Revelation and they can be understood only by people of pure heart”). 44 In his analysis, Crocco follows E. Kaiser, Pierre Abélard critique, Librairie Catholique, Frebourg en Suisse 1901, p. 94-102 and T. Heitz, La philosophie et la foi l’oeuvre Abélard, “Revue des Sciences philosophiques et theologiques”, vol. I 1907, p. 713-715. 42

The Role of Dialectics in Peter Abelard’s Concept of Theology


aided by reason. Thus, a relation between intelligere and credere is possible and manifests itself in the fact that, on the one hand, faith without any attempts at intellectual exploration is meaningless, and on the other hand, that faith itself is beyond the order of experimental knowledge. Another verb: comprehendere was used by Abelard to express total understanding of an unexperiancable reality, some vital knowledge about divine mysteries that radically exceed the power of human perception. Then, the activity of reason in the field of faith was for Abelard that of intelligere but not comprehendere in the case of the revealed truths. *** In the introduction to this paper I raised a question: To what extent and in what dimensions – according to Abelard – can dialectics help in pursuing theology? It is impossible to fully answer this question without making reference to Abelard’s thought. In order to judge his contribution to a better understanding of the relation between logic and theology in the history of these disciplines, we should bear in mind that the context for his teaching was that of a dispute between dialecticians and antidialecticians, the latter of whom were in a privileged position. This context made it necessary to exercise great caution in formulating any judgements concerning the status of dialectics, so as to prevent its disappearance from the field of orthodox theology. In this case, Abelard was successful only to some extent. Although a few of his particular solutions were condemned, the method itself was presented in such a cohesive and interesting way that it survived and became the basis for the development of modern theology. The multidimensionality of dialectics in Abelard’s thought makes this discipline widely applicable in the field of faith. It has its role in apologetics as well as in the development and purification of the Christian doctrine. At the same time, we must remember about its limitations in the creation of the theological system. God cannot be


Damian Wąsek

fully comprehended, and therefore, reason must be accompanied by humility. It is worth mentioning in this context that there are few texts concerning the inadequacy of the rational order as compared to that of the revelation, whereas texts on the competence of intellect are present in most of Abelard’s writings (this uneven distribution of emphasis was most probably one of the reasons why his theology was condemned). What are current tasks for logic at the point where it meets theology? To what extent and in what dimensions can these disciplines cooperate? The analyses presented above bring to mind two fields of cooperation that seem very important from the theological point of view. The first of them is concern for the credibility of Christianity. In order for religion – treated as a system of truths – to be credible, it must form a coherent system. Logic makes it possible to expose contradictions and find ‘weak points’ of the doctrine in terms of methodological cohesion. Dialogue with modern atheism (especially that based on empirical sciences) is the other field of cooperation between logic and theology. Logical instruments can be used to test, in terms of logical veracity, conclusions formed by atheists and their charges against the revelation, as well as to specify the competence of the secular sciences in the field of theology.

Marek Porwolik Cardinal Stefan Wyszyński University in Warsaw

Formalizations of the Argument Ex Causa Efficientis Presented by Fr. Bocheński* 1. Introduction


is impossible to talk about the attempts to utilize logic in theology without mentioning the interwar period and the activities of the so-called Cracow Circle. It was created by Fr. Jan Salamucha, Mr. Jan Drewnowski, Mr. Bolesław Sobociński, and Fr. Józef Maria Bocheński. The death of the first of these logicians in the Warsaw Uprising ended the official activities of the Circle. After many years, the last of the mentioned men returned to the research into the applications of logic in theology at the end of his life. The fruit of this research was the formalization of St. Thomas Aquinas’ Five Ways. The return to this interesting problem had further stages, although less known to the public. Namely, Fr. Bocheński formalized Program studiów o Bogu (a programme of studies about God), in which he specified the objectives of the application of logic in theology. He mentioned the need to create such a programme during the ceremony at the Warsaw Theological Academy on 15 October 1990, when he was awarded the degree of doctor honoris causa.1 The programme was published in 2003 in the book Gottes Dasein und Wesen. Logi-

This contribution was made possible through the support of a grant “The Limits of Scientific Explanation” from the John Templeton Foundation. 1 J.M. Bocheński, O współczesnym stanie i zadaniach teologii filozoficznej, “Studia Philosophiae Christianae” 1991, 27/2, p. 103-107. *


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sche Studien zur Summa Theolgiae I, qq. 2-11.2 The book involved the corrected versions of the formalization of the Five Ways, as well as the formalizations of the subsequent questions from Summa Theologiae, which were supposed to constitute the implementation of the above-mentioned programme. The subject of our reflection is the comparison, the critical analysis, and the completion of the formalization of the Second Way, presented by the philosopher of Fribourg. It is the very way that was considered the only ‘efficient’ one among all the Five Ways. Besides, on the basis of the description of God in the Second Way, the philosopher tried to construct a preliminary axiomatic system for the description of the theory of the Absolute. In addition to the materials mentioned above, the source texts for our study are the typescripts of the book Gottes Dasein und Wesen: one German from 1989, and one Polish from 1993. The latter is more interesting as it was translated by Fr. Bocheński himself, and it is more recent than the German version submitted (in 1990 according to the book; in 1991 according to the Polish manuscript) in the publishing house in Munich. Because of numerous inaccuracies and mistakes, the book with the emendations, which were not authorized by Fr. Bocheński, was published after more than 10 years following the submission of the manuscript in the otherwise favourably predisposed publishing house. The five source texts include three versions of the formalization of the discussed way. Our main aim is to analyze these texts.3

J.M. Bocheński, Gottes Dasein und Wesen. Logische Studien zur Summa Theolgiae I, qq. 2-11, Munich 2003, p. 17-28. 3 I. a) J.I.M. Bocheński, Die fünf Wege, “Freiburger Zeitschrift für Philosophie und Theologiae” 1989, 36/3, p. 235-265. b) J.M. Bocheński, Pięć dróg, [in:] J.M. Bocheński, Logika i filozofia. Wybór pism, transl. J. Miziński, Warsaw 1993, p. 469-503. II. a) J.M. Bocheński, Gottes Dasein und Wesen. Mathematisch-logische Studien zur Summa Theolgiae I, qq. 2-13, Freiburg 1989, typescript. b) A typescript of the Polish translation II a) from 1993, prepared by Fr. Bocheński (without the title page). 2

Formalizations of the Argument Ex Causa Efficientis...


2. The formalized text For Fr. Bocheński, the subject of the formal analysis is the original Latin text of Summa Theologiae.4 Each sentence is written separately and marked with a number, which shows its place in the very text. The first figure points to the chapter (quaestionem), the second to the section (articulum), the third figure is the paragraph (if applicable), and the other figures point to the place of the sentence within the paragraph, or if there is only one paragraph, to the place in the section.5 In order to facilitate understanding, the English translation is provided next to the Latin text.6 No.

Latin text

English text


invenimus enim in istis sensibilibus esse ordinem causarum efficientium

In the world of sense we find there is an order of efficient causes.


nec tarnen invenitur nec est possibile, quod aliquid sit causa efficiens sui ipsius;

There is no case known (neither is it, indeed, possible) in which a thing is found to be the efficient cause of itself;


quia sic esset prius seipso quod est impossibile

for so it would be prior to itself, which is impossible.


Non autem est possibile quod in causis efficientibus procedatur in infinitum.

Now in efficient causes it is not possible to go on to infinity,


quia in omnnibus causis efficientibus ordinatis primum est causa medii et medium est causa ultimi, ...

because in all efficient causes following in order, the first is the cause of the intermediate cause, and the intermediate is the cause of the ultimate cause, …

III. J.M. Bocheński, Gottes Dasein und Wesen. Logische Studien zur Summa Theolgiae I, qq. 2-11, Munich 2003. 4 Thomas Aquinas, Thomae Aquinatis Opera omnia (ed. Leonina), IV, Roma 1882. 5 Ia, p. 242; Ib, p. 477; IIa, p. 22; IIb, p. 13; III, p. 35. 6 Thomas Aquinas, Summa Theologiae, transl. by Fathers of the English Dominican Province, Westminster 1981.

Marek Porwolik

162 2.3206.

remota autem causa removetur effectus.

Now to take away the cause is to take away the effect.


ergo si non fuerit primum in causis efficientibus, non erit ultimum nec medium.

Therefore, if there be no first cause among efficient causes, there will be no ultimate, nor any intermediate cause.

2.3208. Sed si procedatur in infmitum in causis efficientibus, non erit prima causa efficiens:

But if in efficient causes it is possible to go on to infinity, there will be no first efficient cause,


et sic non erit nec effectus ultimus, nec causae efficientes mediae.

neither will there be an ultimate effect, nor any intermediate efficient causes.


quod patet esse falsum.

all of which is plainly false.


Ergo est necesse ponere aliquam causam efficientem primam:

Therefore it is necessary to admit a first efficient cause,


quam omnes Deum nominant.

to which everyone gives the name of God.

3. The rules of formalization Before conducting the very formal analysis, Fr. Bocheński lists the following applied abbreviations:7 CA(x,y) =: x est causatum ab y (x is caused by y) PR(x,y) =: x est prius y (x is before y) v(x) =: x est causatum (x is caused) Cp(x) =: x est causa prima (x is the first cause) D(x) =: x est Deus (x is God) R* ε Inf =: die durch R gebildete Reihe ist unendlich (the series created by R is infinite).8

Ia, p. 237-238; Ib, p. 471-473; IIa, p. 22-24; IIb, p. 13-15; III, p. 36-39. In IIb Fr. Bocheński explains the abbreviation in the following way: “a series created by R is infinite”. Cf. IIb, p. 15. Besides, in I and in IIa, instead of R* there is R* and in Ib and IIa instead of ε there is ∈. 7 8

Formalizations of the Argument Ex Causa Efficientis...


In the various versions of his formal commentary, Fr. Bocheński utilizes the rules of inference,9 which are marked in the text with the subsequent letters of the Latin alphabet. When listing the rules, the author provides the moment in the formalization where the rule has been applied. This moment is specified by the number of the verse in the formal text. The verse has been given in brackets. Because of the length limitations of the present study, the variety of the specifications in the various versions of the text, and the numerous inaccuracies in quoting them, those rules will not be analyzed in the present paper. The formal language used by Fr. Bocheński is taken from Principia Mathematica.10 In version III it is applied in a modified form, similar to the symbols used nowadays. In our analysis of the Second Way, the modernized version of formalization will be provided. Two of the schemas applied in the formalization belong to the Classical Propositional Logic. The other schemas belong to the Classical First Order Predicate Logic with Identity. In the commentary to the list of the schemas, Fr. Bocheński suggests that St. Thomas unjustifiably considered them to be syllogistic moods, although he claimed that in some cases “a syllogistic interpretation is at least possible”. Besides, in his opinion, some schemas of the Classical First Order Predicate Logic with Identity can be replaced by some schemas of the Classical Propositional Logic. ‘For reasons of safety’, however, he expressed them in the Predicate Logic. All the rules are the elementary rules for him.11 Apart from the use of the various rules of inference, there are certain discrepancies between the places of their use, which are shown by the very list of the rules, and their real applications in the three formalizations. The list refers to all the three versions of the text. The reasons for this situation can be found in both the inaccuracy of the author, as well as the evolution of the very text of formalization, which was difficult to follow for the commentaries, whose role was secondary. Ia, p. 238-240; Ib, p. 473-475; IIa, p. 26-30; IIb, p. 16-20; III, p. 41-48. B. Russell, A.N. Whitehead, Principia Mathematica, Cambridge 1950, p. XVI. 11 IIa, p. 30; IIb, p. 20; III p. 48. This commentary is absent from version I. 9


Marek Porwolik


4. The formalized notation and its analysis Fr. Bocheński provides the formalization of the Second Way in the following versions:12 1. I,II,III: ∀ ∀ [CA( x, y ) ∧ x = y → PR( x, x)]


2. I,II,III: ∀x ~ PR( x, x)


3. I,II,III: ∀ [v( x) → ∃ CA( x, y )]


4. I,II,III: ∃x v(x)


x y





∃ ∃{CA( x, y ) ∧ x ≠ y ∧ ∃ [CA( y, z ) ∧ y ≠ z ]} ont14 x y


→ ~ (CA * ε Inf ) 6. I,II: III:

∀ ∀{CA( x, y ) ∧ x ≠ y → ∃ [CA( y, z ) ∧ y ≠ z ]} ont x y


∀ ∀{CA( x, y ) ∧ x ≠ y ∧ v( y ) x y


→ ∃ [CA( y, z ) ∧ y ≠ z ]} z

7. I,II,III:

~ (CA * ε Inf ) → ∃ Cp ( x) x


8. I,II,III: ∀x [Cp ( x) → D( x)]


9. I,II,III: ∀x ∀y [CA( x, y ) → x ≠ y ]

(1, 2)

Because of mistakes and the variety of the symbols used, the rules of inference applied by Fr. Bocheński are omitted. Ia, p. 250-251; Ib, p. 487; IIa, p. 38; IIb, p. 26; III, p. 59. 13 Versions IIa and IIb only differ in the way of writing symbol R*. As in the case of I, also in IIa R* is used. In order not to multiply the versions of formalization, in the case of II the notation was unified. 14 In version I, verses 5 and 6 appear in the reversed order. The form of the verse used above comes from Ia. In Ib there is an expression, which can be interpreted differently, without any reference to Ia. We assume that it is the consequence of a mistake in printing. 12

Formalizations of the Argument Ex Causa Efficientis...


10. I,II,III:

∀{v( x) → ∃ [CA( x, y ) ∧ x ≠ y ]}

(3, 9)

11. I,II,III:

∃ ∃ [CA( x, y ) ∧ x ≠ y ]

(10, 4)

12. I,II,III:

~ (CA * ε Inf )

(5, 6, 11)

13. I,II,III:

∃ Cp (x)

(7, 12)

14. I,II,III:

∃ D(x)

(8, 13)



x y



Besides the above-mentioned discrepancies in the form of the notation and the use of the various schemas of inference, it is possible to say that the differences between the versions of formalization only refer to the expression in line 6. In versions I and II, it has the following form: ∀ ∀{CA( x, y ) ∧ x ≠ y → ∃ [CA( y, z ) ∧ y ≠ z ]} x y


The form states that if a certain x is caused by a certain y, which is different from x, then there exists such a z, different from y, which causes y. In version III there is an additional condition put on y; namely, such that y is also caused: ∀ ∀{CA( x, y ) ∧ x ≠ y ∧ v( y ) → ∃[CA( y, z ) ∧ y ≠ z ]} x y


In this way, it is possible to avoid a controversial thesis saying that the first cause is also brought about by something different from it. The thesis results from the rule, not expressed in the very formalization, such that the first cause, whatever its interpretation, is the effective cause of at least one x. The modification of verse 6 in version III may prove that such a relationship was taken into account. In his commentary to the formalization of all the Five Ways, Fr. Bocheński divides the premises into four classes:

Marek Porwolik


a) definitions b) empirical sentences c) philosophical sentences (sentences taken from Aristotle’s philosophy) d) formal-logical sentences. In works I-III, the philosopher collates the premises in a table regarding all the Five Ways. Let us analyze the results from the table, regarding the Second Way.15 Name Definitions Empirical sentences Philosophical sentences Logical sentences

Version I 2 1 5 0

Version II 3 1 4 0

Version III 2 1 5 0

It is possible to notice the difference in the number of the premises of each kind. Interestingly, version III returns with the ostensibly wrong data given in version I. In fact, Fr. Bocheński always mentions 3 definitions, 1 empirical sentence, and 4 philosophical (ontological) sentences. It can be a trace of the hesitations of the author with regard to the placing of certain premises in the four categories. This seems to be confirmed by the further analysis of the text, where Fr. Bocheński tries to explain again all the premises used in the formalizations of all the Five Ways.16 In this place of his work, in all the versions (also in version II), Fr. Bocheński mentions the relationships from verses 7 and 8 as definitions. He omits the following relationship: ∀ [v( x) → ∃ CA( x, y )] x

15 16


Ia, p. 259; Ib p. 496; IIa, p. 45; IIb, p. 32; III, p. 69. Ia p. 259-265; Ib p. 497-502; IIa, p. 45-49; IIb, p. 32-35; III, p. 69-75.

Formalizations of the Argument Ex Causa Efficientis...


which appears in verse 3 and is marked in each version as ‘def’. An identical relationship can be found a few pages further, among the philosophical assumptions. It is the same in version II. In the other versions, Fr. Bocheński provides its modified form:

∀ {v( x) → ∃ [CA( x, z ) ∧ z ≠ x]} x


Let us have a closer look at the philosophical assumptions. There are 21 assumptions named “philosophical”. Five of them refer to the Second Way.17 Fr. Bocheński numbers the subsequent assumptions, and provides the number of the sentence from the Latin text of Summa Theologiae, to which he refers in the given place. The assumptions in quastion have numbers 7-11. They can be found after the last assumption regarding the First Way, and before the first assumption regarding the Third Way. Fr. Bocheński lists the assumptions from all the ways. In Ia, IIa, III, and partially in IIb they are separate, and it is stated which way they come from. It is important because on the list of assumptions next to expressions 4, 9, 10, and 15, there is no number of the sentence from Summa, which is referred to by the given expression. However, the place where the assumption was used and its content allow one to place it in the correct via. Among the philosophical assumptions, Fr. Bocheński distinguishes four classes, among which he places the assumptions regarding the Second Way:18 a) analytic sentences resulting from the meaning of words (7, 8), b) different form of the rule of causality (9, 10), c) the rules of finiteness of the relevant series (11), d) synthetic sentences (11).19 One of them, as it has already been mentioned, appears as a definition in the formalization. 18 Ia p. 262-263; Ib p. 500-501; IIa, p. 47-48; IIb, p. 34; III p. 73-74. 19 In I-III as an example of a synthetic sentence, Fr. Bocheński provides assumption 11, but while moving to the description of its content, he actually describes assumption 12 (“what is unnecessary does not exist at a certain moment” Ib, p. 501). In III in the very description of the listed synthetic sentences, the number of the assumption has 17

Marek Porwolik


Besides, sentences 9, 10, and 11 are ontological sentences in the views of the philosopher of Fribourg. Despite distinguishing four classes of philosophical sentences, in the formalizations, Fr. Bocheński only utilizes the abbreviation ‘ont’. Let us have a closer look at the specific assumptions quoted by Fr. Bocheński.20 Assumption 7. (2.3202/3)21

∀ ∀ [CA( x, y ) → PR( y, x)] 22 x y

Die Ursache ist vor dem Verursachten. (Ia; IIa; III) Przyczyna jest przed skutkiem. (Ib; IIb) The cause is before the effect. To make things clearer, in each assumption we will quote the assumption from the formalization carried out by Fr. Bocheński. In his view, the philosophical assumptions gathered here came from his formalization. The premise in the Second Way reads as follows: ∀ ∀ [CA( x, y ) ∧ x = y → PR( x, x)] x y

First of all, in his list of assumptions, Fr. Bocheński provides a relationship which is stronger than the one found in the very formalization. The latter seems to be closer to what is found in the relevant sentences of Summa. It is also necessary, as done by the author, to refer to both sentences 2.3202 i 2.3203 in versions II and III, and not only to the last one. Thus, in his list of philosophical assumptions, been changed from 11 to 12, although when mentioning them, number 11 remains. This gives rise to the question how Fr. Bocheński treated assumption 11. The change of the numbers can be simply a spelling mistake or point to a different, shorter list of assumptions. 20 The assumptions have been given in the contemporary notation. 21 In Ia and Ib Fr. Bocheński only refers to 2.3203. 22 In I-III there is: ∀∀ [CA( x, y) → PR( x, y)], which we treat as an obvious mistake. x y

Formalizations of the Argument Ex Causa Efficientis...


Fr. Bocheński provides not only a certain philosophical assumption present also in the very formalization (as was its original purpose), but a more general relationship. It is possible that the relationship was present in earlier versions of the formalization, which explains its presence in the above-mentioned list. The assumption constitutes an analytic truth. Fr. Bocheński mentioned that in his commentary after the presentation of the very formalization of the Second Way. Now, he reminds about it for the second time. Assumption 8. (2.3203)

∀ ~ PR( x, x) x

Kein Gegenstand ist vor sich selbst. (Ia; IIa; III) Żaden przedmiot nie jest przed sobą samym. (Ib) Nic nie jest przed sobą samym. (IIb) Nothing exists before itself. The premise from the formalization is:

∀ ~ PR( x, x) x

With reference to this assumption, we can see the total correspondence between the expressions from the formalization and from the list of assumptions as well as the text of Summa. For Fr. Bocheński, it is also an analytic truth. Assumption 9. I, III:

∀{v( x) → ∃ [CA( x, z ) ∧ z ≠ x]} x


Alles, was verursacht ist, wird durch etwas anderas verursacht. (Ia; III)

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Wszystko, co jest spowodowane, spowodowane jest przez coś innego. (Ib)23 Everything that is caused, is caused by something else. The premise from the formalization is: ∀ [v( x) → ∃ CA( x, y )] x



∀[v( x) → ∃ CA( x, z )] x


Alles, was verursacht ist, wird durch etwas anderas verursacht. (IIa) Dla każdego skutku istnieje przyczyna. (IIb) For every effect there exists its cause. The premise from the formalization is: ∀ [v( x) → ∃ CA( x, y )] x


The relationship quoted among the philosophical assumptions, expressing the rule of causality, and marked with number 9, exists on the list of assumptions in two versions and with two different commentaries. The problem lies in the manuscripts. The relationship provided there on the basis of the Classical First Order Predicate Logic with Identity is weaker than in I and III. In version IIa, the relationship was accompanied by the commentary from version I and III, which was inadequate to the formal notation used in this version. The relationship from I and III cannot be found in the very formalization, but it is possible to find the relationship from II. It exists there as a definition.


In the original text (Ia and Ib) the following expression is used: ( x) : v( x). ⊃ (∃z ) . CA( y, z ) . ~ ( x = y )

It states that if something is caused, then something else different from it is caused by something. Such a relationship fails to meet its description. Thus, it has been treated as a spelling mistake.

Formalizations of the Argument Ex Causa Efficientis...


Besides, in the case of this assumption, Fr. Bocheński does not provide the fragment from Summa, to which he refers. In the commentary to the list of assumptions, he only mentions that assumptions 4 and 10 are assumed in Aquinas’ text tacitly. That is why, in their case there is no specific sentence or sentences to support the relevant formal expressions. However, also in the cases of assumptions 9 and 15 no references to St. Thomas’ text are made. To support this assumption, it is only possible to find the following sentence in the text of the Second Way: (2.3206) Now to take away the cause is to take away the effect. The sentence says that everything that constitutes the effect has its cause. It would be in line with everything that is found in the expressions from the formalization. However, on the basis of this sentence it is impossible to decide that the cause differs from the effect. Such a claim can be found on the list of assumptions in versions I and III. Assumption 10. ∀ ∀ [CA( x, y ) → ∃ CA( y, z )] I, II: x y


Wenn etwas verursacht wird, dann gibt es etwas, das den Verursachenden (selbst) verursacht. (Ia; IIa) Jeśli coś jest spowodowane, to istnieje coś, co powoduje powodującego. (Ib) Każda przyczyna ma przyczynę. (IIb) If something is caused, then there exists something else that causes the causing. (Every cause has a cause). The premise from the formalization is: ∀ ∀{CA( x, y ) ∧ x ≠ y → ∃ [CA( y, z ) ∧ y ≠ z ]} x y



∀ ∀ [CA( x, y ) ∧ v( y ) → ∃ CA( y, z )] x y


Marek Porwolik


Wenn etwas durch einen Verursachten verursacht wird, dann gibt es etwas, das den Verursachenden (selbst) verursacht. (III) The premise from the formalization is: ∀ ∀{CA( x, y ) ∧ x ≠ y ∧ v( y ) → ∃ [CA( y, z ) ∧ y ≠ z ]} x y


Commenting on his classification of the philosophical assumptions, Fr. Bocheński points to premise 10 as assumed tacitly in Summa. It can be the justification for the absence of any reference to Aquinas’ text in this very assumption. Regardless of the version of the text, Fr. Bocheński mentions the assumption that is absent from the formalization, despite the fact that the list of assumptions was supposed to organize those assumptions which were utilized in the formalization. Again, there are different versions of the formal text as well as the commentary. Apart from that, expression from I and II is stronger than that used in III, because the latter logically results from the former on the basis of the Classical First Order Predicate Logic with Identity. For that reason, the verbal commentary in III has been corrected. Obviously, the relationship from III can be considered closer to the thought of Aquinas, so the improvement of assumption 10 is expected, but it is still only correcting the assumption which is absent from the very formalization. Fr. Bocheński thinks that this premise, similarly to premise 9, constitutes a sentence expressing the rule of causality. Besides, we notice that assumption 10 from version I or II results in the fact that the first cause has a cause, which is not required by us. Assumption 11. (2.3204) I, II: {∃ ∃[CA( x, y ) ∧ x ≠ y ] → ∃ ∃ [CA( y, z ) ∧ y ≠ z ]} x y

y z

→ ~ (CA * ε Inf ) III: ∃ ∃{CA( x, y ) ∧ x ≠ y ∧ ∃ [CA( y, z ) ∧ y ≠ z ]} → ~ (CA * ε Inf ) x y


Formalizations of the Argument Ex Causa Efficientis...


Die so gebildete Reihe ist nicht unendlich. (Ia; IIa; III) Tak utworzony szereg nie jest nieskończony. (I; IIb) The series created in this way is not infinite. The premise from the formalization is: ∃ ∃{CA( x, y ) ∧ x ≠ y ∧ ∃ [CA( y, z ) ∧ y ≠ z ]} → ~ (CA * ε Inf ) x y


By calling premise 11 “the rule of finiteness”, Fr. Bocheński indicates the way he wants to interpret the expression ~(CA* ε Inf) in the Second Way. The name used by him shows that the series CA* is simply finite. Fr. Bocheński mentioned that earlier, when commenting on the differences between the First and the Second way. It is interesting that all the versions of formalization include the same formal expression, but on the list of assumptions there is only the formula from version III that corresponds with it. In this way, we find again a certain discrepancy, proving perhaps a certain evolution of Fr. Bocheński’s text. The next question arising concerns the correspondence between the formal expressions and the text of Summa. The antecedent of the expression from the formalization, and from version III of the list of assumption says about the existence of the first effective cause, which is different from both its efficient cause and its effect. The antecedent of the assumption from the list in versions I and II provides the relationship that if there exists the efficient cause of something different from itself, there exists the efficient cause of something different from itself. This expression is an interpretation of a tautology of the Classical Propositional Logic, saying that: p → p. Looking at the text of Summa, which says only about the fact that “it is impossible to follow the series of effective causes infinitely”, it seems that the way of formalization of this fragment carried out by Fr. Bocheński, regardless of the version, coming either from the formalization of the Second Way or from the list of assumptions, differs substantially from the text of Summa. All the more as Aquinas does not mention that the series might be possibly finite.

Marek Porwolik


At the end of our analysis, let us return to the very formalisms and let us consider their correctness with regard to the rules of logical resulting. Versions I and II are correct in this respect. However, it is possible to have some reservations with reference to version III. Because of the change of the assumption in verse 6 of version III, verse 12 does not result from 5, 6 and 11. The difficulty is solved in III in the following commentary, found in annotation 29: Ab (11) sollte folgendermaßen ergänzt werden: 11a

(∃x, y ) (CA( x, y ) . x ≠ y . v( y ) )



(∃y, z ) (CA( y, z ) . z ≠ x )

6, 11a


~ (CA * İ Inf )

11a, 12a, 5

und so weiter. Die Prämisse (11a), wenn falsch, impliziert wegen (11) die Existenz nicht verursachten Ursache, einer causa prima. Also gleichgültig, ob die Annahme (12a) wahr oder falsch ist, gilt (∃x) Cp(x). (A. d. H.) The above completion, which was introduced by the publishers of version III, is supposed to make the proof logically correct. In verse 12 it should be: (∃y, z) (CA(y, z) . z ≠ y). In this way, the correctness was reached at the price of abandoning the way of formalization and argumentation assumed by Fr. Bocheński as well as Aquinas’ course of thinking. Verse 12 in version III does not result logically from verses 5, 6, and 11. The following schema (marked with letter k by Fr. Bocheński), not belonging to the Classical First Order Predicate Logic with Identity, was used in it: ∃ ∃ [ ( x, y ) ∧ ∃ ( y, z )] → p x y


∀ ∀ [ ( x, y ) ∧ ( y ) → ∃ ( y, z )] x


∃ ∃ ( x, y ) x y



Formalizations of the Argument Ex Causa Efficientis...


Let us consider the following interpretation. Let the variables x, y, z, ... run over the set of natural numbers, that is, the valuation v of the individual variables is a function such that: υ: {x, y, z, …}→ N. We accept the interpretation in N = (N, · , >, =) such that: N╞υΦ(x,y) ≡ υ(x)·υ(y) = 2 N╞υΨ(x,y) ≡ υ(x)·υ(y) = 2 and υ(y) > 3 N╞υΧ(x) ≡ υ(x) ≠ υ(x) and p = Δ(z), where: N╞υΔ(z) ≡ υ(z) > υ(z). This interpretation shows that our schema is not a rule of the Classical First Order Predicate Logic with Identity. Because of the defects in version III, presented above, it is justifiable to attempt at emending the version. One of such emendations is suggested below. Version IV: 1.

∀ ∀ [CA( x, y ) ∧ x = y → PR( x, x)]



∀ ~ PR( x, x)



∃ ∃ [CA( x, y ) ∧ v( y )]



∃ ∃{CA( x, y ) ∧ x ≠ y ∧ ∃ [CA( y, z ) ∧ y ≠ z ]}

x y


x y

x y



→ ~ (CA * ε Inf ) 5.

∀ ∀{CA( x, y ) ∧ x ≠ y ∧ v( y ) x y


→ ∃[CA( y, z ) ∧ y ≠ z ]} z


~ (CA * ε Inf ) → ∃ Cp ( x) x


Marek Porwolik



∀ [Cp ( x) → D( x)]



∀ ∀ [CA( x, y ) → x ≠ y ]

(1, 2)


∃ ∃ [CA( x, y ) ∧ x ≠ y ∧ v( y )]

(3, 8)


~ (CA * ε Inf )

(4, 5, 9)


∃ Cp (x)

(6, 10)


∃ D(x)

(7, 11)


x y

x y



In comparison with version III, assumptions 3 and 4 from it were replaced with one saying that there exists such y which causes something, being caused at the same time. First, such an empirical premise makes version III a logically correct formalism. Second, it can be accepted as a kind of interpretation of sentence 2.3201 from Summa, saying about perceiving the order of efficient causes among the sensual things around us.

5. Conclusion The formalizations of the Second Way presented by Fr. Bocheński are provided in three versions. The last of them, published in his book, includes emendations introduced by the publishers, which were not authorized by him. The three versions are different from one another at various levels. Unfortunately, even the latest version is not free from inaccuracies. Although the hilosopher of Fribourg enjoyed preparing lists and comparisons, he made numerous mistakes in them. The mistakes refer to the type and place of the application of the rules of inference, as well as the premises applied in the formalizations. This fact proves that the formal record evolved, but the emendations did not follow the evolution, although it was necessary to introduce

Formalizations of the Argument Ex Causa Efficientis...


them also into the text of the commentary. Only the formalization in the third version needs emendations that allow it to be kept logically correct. We did not attempt either at a broader semantic analysis of the formalizations, or at discussing the manner of translating the text of Summa into the formal language, which can constitute a separate field of analysis. In spite of narrowing of the research, the aim of the study seems to be achieved: it was to show the context of creating the preliminary axiomatization of the theory of the Absolute understood as the first cause, which we find in Fr. Bocheński’s analysis of the Second Way.

Marie Duží VŠB-Technical University Ostrava

Ambiguities in Natural Langauge and Ontological Proofs* 1. Introduction


natural language we encounter many features of ambiguity, where one term/expression has more than one meaning. A logical analysis of such a piece of natural language should translate each of its unambiguous meanings into a logically perfect notation. Frege’s Begriffsschrift was the first major attempt in modern logic to create such a notation (though he primarily intended it for mathematical language).1 There are various origins and various manifestations of ambiguity, not least the cases bearing on quantifier scopes, like ‘Every boy dances with one girl.’ Another sort of examples is ‘John loves his wife, and so does Peter,’ which is ambiguous between Peter loving John’s wife and Peter loving his own wife, because it is ambiguous which property ‘so’ picks up.2 A third, and perhaps less-noticed, sort of ambiguity is pivoted on topic and focus articulation of a sentence. For instance, ‘John only introduced Bill to Sue,’ to use Hajičová’s example,3 This contribution was made possible through the support of a grant “The Limits of Scientific Explanation” from the John Templeton Foundation. 1 See G. Frege, Die Grundlagen der Arithmetik, W. Koebner, Breslau 1884. 2 See S. Neale, This, That, and the Other, [in:] A. Bezuidenhout and M. Reimer (eds), Descriptions and Beyond, Oxford University Press, Oxford 2004, p. 68-182; and also M. Duží and B. Jespersen, Procedural Isomorphism, Analytic Information, and Betaconversion by Value, “Logic Journal of the IGPL” (forthcoming). 3 See E. Hajičová, What We Are Talking About and What We Are Saying About It, [in:] A. Gelbukh (ed.), Computational Linguistics and Intelligent Text Processing, vol. 4919, Springer-Verlag LNCS, Berlin–Heidelberg 2008, p. 241-262. *


Marie Duží

lends itself to two different kinds of construal: ‘John did not introduce other people to Sue except for Bill’ and ‘The only person Bill was introduced to by John was Sue.’ There are two sentences whose semantics, logical properties and consequences only partially overlap. The fourth sort of ambiguity is pivoted on de dicto vs. de re way in which a constituent of a sentence is conceptualised. To put it another way, this sort of ambiguity concerns intensional vs. extensional reading of a sentence. For instance, the sentence ‘God is omnipotent’ can be read in two different ways. On its extensional reading, that is ‘God (whoever this being is) has the property of being omnipotent,’ the sentence presupposes that there is a unique God. If it is not so then the sentence has no truth value. Thus, the extensional variant of the sentence is empirical. We cannot logically prove that there is God. The intensional variant is to be understood like this: ‘Omnipotence is among the requisites of God.’ On this reading, the sentence is analytically true. Topic-focus, de dicto/de re and intensional/extensional ambiguity have much in common, and in this paper I will deal with ambiguities of this kind. I am going to show that these ambiguities also crop up in the ontological proofs of the existence of God and I intend to illustrate how these proofs (with the exception of St. Anselm’s arguments of Proslogion III)4 are flawed by not respecting these ambiguities. Based on an analysis of topic-focus articulation, I proposed a solution5 to the almost a hundred-year old dispute over Strawsonian vs. Russellian definite descriptions.6 The point of departure for this 4 For the analysis of St. Anselm’s arguments see M. Duží, St. Anselm’s Ontological Arguments, “Polish Journal of Philosophy” 2011, vol. V, no. 1, p. 7-37. 5 M. Duží, Strawsonian vs. Russellian Definite Descriptions, “Organon F” 2009, vol. XVI, no. 4, p. 587-614. 6 See, for instance, K.S. Donnellan, Reference and Definite Descriptions, “Philosophical Review” 1966, vol. 77, p. 281-304; K. von Fintel, Would You Believe It? The King of France is Back! (Presuppositions and Truth-Value Intuitions), [in:] M. Reimer, A. Bezuidenhout (eds), Descriptions and Beyond, Clarendon Press, Oxford 2004, p. 315-341; S. Neale, Descriptions, MIT Press Books, Cambridge 1990; B. Russell, On denoting, “Mind” 1905, vol. 14, p. 479-493, and Mr. Strawson on Referring, “Mind” 1957, vol. 66, p. 385-389; P.F. Strawson, On referring, “Mind” 1950, vol. 59, p. 320334, and Identifying Reference and Truth-Values, “Theoria” 1964, vol. 3, p. 96-118.

Ambiguities in Natural Langauge and Ontological Proofs


solution is that sentences of the form ‘The F is a G’ are ambiguous. Their ambiguity is not rooted in a shift of meaning of the definite description ‘the F’. Rather, the ambiguity stems from different topicfocus articulations of such sentences. Russell and Strawson took themselves to be at loggerheads, whereas, in fact, they spoke at cross purposes. The received view still tends to be that there is room for at most one of the two positions, since they are deemed incompatible. And they are, of course, incompatible if they must explain the same set of data. But they should not, in my view. One theory is excellent at explaining one set of data, but poor at explaining the data that the other theory is excellent at explaining; and vice versa. My novel contribution advances the research into definite descriptions by pointing out how progress has been hampered by a false dilemma and how to move beyond that dilemma. The point is this. If ‘the F’ is the topic phrase, then this description occurs with de re supposition and Strawson’s analysis appears to be what is wanted. On this reading the sentence presupposes the existence of the descriptum of ‘the F’. The other option is ‘G’ occurring as the topic and ‘the F’ as focus. This reading corresponds to Donnellan’s attributive use of ‘the F’ and the description occurs with de dicto supposition. On this reading the Russellian analysis gets the truth-conditions of the sentence right. The existence of a unique F is merely entailed. Ancillary to my analysis is a general analytic schema of sentences coming with a presupposition. This analysis makes use of a definition of the if-then-else connective known from programming languages. A broadly accepted view of the semantic nature of this connective is that it is a so-called non-strict function that does not comply with the principle of compositionality. However, the semantic nature of the connective is contested among computer scientists. I showed – and this is also a novel contribution of mine – that there is no cogent reason for embracing a non-strict definition and contextdependent meaning, provided a higher-order logic making it possible to operate on hyperintensions is applied. The framework of Tichý’s Transparent Intensional Logic (TIL) possesses sufficient expressive


Marie Duží

power, and will figure as a background theory throughout my exposition in this paper.7 Tichý’s TIL was developed simultaneously with Montague’s Intensional Logic, IL.8 The technical tools of disambiguation will be familiar from IL, with two exceptions. One is that we λ-bind separate variables w, w1, …, wn ranging over possible worlds and t, t1, …, tn ranging over times. This dual binding is tantamount to explicit intensionalization and temporalization. The other exception is that functional application is the logic both of extensionalization of intensions (functions from possible worlds) and of predication.9 Application is symbolized by square brackets: ‘[…]’. Intensions are extensionalized by applying them to worlds and times, as in [[Intension w] t], abbreviated by subscripted terms for world and time variables: Intensionwt is the extension of the generic intension Intension at 〈w, t〉. Thus, for instance, the extensionalization of a property yields a set (possibly an empty one), and the extensionalization of a proposition yields a truth-value (or no value at all). A general objection to Montague’s IL is that it fails to accommodate hyperintensionality as, indeed, any formal logic interpreted set-theoretically is bound to, unless a domain of primitive hyperintensions is added to the frame. Any theory of natural language analysis needs a hyperintensional (preferably procedural) semantics in order to crack the hard nuts of natural language semantics. In global terms, without procedural semantics TIL is an anti-contextualist (i.e. transparent), explicitly intensional modification of 7 For details on TIL see, in particular, M. Duží, B. Jespersen, and P. Materna, Procedural Semantics for Hyperintensional Logic; Foundations and Applications of Transparent Intensional Logic, series Logic, Epistemology, and the Unity of Science, vol. 17, Springer, Berlin 2010, p. 550; P. Tichý, The Foundations of Frege’s Logic, De Gruyter, New York, Berlin 1988, and by the same author, Collected Papers in Logic and Philosophy, V. Svoboda, B. Jespersen, C. Cheyne (eds), Filosofia, Czech Academy of Sciences, Prague and University of Otago Press, Dunedin 2004. 8 For a detailed critical comparison of TIL and Montague’s IL, see M. Duží et al., Procedural Semantics..., op. cit., p. 85-102, § 2.4.3. 9 For details, see B. Jespersen, Predication and Extensionalization, “Journal of Philosophical Logic” 2008, vol. 37, p. 479-499.

Ambiguities in Natural Langauge and Ontological Proofs


IL. With procedural semantics, TIL rises above the model-theoretic paradigm and joins instead the paradigm of hyperintensional logic and structured meanings. The structure of this chapter is as follows. Section 2 is an introduction to TIL. I introduce the semantic and logical foundations of TIL here. Sections 3 and 4 contain the main results of this study. In Section 3 I propose a solution to the dispute over Strawsonian vs. Russellian definite descriptions. Section 4 introduces the problem of ambiguities stemming from different topic-focus articulation and I present here a solution. Moreover, I present generalization of the method of topic-focus disambiguation to sentences containing not only definite descriptions but also general terms occurring with different suppositions. To this end I make use of the strict analysis of the if-then-else function that is defined in paragraph 4.1. Section 5 presents a common fault that can be found in ontological proofs of God’s existence, with the exception of Anselm’s argument of Proslogion III, where St. Anselm carefully avoids this mistake. Finally, Section 6 summarizes the results.

2. The foundations of TIL TIL is an overarching semantic theory for all sorts of discourse, whether colloquial, scientific, mathematical or logical. The theory is a procedural (as opposed to denotational) one, according to which sense is an abstract, extra-linguistic procedure detailing what operations to apply to what procedural constituents to arrive at the product (if any) of the procedure. Such procedures are rigorously defined as TIL constructions. The semantics is tailored to the hardest case, as constituted by hyperintensional contexts, and generalized from there to simpler intensional and extensional contexts. This entirely anticontextual and compositional semantics is, to the best of my knowledge, the only one that deals with all kinds of context in a uniform way. Thus we can characterize TIL as an extensional logic of hyper-


Marie Duží

intensions.10 The sense of an empirical sentence is an algorithmically structured construction of the proposition denoted by the sentence. The denoted proposition is a flat, or unstructured, mapping with domain in a logical space of possible worlds. Our motive for working ‘top-down’ has to do with anti-contextualism: any given unambiguous term or expression (even one involving indexicals or anaphoric pronouns) expresses the same construction as its sense whatever sort of context the term or expression is embedded in. And the meaning of an expression determines the respective denoted entity (if any), but not vice versa. The denoted entities are (possibly 0-ary) functions understood as set-theoretical mappings. Thus, we strictly distinguish between a procedure (construction) and its product (here, a constructed function), and between a function and its value. What makes TIL suitable for the job of disambiguation is the fact that the theory construes the semantic properties of the sense and denotation relations as remaining invariant across different sorts of linguistic contexts.11 Thus logical analysis disambiguates ambiguous expressions in such a way that an ambiguous expression is furnished with more than one context-invariant meaning, that is TIL construction. However, logical analysis cannot dictate which disambiguation is the intended one. It falls to pragmatics to select the intended one. The context-invariant semantics of TIL is obtained by universalizing Frege’s reference-shifting semantics custom-made for ‘indirect’ contexts.12 The upshot is that it becomes trivially true that all contexts are transparent, in the sense that pairs of terms that are co-denoting outside an indirect context remain co-denoting inside an indirect context and vice versa. In particular, definite descripFor the most recent application, see M. Duží and B. Jespersen, Transparent Quantification into Hyperpropositional Contexts de re, “Logique et Analyse” 2012, vol. 220 (forthcoming). 11 Indexicals being the only exception: while the sense of an indexical remains constant, its denotation varies in keeping with its contextual embedding. See M. Duží et al. Procedural Semantics..., op. cit., § 3.4. 12 See G. Frege, Über Sinn und Bedeutung, “Zeitschrift für Philosophie und philosophische Kritik” 1892, vol. 100, p. 25-50. 10

Ambiguities in Natural Langauge and Ontological Proofs


tions that only contingently describe the same individual never qualify as co-denoting. Our term for the extra-semantic, factual relation of contingently describing the same entity is ‘reference’, whereas ‘denotation’ stands for the intra-semantic, pre-factual relation between two words that pick out the same entity at the same world/time-pairs. The syntax of TIL is Church’s (higher-order) typed λ-calculus, but with the all-important difference that the syntax has been assigned a procedural (as opposed to denotational) semantics. Thus, abstraction transforms into the molecular procedure of forming a function, application into the molecular procedure of applying a function to an argument, and variables into atomic procedures for arriving at their values. Furthermore, TIL constructions represent our interpretation of Frege’s notion of Sinn (with the exception that constructions are not truth-bearers; instead, some present either truth-values or truthconditions) and are kindred to Church’s notion of concept. Constructions are linguistic senses as well as modes of presentation of objects and are our hyperintensions. While the Frege-Church connection makes it obvious that constructions are not formulae, it is crucial to emphasize that constructions are not ‘functions(-in-extension)’, either. They might be explicated as Church’s ‘functions-in-intension’, but we do not use the term ’function-in-intension’, because Church did no define it (he only characterized functions-in-intension as rules for presenting functions-in-extension). Rather, technically speaking, some constructions are modes of presentation of functions, including 0-place functions such as individuals and truth-values, and the rest are modes of presentation of other constructions. Thus, with constructions of constructions, constructions of functions, functions, and functional values in our stratified ontology, we need to keep track of the traffic between multiple logical strata. The ramified type hierarchy does just that. What is important about this traffic is, first of all, that constructions may themselves figure as functional arguments or values. Thus we consequently need constructions of one order higher in order to present those being arguments or values of functions. With

Marie Duží


both hyperintensions and possible-world intensions in its ontology, TIL has no trouble assigning either hyperintensions or intensions to variables as their values. However, the technical challenge of operating on constructions requires two (occasionally three) interrelated, non-standard devices. The first is Trivialization, which is an atomic construction, whose only constituent part is itself. The second is the function Sub (for ‘substitution’). (The third is the function Tr, for ‘Trivialization’, which takes an object to its Trivialization.) We say that Trivialization is used to mention other constructions.13 The point of mentioning a construction is to make it, rather than what it presents, a functional argument. Hence for a construction to be mentioned is for it to be Trivialized; in this way the context is raised up to a hyperintensional level. Our neo-Fregean semantic schema, which applies to all contexts, is this triangulation: Expression

Construction expresses

Denotation constructs

denotes The most important relation in this diagram is between an expression and its meaning, i.e. a construction. Once constructions have been defined, we can logically examine them; we can investigate a priori what (if anything) a construction constructs and what is entailed by it. Thus meanings (i.e. constructions) are semantically primary, denotations secondary, because an expression denotes an object (if any) via its meaning, that is a construction expressed by the expression. Once a construction is explicitly given as a result of logical analysis, the entity (if any) it constructs is already implicitly The use/mention distinction normally applies only to words; in TIL it applies to the meanings of words (i.e. constructions). See M. Duží et al., Procedural Semantics..., op. cit., §2.6. In theory, a construction may be mentioned by another construction than Trivialization, but in this paper we limit ourselves to Trivialization. 13

Ambiguities in Natural Langauge and Ontological Proofs


given. As a limiting case, the logical analysis may reveal that the construction fails to construct anything by being improper. In order to put our framework on a more solid ground, we now present particular definitions. First, we set out the definitions of firstorder types (regimented by a simple type theory), constructions, and higher-order types (regimented by a ramified type hierarchy), which taken together form the nucleus of TIL, accompanied by some auxiliary definitions. The type of first-order objects includes all objects that are not constructions. Therefore, it includes not only the standard objects of individuals, truth-values, sets, etc., but also functions defined on possible worlds (i.e. the intensions germane to possible-world semantics). Sets, for their part, are always characteristic functions and insofar extensional entities. But the domain of a set may be typed over higher-order objects, in which case the relevant set is itself a higherorder object. Similarly for other functions, including relations, with domain or range in constructions. That is, whenever constructions are involved, we find ourselves in the ramified type hierarchy. The definition of the ramified hierarchy of types decomposes into three parts: firstly, simple types of order 1; secondly, constructions of order n; thirdly, types of order n + 1. Definition 1 (types of order 1). Let B be a base, where a base is a collection of pair-wise disjoint, non-empty sets. Then: (i) Every member of B is an elementary type of order 1 over B. (ii) Let α, β1, ..., βm (m > 0) be types of order 1 over B. Then the collection (α β1 ... βm) of all m-ary partial mappings from β1 × ... × βm into α is a functional type of order 1 over B. (iii) Nothing is a type of order 1 over B, unless it so follows from (i) and (ii).


Marie Duží

Definition 2 (construction). (i) The Variable x is a construction that constructs an object X of the respective type dependently on a valuation v; x v-constructs X. (ii) Trivialization: Where X is an object whatsoever (an extension, an intension or a construction), 0X is the construction Trivialization. It constructs X without any change. (iii) The Composition [X Y1…Ym] is the following construction. If X v-constructs a function f of a type (αβ1…βm), and Y1, …, Ym v-construct entities B1, …, Bm of types β1, …, βm, respectively, then the Composition [X Y1…Ym] v-constructs the value (an entity, if any, of type α) of f at the tuple-argument 〈B1, …, Bm〉. Otherwise the Composition [X Y1…Ym] does not v-construct anything and so is v-improper. (iv) The Closure [λx1…xm Y] is the following construction. Let x1, x2, …, xm be pair-wise distinct variables v-constructing entities of types β1, …, βm and Y a construction v-constructing an α-entity. Then [λx1 … xm Y] is the construction λ-Closure (or Closure). It v-constructs the following function f of the type (αβ1…βm). Let v(B1/x1,…,Bm/xm) be a valuation identical with v at least up to assigning objects B1/β1, …, Bm/βm to variables x1, …, xm. If Y is v(B1/x1,…,Bm/xm)-improper (see iii), then f is undefined at the argument 〈B1, …, Bm〉. Otherwise the value of f at 〈B1, …, Bm〉 is the α-entity v(B1/x1,…,Bm/xm)-constructed by Y. (v) The Single Execution 1X is the construction that either v-constructs the entity v-constructed by X or, if X v-constructs nothing, is v-improper (yielding nothing relative to v). (vi) The Double Execution 2X is the following construction. Where X is any entity, the Double Execution 2X is v-improper (yielding nothing relative to v) if X is not itself a construction, or if X does not v-construct a construction, or if X v-constructs a v-improper construction. Otherwise, let X v-construct a construction Y and Y v-construct an entity Z: then 2X v-constructs Z. (vii) Nothing is a construction, unless it so follows from (i) through (vi).

Ambiguities in Natural Langauge and Ontological Proofs


Definition 3 (ramified hierarchy of types). T1 (types of order 1). See Definition 1. Cn (constructions of order n) i) Let x be a variable ranging over a type of order n. Then x is a construction of order n over B. ii) Let X be a member of a type of order n. Then 0X, 1X, 2X are constructions of order n over B. iii) Let X, X1,..., Xm (m > 0) be constructions of order n over B. Then [X X1... Xm] is a construction of order n over B. iv) Let x1,...xm, X (m > 0) be constructions of order n over B. Then [λx1...xm X] is a construction of order n over B. v) Nothing is a construction of order n over B, unless it so follows from Cn (i)-(iv). Tn+1 (types of order n + 1). Let ∗n be the collection of all constructions of order n over B. Then i) ∗n and every type of order n are types of order n + 1. ii) If m > 0 and α, β1,...,βm are types of order n + 1 over B, then (α β1 ... βm) (see T1 ii)) is a type of order n + 1 over B. iii) Nothing is a type of order n + 1 over B, unless it so follows from Tn+1 (i) and (ii). Remark. For the purposes of natural-language analysis, we are currently assuming the following base of ground types, which is part of the ontological commitments of TIL: ο: the set of truth-values {T, F} ι: the set of individuals (the universe of discourse) τ: the set of real numbers (doubling as discrete times) ω: the set of logically possible worlds (the logical space). Empirical languages incorporate an element of contingency, because they denote empirical conditions that may or may not be satisfied at some world/time pair of evaluation. Non-empirical languages (in particular the language of mathematics) have no need for an additional category of expressions for empirical conditions. We model these


Marie Duží

empirical conditions as possible-world intensions. They are entities of type (βω): mappings from possible worlds to an arbitrary type β. The type β is frequently the type of the chronology of α-objects, i.e. a mapping of type (ατ). Thus α-intensions are frequently functions of type ((ατ)ω), abbreviated as ‘ατω’. Extensional entities are entities of a type α where α ≠ (βω) for any type β. Examples of frequently used intensions are: propositions of type οτω, properties of individuals of type (οι)τω, binary relations-in-intension between individuals of type (οιι)τω, individual offices/roles of type ιτω. Our explicit intensionalization and temporalization enables us to encode constructions of possible-world intensions, by means of terms for possible-world variables and times, directly in the logical syntax. Where variable w ranges over possible worlds (type ω) and t over times (type τ), the following logical form essentially characterizes the logical syntax of any empirical language: λwλt […w….t…]. Where α is the type of the object v-constructed by […w….t…], by abstracting over the values of variables w and t, we construct a function from worlds to a partial function from times to α, that is a function of type ((ατ)ω), or ‘ατω’ for short. Logical objects like truth-functions and quantifiers are extensional: ∧ (conjunction), ∨ (disjunction) and ⊃ (implication) of type (οοο), and ¬ (negation) of type (οο). The quantifiers ∀α, ∃α are typetheoretically polymorphous functions of type (ο(οα)), for an arbitrary type α, defined as follows. The universal quantifier ∀α is a function that associates a class A of α-elements with T if A contains all elements of the type α, otherwise with F. The existential quantifier ∃α is a function that associates a class A of α-elements with T if A is a non-empty class, otherwise with F. Another logical object we need is a partial polymorphic function Singularizer Iα of type (α(οα)). A singularizer is a function that associates a singleton S with the only member of S, and is otherwise (i.e. if S is an empty set or a multi-element set) undefined. Below, all type indications will be provided outside the formulae in order not to clutter the notation. Furthermore, ‘X/α’ means that

Ambiguities in Natural Langauge and Ontological Proofs


an object X is (a member) of type α. ‘X →v α’ means that the type of the object v-constructed by X is α. We write ‘X → α’ if what is v-constructed does not depend on a valuation v. This holds throughout: w →v ω and t →v τ. If C →v ατω then the frequently used Composition [[C w] t], which is the intensional descent (a.k.a. extensionalization) of the α-intension v-constructed by C, will be encoded as ‘Cwt’. When using constructions of truth-functions, we often omit Trivialisation and use infix notation to conform to standard notation in the interest of better readability. Also, when using constructions of identities of α-entities, =α/(οαα), we omit Trivialization, the type subscript, and use infix notion when no confusion can arise. For instance, instead of ‘[0⊃ [0=ι a b] [0=((οτ)ω) λwλt [Pwt a] λwλt [Pwt b]]]’, where =ι/(οιι) is the identity of individuals and =((οτ)ω)/(οοτωοτω) the identity of propositions; a, b constructing objects of type ι, P objects of type (οι)τω, we write ‘[[a = b] ⊃ [λwλt [Pwt a] = λwλt [Pwt b]]]’. We invariably furnish expressions with procedural structured meanings, which are explicated as TIL constructions. The analysis of an unambiguous sentence thus consists in discovering the logical construction encoded by a given sentence. The TIL method of analysis consists of three steps: 1. Type-theoretical analysis, i.e. assigning types to the objects that receive mention in the analysed sentence. 2. Type-theoretical synthesis, i.e. combining the constructions of the objects ad (1) in order to construct the proposition of type οτω denoted by the whole sentence. 3. Type-theoretical checking, i.e. checking whether the proposed analysans is type-theoretically coherent.

Marie Duží


To illustrate the method, let us analyse the sentence (A)

“The baptism site of Jesus is Bethany beyond the Jordan”.

Ad 1. The sentence talks about the baptism site, Jesus and Bethany beyond the Jordan. Thus, we have: Baptism_site_of/(ιι)τω: an empirical function assigning to an individual another individual, viz. the baptism site; Jesus/ι; Bethany(_beyond_the_ Jordan)/ι. Ad 2 and 3. Now, we have to compose the constructions of the objects ad 1 so that to construct the proposition denoted by the whole sentence, and respect type-theoretical constraints. Here is how. First, the definite description ‘the baptism site of Jesus’ does not denote a particular individual; rather, it denotes an individual role of type ιτω. Jesus could have been baptised anywhere else or nowhere. The only testimony that Jesus was baptised is the Bible: “Then Jesus came from Galilee to John at the Jordan to be baptised by him”. (Matthew 3:13) Moreover, that at this very spot history was made was only recently discovered after becoming lost for centuries. To construct this role, we first extensionalise the intension Baptism_site_of by composing it with w, [0Baptism_site_of w] → ((ιι)τ), and t, [[0Baptism_site_of w] t] → (ιι), or 0Baptism_site_ofwt → (ιι) for short. Applying this function to Jesus/ι we obtain another individual: [0Baptism_site_ofwt 0Jesus] → ι. Finally, by abstracting over the values of variables w and t we construct the role/office: λwλt [0Baptism_site_ofwt 0Jesus] → ιτω.

Ambiguities in Natural Langauge and Ontological Proofs


Our sentence states that this office is occupied by Bethany. Thus we must extensionalise the above office again to obtain an individual and apply the identity function to this individual and Bethany: [[0Baptism_site_ofwt 0Jesus] = 0Bethany]. Since the sentence is empirical (it might have been otherwise), we again abstract over the values of w, t to obtain the proposition: (A*)

λwλt [[0Baptism_site_ofwt 0Jesus] = 0Bethany].

Gloss: (A*) is the meaning expressed by the sentence (A). Hence the sentence encodes an instruction how in any possible world w at any time t to evaluate its truth-conditions. Read this instruction like this: in any possible world (λw), at any time (λt), apply the function Baptism_site_ofwt to Jesus to obtain the particular site and check whether this spot is identical with Bethany. This is exactly what has happened as the letters of authentication testify: “The overwhelming biblical, archaeological, and historical evidence has led many religious leaders throughout the world to recognize this site as the authentic site of Jesus’ baptism”.14

3. Definite descriptions: Strawsonian or Russellian? Now I am going to propose a solution to the well-known Strawson-Russell standoff. In other words, I am going to analyse the phenomena of presupposition and entailment connected with using definite descriptions with supposition de dicto or de re, and I will show how the topic-focus distinction determines which of the two cases applies. Before presenting a summary of the de dicto/de re distinction, let me briefly introduce three kinds of context in which a construction can occur. They are characterized as follows: 14


Marie Duží



hyperintensional contexts: the kind of context in which a construction is not used to v-construct an object; rather, it is itself mentioned as a functional argument (though a hyperintension of one order higher needs to be used to mention this lower-order construction), ii) intensional contexts: this is a context generated by a construction that is used to present a function (mapping, possibly of 0-arity) without presenting a particular value of the function; moreover, the construction does not occur within another hyperintensional context, because a higher-order context is dominant over a lower one, iii) extensional contexts: a construction is used to produce a particular value of the v-constructed function at a given argument; moreover, the construction does not occur within another intensional or hyperintensional context. Using a construction of an intension either with de dicto or de re supposition is closely connected with the intensional or extensional occurrence of a constituent. Thus, the de dicto/de re distinction can be briefly characterized like this. The distinction concerns constructions that construct intensions. The schema of such a construction is λwλt [… w … t …]. Now if a construction C occurs in the Composition [… w … t …] intensionally, then we say that C occurs with de dicto supposition, if C occurs in this Composition extensionally, then C occurs with de re supposition. Examples. Consider, for instance, the sentences: (1) ‘The Pope is a German.’ (2) ‘Joseph Ratzinger became Pope on April 19, 2005.’ Sentence (1) expresses the construction: (1’) λwλt [0Germanwt 0Popewt], whereas (2) expresses:

Ambiguities in Natural Langauge and Ontological Proofs


λwλt [0Pastt λc ∃t’[[c t’] ∧ [0Becomewt’ 0Ratzinger 0Pope]] 0 April19]. Types: German/(οι)τω; Pope/ιτω; Past/(ο(ο(οτ))(οτ))τ; Become/ (οιιτω)τω; Ratzinger/ι; April19/(οτ); c→(οτ); t, t’→τ.


The Trivialization 0Pope→ιτω occurs extensionally in [0Germanwt 0 Popewt], because it is used to v-construct the value of an ι-office. Thus, 0Pope occurs with de re supposition in (1’), as only the value v-constructed by [0Germanwt 0Popewt] matters, the other values being irrelevant. On the other hand, in (2’) 0Pope occurs intensionally. The constituent 0Pope is used to construct an ι-office rather than a particular value. The whole papal office matters in the truth-conditions of the proposition constructed by (2’) rather than just a particular value. Thus, 0Pope occurs with de dicto supposition in (2’). Consequently, for (1) the two principles de re are valid. If Ratzinger is the Pope, then from (1) we may validly infer that Ratzinger is a German, whereas we cannot validly infer from (2) that Ratzinger became Ratzinger. Moreover, (1) not only implies but even presupposes – unlike (2) – that the Pope should exist.

3.1. Topic-focus ambiguity When used in a communicative act, a sentence communicates something (the focus F) about something (the topic T). Thus, the schematic structure of a sentence is F(T). The topic T of a sentence S is often associated with a presupposition P of S such that P is entailed both by S and non-S. On the other hand, the clause in the focus usually occasions a mere entailment of some P by S. To give an example, consider the sentence ‘Our defeat was caused by John’.15 There are two possible readings of this sentence.

This and some other examples were taken from E. Hajičová, What We Are Talking About..., op. cit. 15


Marie Duží

Taken one way, the sentence is about our defeat, conveying the snippet of information that it was caused by John. In such a situation, the sentence is associated with the presupposition that we were defeated. Indeed, the negated form of the sentence, ‘Our defeat was not caused by John’, also implies that we were defeated. Thus, ‘our defeat’ is the topic and ‘was caused by John’ the focus clause. Taken the other way, the sentence is about the topic John, ascribing to him the property that he caused our defeat (focus). Now, the scenario of truly asserting the negated sentence can be, for instance, the following. Though it is true that John has a reputation for being rather a bad player, Paul was in excellent shape and so we won. Or, another scenario is possible. We were defeated, only not because of John but because the whole team performed badly. Hence, our being defeated is not presupposed by this reading, it is only entailed. Schematically, if ╞ is the relation of entailment, then the logical difference between a mere entailment and a presupposition is this: P is a presupposition of S: (S╞ P) and (non-S╞ P) Corollary: If P is not true, then neither S nor non-S is true. Hence, S has no truth-value. P is (S╞ P) and neither (non-S╞ P) nor (non-S╞ non-P) Corollary: If S is not true, then we cannot deduce anything about the truth-value of P. More precisely, the entailment relation obtains between hyperpropositions P, S; i.e. the meaning of P is analytically entailed or presupposed by the meaning of S. Thus: ╞/((ο∗n∗n) is defined as follows. Let CS, CP be constructions assigned to sentences S, P, respectively, as their meanings. Then S entails P

Ambiguities in Natural Langauge and Ontological Proofs


(CS╞ CP) iff the following holds:16 ∀w∀t [[0Truewt CS] ⊃ [0Truewt CP]]. Since we work with properly partial functions, we need to apply the propositional property True/(οοτω)τω, which returns T for those 〈w, t〉-pairs at which the argument proposition is true, and F in all the remaining cases. There are two other propositional properties: False and Undef, both of type (οοτω)τω. The three properties are defined as follows. Let P be a propositional construction (P/∗n → οτω). Then: [0Truewt P] v-constructs the truth-value T iff Pwt v-constructs T, otherwise F. 0 [ Falsewt P] v-constructs the truth-value T iff [¬Pwt] v-constructs T, otherwise F. 0 [ Undefwt P] v-constructs the truth-value T iff [¬[0Truewt P] ∧ ¬[0Falsewt P]] v-constructs T, otherwise F. Thus, we have: ¬[0Undefwt P] = [[0Truewt P] ∨ [0Falsewt P]] ¬[0Truewt P] = [[0Falsewt P] ∨ [0Undefwt P]] ¬[0Falsewt P] = [[0Truewt P] ∨ [0Undefwt P]] Hence, though we work with truth-value gaps, we do not work with a third truth-value, and our logic is, in this weak sense, bivalent.

3.2. The King of France revisited To illustrate the topic-focus ambiguity, I will now briefly summarise my proposed solution of 100-year dispute over Russellian vs.

For the general definition of entailment and the difference between analytical and logical entailment, see M. Duží, The Paradox of Inference and the Non-triviality of Analytic Information, “Journal of Philosophical Logic” 2010, vol. 39, no. 5, p. 473-510. 16

Marie Duží


Strawsonian definite descriptions.17 To this end, I will use the notoriously invoked sentence ‘The King of France is bald’. On its perhaps most natural reading, the sentence predicates the property of being bald (the focus) of the individual (if any), that is the present King of France (the topic). Yet there is another, albeit less natural, reading of the sentence. Imagine that the sentence is uttered in a situation when we are talking about baldness and somebody asks ‘Who is bald?’. The answer might be ‘Well, among those who are bald, there is the present King of France’. If you got such an answer, you would most probably protest, ‘This cannot be true, for there is no King of France now’. On such a reading, the sentence is about baldness (topic), claiming that this property is instantiated, among others, by the King of France (focus). Since there are no rigorous grammatical rules in English to distinguish between the two variants, the input of our logical analysis is the result of a linguistic analysis, where the topic and focus of a sentence are made explicit.18 Thus, I will mark the topic clause in italics. The two readings of the above sentence are: (S) (R)

‘The King of France is bald.’ ‘The King of France is bald.’

I am going to show that the two readings are not equivalent, because they have different truth-conditions. (S) is the variant with Strawsonian truth-conditions and (R) the variant that complies with Russellian truth-conditions.

See M. Duží, Strawsonian vs. Russellian Definite Descriptions, op. cit.; and also M. Duží, Resolving Topic-Focus Ambiguities in Natural Language, [in:] Muhammad Tanvir Afzal (ed.) Semantics in Action – Applications and Scenarios, InTech Europe, Croatia 2012, p. 239-266. 18 For instance, the Prague linguistic school created The Prague Dependency Treebank for the Czech language, which contains a large number of Czech texts with complex and interlink annotation on different levels. The tectogrammatical representation contains the semantic structure of sentences with topic-focus annotators. For details, see http://ufal.mff.cuni.cz/pdt2.0/. 17

Ambiguities in Natural Langauge and Ontological Proofs


The analysis of (S) is obtained in a very similar way as described in Section 2: λwλt [0Baldwt λwλt [0King_ofwt 0France]wt]. Types: Bald/(οι)τω; King_of/(ιι)τω; France/ι; λwλt [0King_ofwt 0 France] → ιτω: the individual office of the King of France. The meaning of ‘the King of France’, viz. λwλt [0King_ofwt 0 France], occurs in (S) with de re supposition, because the object of predication is the unique value (in a 〈w, t〉-pair of evaluation) of the office rather than the office itself.19 The following two de re principles are satisfied: the principle of existential presupposition and the principle of substitution of co-referential expressions. Thus, the following arguments are valid (though not sound): The King of France is/is not bald The King of France exists The King of France is bald The King of France is Louis XVI Louis XVI is bald. Here are the proofs. (a) Existential presupposition: First, existence is here a property of an individual office rather than of some non-existing individual (whatever it might mean for an individual not to exist). Thus we have Exist/(οιτω)τω. To prove the validity of the first argument, we define Exist/(οιτω)τω as the property of an office’s being occupied at a given world/time pair: Exist =of λwλt λc [0∃λx [x =i cwt]], i.e. [0Existwt c] =o [0∃λx [x =i cwt]].


For details on de dicto vs. de re supposition, see M. Duží et al., Procedural Semantics..., op. cit., esp. §§ 1.5.2 and 2.6.2; and also M. Duží, Intensional Logic and the Irreducible Contrast Between de dicto and de re, “ProFil” 2004, vol. 5, no. 1, p. 1-34. 19


Marie Duží

Types: ∃/(ο(οι)): the class of non-empty classes of individuals; c →v ιτω; x →v ι; =o/(οοο): the identity of truth-values; =of /(ο(οιτω)τω(οιτω)τω): the identity of properties of individual offices; =i/(οιι): the identity of individuals, x →v ι. Now let Empty/(ο(οι)) be the singleton containing the empty set of individuals, and Improper/(ο∗1)τω the property of constructions of being v-improper at a given 〈w, t〉-pair, Louis/ι, the other types as above. Then at any 〈w, t〉-pair the following proof steps are truthpreserving: 1) 2) 3) 4) 7)

assumption (¬)[0Baldwt λwλt [0King_ofwt 0France]wt] 0 0 0 0 by Def. 2, iii) ¬[ Improperwt [λwλt [ King_ofwt France]wt]] 0 0 0 ¬[ Empty λx [x =i [λwλt [ King_ofwt France]]wt]] (2), by Def. 2, iv) [0∃λx [x =i [λwλt [0King_ofwt 0France]]wt]] EG 0 0 0 by Def. of Exist. [ Existwt [λwλt [ King_ofwt France]]]

(b) Substitution: 1) [0Baldwt λwλt [0King_ofwt 0France]wt] assumption 0 0 0 assumption 2) [ Louis =i λwλt [ King_ofwt France]wt] 0 0 substitution of identicals. 3) [ Baldwt Louis] As explained above, the sentence (R) is not associated with the presupposition that the present King of France exists, because ‘the King of France’ occurs now in the focus clause. The truth-conditions of the Russellian ‘The King of France is bald’ are these: • True, if among those who are bald there is the King of France • False, if among those who are bald there is no King of France (either because the present King of France does not exist or because the King of France is not bald). Thus the two readings (S) and (R) have different truth-conditions, and they are not equivalent, albeit they are co-entailing. The reason is this. Trivially, a valid argument is truth-preserving from premises to conclusion. However, due to partiality, the entailment relation may

Ambiguities in Natural Langauge and Ontological Proofs


fail to be falsity-preserving from conclusion to premises. As a consequence, if A, B are constructions of propositions such that A╞ B and B╞ A, then A, B are not necessarily equivalent in the sense of constructing the same proposition. The propositions they construct may not be identical, though the propositions take the truth-value T at exactly the same world/times, because they may differ in such a way that at some 〈w, t〉-pair(s) one takes the value F, while the other is undefined. The pair of meanings of (S) and (R) is an example of such co-entailing, yet non-equivalent hyperpropositions. If the value of the proposition constructed by the meaning of (S) is T, then so is the value of the proposition constructed by the meaning of (R), and vice versa. But, for instance, in the actual world now the proposition constructed by (S) has no truthvalue, whereas the proposition constructed by (R) takes value F. Russell argued for his theory:20 The evidence for the above theory is derived from the difficulties which seem unavoidable if we regard denoting phrases as standing for genuine constituents of the propositions in whose verbal expressions they occur. Of the possible theories which admit such constituents the simplest is that of Meinong. This theory regards any grammatically correct denoting phrase as standing for an object. Thus ‘the present King of France’, ‘the round square’, etc., are supposed to be genuine objects. It is admitted that such objects do not subsist, but nevertheless they are supposed to be objects. This is in itself a difficult view; but the chief objection is that such objects, admittedly, are apt to infringe the law of contradiction. It is contended, for example, that the existent present King of France exists, and also does not exist; that the round square is round, and also not round, etc. But this is intolerable; and if any theory can be found to avoid this result, it is surely to be preferred.


B. Russell, On denoting, op. cit., p. 3.


Marie Duží

We have such a theory at hand, viz. TIL. Moreover, TIL makes it possible to avoid the other objections against Russell’s analysis as well. Russellian rephrasing of the sentence ‘The King of France is bald’ is this: ‘There is a unique individual such that he is the King of France and he is bald’. This sentence expresses the construction: (R*) λwλt [0∃λx [x =i [λwλt [0King_ofwt 0France]wt] ∧ [0Baldwt x]]].21 TIL analysis of the ‘Russellian rephrasing’ does not deprive ‘the King of France’ of its meaning. The meaning is invariably, in all contexts, the Closure λwλt [0King_ofwt 0France]. Thus, the second objection to the Russellian analysis is not pertinent here. Moreover, even the third objection is irrelevant, because in (R*) λwλt [0King_ofwt 0France] occurs intensionally, unlike in the analysis of (S) where it occurs extensionally.22 The existential quantifier ∃ applies to sets of individuals, rather than a particular individual. The proposition constructed by (R*) is true if the set of individuals who are bald contains the individual who occupies the office of King of France, otherwise it is simply false. The truth conditions specified by (R*) are Russellian. Thus, we might be content with (R*) as an adequate analysis of the Russellian reading (R). Yet we should not be. The reason is this. Russell’s analysis has another defect: it does not comply with Carnap’s principle of subject-matter, which states, roughly, that only those entities that receive mention in a sentence can become constituents of its meaning.23 In other words, (R*) is not the literal analysis of the sentence ‘The King of France is bald’, because existence and conjunction do not reNote that in TIL we do not need the construction corresponding to ∀y (Fy ⊃ x=y) specifying the uniqueness of the King of France, because it is inherent in the meaning of ‘the King of France’. This holds also for languages like Czech or Polish, which lack grammatical articles. The meaning of descriptions ‘the King of France’, ‘král Francie’, ‘Król Francji’ is a construction of an individual office of type ιτω occupied in each 〈w, t〉-pair by at most one individual. 22 For the definition of extensional, intensional and hyperintensional occurrence of a construction, see M. Duží at al., Procedural Semantics..., op. cit., § 2.6. 23 See R. Carnap, Meaning and Necessity, Chicago University Press, Chicago 1947, § 24.2, § 26. 21

Ambiguities in Natural Langauge and Ontological Proofs


ceive mention in the sentence. Russell did avoid the intolerable result that the King of France both does and does not exist, but the price he paid is too high, because his rephrasing of the sentence is too loose a reformulation of it. TIL, as a hyperintensional, typed partial λ-calculus, is in a much better position to solve the problem. From the logical point of view, the two readings differ in the way their respective negated form is obtained. Whereas the Stawsonian negated form is ‘The King of France is not bald’, which obviously lacks a truth-value if the King of France does not exist, the Russellian negated form is ‘It is not true that the King of France is bald’, which is true at those 〈w, t〉-pairs where the office is not occupied. Thus, in the Strawsonian case, the property of not being bald is ascribed to the individual, if any, that occupies the royal office. The meaning of ‘the King of France’ occurs with de re supposition, as we have seen above. On the other hand, in the Russellian case the property of not being true is ascribed to the whole proposition that the King is bald, and thus (the same meaning of) the description ‘the King of France’ occurs with de dicto supposition. Hence, we simply ascribe the property of being or not being true to the whole proposition. To this end, we apply the propositional property True/(οοτω)τω defined above. Now, the analysis of the sentence (R) is this construction: (R’)

λwλt [0Truewt λwλt [0Baldwt λwλt [0King_ofwt 0France]wt]]

Now the negation of (R’) (R’_neg) λwλt ¬[0Truewt λwλt [0Baldwt λwλt [0King_ofwt 0France]wt]] does that the King of France exists, which is just as it should be. (R’_neg) constructs the proposition non-P that takes the truth-value T if the proposition that the King of France is bald takes the value F (because the King of France is not bald) or is undefined (because the King of France does not exist).

Marie Duží


In summary, Russell and Strawson took themselves to be at loggerheads; whereas, in fact, they spoke at cross purposes. The received view still tends to be that there is room for at most one of the two positions, since they are deemed incompatible. And they are, of course, incompatible if they must explain the same set of data. But they should not, in my view. One theory is excellent at explaining one set of data, but poor at explaining the data that the other theory is excellent at explaining, and vice versa. My novel contribution advances the research into definite descriptions by pointing out how progress has been hampered by a false dilemma and how to move beyond that dilemma. The sentence ‚The baptism site of Jesus is Bethany beyond the Jordan‘ is also ambiguous. We analysed this sentence in Section 2 as expressing the construction: λwλt [λwλt [0Baptism_site_ofwt 0Jesus]wt = 0Bethany] or (A*) for short. This analysis corresponds to the Strawsonian reading: ‘The baptism site of Jesus is Bethany beyond the Jordan’, with ‘the baptism site of Jesus’ as the topic clause. On this reading, the sentence presupposes that the Jesus’ baptism site exists, and consequently that Jesus had been baptised. The other variant is Russellian: ‘The baptism site of Jesus is Bethany beyond the Jordan.’ or in a bit more disambiguating way: ‘Bethany beyond the Jordan is the baptism site of Jesus.’ This variant could be used for instance in such a situation when one asks ‘What is so interesting about the Bethany beyond the Jordan,

Ambiguities in Natural Langauge and Ontological Proofs


why so many pilgrims come here?’. The answer would be ‘You don’t know? This spot is the baptism site of Jesus!’. But somebody might protest ‘Oh no, it is not true, Jesus was not baptised’, or another objection might be ‘No, it is not true, Bethany is not the site of Jesus’ baptism, it happened somewhere else’. Hence now the sentence merely entails that the site of Jesus’ baptism exists. The sentence might be false either because there is no site of Jesus’ baptism (that is that Jesus was not baptised) or because the spot where Jesus was baptised is located somewhere else (most probably at the Jordan river). The analysis of this second, Russellian variant is this: (A**) λwλt [0Truewt λwλt [0Baldwt λwλt [0King_ofwt 0France]wt]]. The negated sentence is then analysed as expressing the construction: (A**N) λwλt ¬[0Truewt λwλt [0Baldwt λwλt [0King_ofwt 0France]wt]].

4. Topic-focus ambivalence in general The same topic-focus ambuiguity crops up almost in all sentences, not only those containing definite description. Consider the sentence (B)

‘All Apostles of Jesus were Galilean Jews’.

Until now we have utilised the singularity of definite descriptions like ‘the King of France’, ‘the baptism site of Jesus’ that denote functions of type ιτω. If the King of France does not exist in some particular world W at some particular time T, the office is not occupied and the function does not have a value at 〈W, T〉. Due to the partiality of the office constructed by λwλt [0King_ofwt 0France] and the principle of

Marie Duží


compositionality, the respective analyses construct properly partial propositions associated with some presuppositions, as desired. Now I am going to generalize the topic-focus phenomenon to sentences containing general terms. To get started, let us analyse the sentence: (B)

‘All Apostles of Jesus were Galilean Jews.’

There are (at least) two non-equivalent variants: (B1) (B2)

‘All Apostles of Jesus were Galilean Jews.’ ‘All Apostles of Jesus were Galilean Jews.’

(B1) is connected with the presupposition that there were some Apostles of Jesus and it only entails that Galilean Jews existed, because the negated sentence ‘Some Apostles of Jesus were not Galilean Jews’ also entails that there were Apostles of Jesus. On the other hand, (B2) presupposes that Galilean Jews existed and merely entails that Jesus had some Apostles. Note however, a classical regimentation of (B1) or (B2) in the language of the first-order predicate logic (FOL). In FOL we cannot distinguish the two variants, and moreover, in FOL we cannot get the truth-conditions of such a sentence that comes attached with a presupposition right. The FOL regimentation would be a formula like: “∀x [JA(x) ⊃ GJ(x)]”, with the intended interpretation assigning to the predicate JA the set of Jesus’ Apostles and the set of Galilean Jews to GJ.24 But this formula is true under every interpretation assigning an empty set of individuals to the predicate JA and false under interpretation assigning a non-empty set to JA and an empty set to GJ. In other words, FOL 24

There is another flaw in this analysis, of course: which sets would those be?

Ambiguities in Natural Langauge and Ontological Proofs


does not make it possible to render the truth-conditions of a sentence equipped with a presupposition, because FOL is a logic of total functions. We need to apply a richer logical system in order to express the instructions how to evaluate the truth-conditions of (B1) and (B2) in the above described way. Let us start with (B1). By reformulating the specification of the truth-conditions of (B1) in a rather technical jargon of English, we get: ‘If Jesus had Apostles then check whether all of them were Galilean Jews, else fail to produce a truth-value.’ We now analyse the particular constituents of this instruction. As always, we start with assigning types to the objects that receive mention in the sentence: Apostle_of((οι)ι)τω: an empirical function that dependending on states-of-affairs assigns to an individual a set of individuals, his Apostles; Jesus/ι; Galilean/((οι)τω(οι)τω): a property modifier, that is a function that assigns to a property of individual (here of being a Jew) another property of individuals (here being a Galilean Jew);25 ∃/(ο(οι)); All/((ο(οι))(οι)): a restricted general quantifier that assigns to a given set the set of all its supersets; [0Galilean 0Jew] → (οι)τω. The presupposition that Jesus had Apostles receives the analysis:26 λwλt [0∃λx [[0Apostle_ofwt 0Jesus] x]]. Now, the literal analysis of the sentence ‘All Jesus’s Apostles were Galilean Jews’ on its neutral reading (that is, without existential presupposition), is best obtained by using the restricted quantifier All, A detailed analysis of property modifiers can be found in M. Duží at al., Procedural Semantics..., op. cit., § 4.4. 26 We now ignore the past tense. This simplification is irrelevant for the analysis of the problem of ambiguity. For details on the analysis of sentences in past, present and future tense, see, however, M. Duží et al., Procedural Semantics, op. cit., § 2.5.2. 25


Marie Duží

because using a general quantifier ∀ would involve implication that does not receive mention in the sentence. Composing the quantifier with the set of Jesus’ Apostles at the world/time pair of evaluation, [0All [0Apostle_ofwt 0Jesus]], we obtain the set of all supersets of Jesus’ Apostles in w at t. The sentence claims that the population of those who are Galilean Jews is one such supersets: λwλt [[0All [0Apostle_ofwt 0Jesus]] [0Galilean 0Jew]wt]. The schematic analysis of sentence (B1) on its topic-like reading that comes with the presupposition that Jesus had Apostles translates into this procedure: (B*1)

λwλt [If ∃x [[0Apostle_ofwt 0Jesus] x] then [[0All [0Apostle_ ofwt 0Jesus]] [0Galilean 0Jew]wt] else Fail.

The formula of analysis of (B2) is obtained in a similar way: (B*2)

λwλt [If ∃x [[0Galilean 0Jew]wt x] then [[0All [0Apostle_ofwt 0 Jesus]] [0Galilean 0Jew]wt] else Fail.

To finish the analysis, we must define the if-then-else-fail function. This I am going to do in the next subsection.

4.1. If-then-else function In a programming language the if-then-else conditional forces a programme to perform different actions depending on whether the specified condition is evaluated true or else false. This is always achieved by selective altering of the control flow based on the specified condition. However, an analysis in terms of material implication, ⊃, or even ‘exclusive or’ as known from propositional logic, is not adequate. The reason is as follows. Since propositional lo-

Ambiguities in Natural Langauge and Ontological Proofs


gic is strictly compositional, both the ‘then clause’ and the ‘else clause’ are always evaluated. For instance, it might seem that the instruction expressed by ‘The only number n such that if 5 = 5 then n equals 1, else n equals the result of 1 divided by 0’ would receive the analysis: [0Iτ λn [[[05=05] ⊃ [n=01]] ∧ [¬[05=05] ⊃ [n=[0Div 01 00]]]]]. Types: Iτ/(τ(οτ)); n →v τ; 0, 1, 5/τ; Div/(τττ): the division function. But the output of the above procedure should be the number 1 because the else clause is never executed. However, due to the strict principle of compositionality that TIL observes, the above analysis fails to produce anything, the construction being improper. For, the Composition [0Div 01 00] does not produce anything: it is improper because the division function takes no value at the argument 〈1, 0〉. Thus [n = [0Div 01 00]] is v-improper for any valuation v, because the identity relation = does not receive a second argument, and so any other Composition containing the improper Composition [0Div 01 00] as a constituent also comes out v-improper. The underlying principle is that partiality is strictly propagated up. This is the reason why the if-then-else connective is often said to denote a non-strict function not complying with the principle of compositionality. However, as I wish to argue, there is no cogent reason to settle for non-compositionality. I suggest applying a mechanism known in computer science as lazy evaluation. The procedural semantics of TIL operates smoothly even at the hyperintensional level of constructions. Thus, it enables us to specify a definition of if-then-else that meets the compositionality constraint. The analysis of: If P then C, else D reveals a procedure that decomposes into two phases. First, on the basis of the condition P, select one of C, D as the procedure to be ex-

Marie Duží


ecuted. Second, execute the selected procedure. The first phase, viz. selection, is realized by the Composition: [0I* λc [[P ⊃ [c = 0C]] ∧ [¬P ⊃ [c = 0D]]]]. Types: P →v ο (the condition of the choice between the execution of C or of D); C, D/∗n; variable c →v ∗n; I*/(∗n(ο∗n)): the singularizer. The product of the Composition [[P ⊃ [c=0C]] ∧ [¬P ⊃ [c=0D]]] is v-constructed as follows. If P v-constructs T, then the variable c receives as its value the construction C, and if P v-constructs F, then the variable c receives the construction D as its value. In either case, the set v-constructed by λc [[P ⊃ [c=0C]] ∧ [¬P ⊃ [c=0D]]] is a singleton whose element is a construction. Applying I* to this set returns as its value the only member of the set, i.e. either C or D.27 Second, the chosen construction c is executed. To execute it, we apply Double Execution; see Def. 2, vi). As a result, the schematic analysis of ‘If P then C, else D’ turns out to be: (*)

[ I* λc [[P ⊃ [c=0C]] ∧ [¬P ⊃ [c=0D]]]].

2 0

Note that the evaluation of the first phase does not involve the execution of either of C or D. In this phase, these constructions figure only as arguments of other functions. In other words, we operate at hyperintensional level. The second phase of execution turns the level down to intensional or extensional one. Thus, we define: Definition 4 (If-then-else, if-then-else-fail). Let p/∗n →v ο; c, d1, d2/∗n+1 → ∗n; 2c, 2d1, 2d2 →v α. Then the polymorphic functions if-then-else and if-then-else-fail of types (αο∗n∗n), (αο∗n), respectively, are defined as follows:

In case P is v-improper, the singleton is empty and no construction is selected to be executed so the execution aborts.


Ambiguities in Natural Langauge and Ontological Proofs


If-then-else = λp d1 d2 2[0I* λc [[p ⊃ [c = d1]] ∧ [¬p ⊃ [c = d2]]]] 0 If-then-else-fail = λp d1 2[0I* λc [p ∧ [c = d1]].


By the definition of If-then-else, the product of this function is obtained as follows: If p v-constructs T, then the construction C is selected as the value of c. If p v-constructs F, then [p ∧ [c=d1]] v-constructs F, hence the class v-constructed by λc [p ∧ [c=d1]] is empty. In this case, the singularizer I* is improper by not producing any value, just as it should be. Now, we are ready to specify the general analytic schema of an (empirical) sentence S associated with a presupposition P. In a technical jargon of English, the evaluation instruction can be formulated as follows: At any 〈w, t〉-pair do this: if Pwt is true then evaluate Swt, else Fail (to produce a truth-value). Let P/∗n → οτω be a construction of a presupposition, S/∗n → οτω the meaning of the sentence S and c/∗n+1 →v ∗n a variable. Then, the corresponding TIL construction is this: λwλt [0If-then-else-fail Pwt 0Swt] = λwλt 2[0I*λc [Pwt ∧ [c = 0Swt]]]. Now we are in the position to refine the analyses of (B*1) and (B*2): (B*1) (B*2)

λwλt 2[0I*λc [∃x [[0Apostle_ofwt 0Jesus] x] ⊃ [c = 0[[0All [0Apostle_ofwt 0Jesus]] [0Galilean 0Jew]wt]]]] λwλt 2[0I*λc [∃x [[0Galilean 0Jew]wt x] ⊃ [c = 0[[0All [0Apostle_ofwt 0Jesus]] [0Galilean 0Jew]wt]]]].

In the interest of better readability, I will use a more standard notation. Hence, instead of either λwλt [0If-then-else-fail Pwt 0Swt] or λwλt 2[0I*λc [Pwt ⊃ [c = 0Swt]]] I will simply write λwλt [If Pwt then Swt else Fail].

Marie Duží


5. A common fault in ontological proofs At the outset of this paper I mentioned the ambiguity pivoted on de dicto and de re way in which a constituent of a sentence is conceptualised. We have seen that the topic clause occurs with de re supposition, whereas the focus clause with de dicto supposition. To put it in another way, this sort of ambiguity concerns intensional vs. extensional reading of a sentence. For instance, the argument of Descartes’ ontological proof, in its very simple form, can be formulated as follows: The essence of God-office is formed by all positive perfections. Existence is a positive perfection. Hence, God exists.

What is wrong here? There are two problems or two flaws: a) There is a confusion of the intensional and extensional level of abstraction: existence is not a property of individuals. Rather, it is a property of the office itself. Thus, existence cannot be a requisite of God-office, it is a second degree intension that is ascribable at the intensional level. However, requisites of an individual office are properties of individuals that are ascribable at the extensional level. b) Descartes applies an invalid inference rule: Existence is a requisite of God’s office. Hence God has the property of existence. As we will see now, there is a missing assumption on the occupancy of the office. Yet adding such an assumption makes the proof circular and thus futile. To explain the second flaw, we need to define and explicate some notions.

Ambiguities in Natural Langauge and Ontological Proofs


Definition 5 (requisites). Let X, Y be intensional constructions such that X/∗n → ατω, Y/∗n → (οα)τω. Then: [0Req Y X] = ∀w∀t [[0Existwt X] ⊃ [0Truewt λwλt [Ywt Xwt]]]. Gloss. definiendum as ‘Y is a requisite of X’, and definiens as ‘Necessarily, if X is occupied at some 〈w, t〉, then it is true that whatever occupies X at 〈w, t〉 has the property Y at this 〈w, t〉.' Example. Requisites of the American presidential office are individual properties like to be a human being, to be properly elected, to be inaugurated, etc. The sentence ‘The King of France is a king’ is ambiguous between two readings – one necessarily true, the other contingently without a truth-value – as Tichý points out.28 The former is the requisite (i.e. de dicto) reading: [0Req 0King λwλt [0King_ofwt 0France]]. Types: King/(οι)τω; King_of/(ιι)τω; France/ι. If true, it is necessarily so, regardless of whether or not some 〈w, t〉 lacks an occupant of λwλt [0King_ofwt 0France]. The other reading is the following de re one: λwλt [0Kingwt λwλt [0King_ofwt 0France]wt]. If true, it is so only because somebody occupies the King’s office constructed by λwλt [0King_ofwt 0France] at 〈w, t〉 and its occupant is in the extension of King at 〈w, t〉.

P. Tichý, Existence and God, “Journal of Philosophy” 1979, no. 76, p. 408; reprinted in Collected Papers in Logic and Philosophy, op. cit., p. 360. 28

Marie Duží


When defining a requisite of an office X, the antecedent condition on X being occupied is required. Otherwise we shall have the following invalid argument on our hands:29 (*)

P is a requisite of office O The occupant of O instantiates P.

This inference pattern is fallacious, for the premise may be true even if O is vacant, in which case the conclusion, so far from being true, is vacuous (i.e. lacks a truth value).30 Applying this invalid inference pattern, we might easily ‘prove’ the existence of the King of France. Here is how: 1) The King of France is a king. 1*) [0Req 0King λwλt [0King_ofwt 0France]] (necessarily true, de dicto)  2) The current King of France is a king. (applying (*)) 2*) λwλt [0Kingwt λwλt [0King_ofwt 0France]wt] (contingently true, de re)  3) The King of France exists. (existential generalization) 0 0 0 3*)λwλt [ Existwt λwλt [ King_ofwt France]]. The step from (2) to (3) is valid, because in any world w at any time t, the step is truth-preserving. Here is the proof: assumption 2a) [0Kingwt λwλt [0King_ofwt 0France]wt] 0 0 2b) λwλt [ King_ofwt France]wt v-proper by Definition of Composition 0 0 2c) λx [λwλt [ King_ofwt France]wt = x] constructs a non-empty class (from 2b)

29 30

Ibidem, p. 408ff; 2004, p. 360ff. Ibidem, p. 408, p. 360, respectively.

Ambiguities in Natural Langauge and Ontological Proofs

2d) [0∃λx [λwλt [0King_ofwt 0France]wt = x]] 2e) [0Existwt λwλt [0King_ofwt 0France]]


existential generalization by Definition of Exist.

However, the step from (1) to (2) is invalid, because it is the step from a de dicto reading to the de re reading. Whereas the de re reading not only entails, but also presupposes the existence of the King of France, the de dicto reading is necessarily true regardless whether the King exists. A valid inference rule can be obtained by adding an extra premise to the effect that the relevant office is occupied:


P is a requisite of office O Office O is occupied The occupant of O instantiates P.

Proof: Let P/∗n → (οα)τω; O/∗m → βτω. i) [0Req P O] assumption 0 assumption ii) [ Existwt O] 0 0 iii) ∀w∀t [[ Existwt O] ⊃ [ Truewt λwλt [Pwt Owt]]] Definition 5 0 0 ∀E iv) [[ Existwt O] ⊃ [ Truewt λwλt [Pwt Owt]]] 0 modus ponens ii), iv). v) [ Truewt λwλt [Pwt Owt]] Now we turn to the definition of essence as a set of requisites. However, this drags type-theoretic complications along, since the requisites of an intension may well be of different types. In order to make the essence homogeneous, we define the essence of an intension as the set of properties that are necessary for an object to satisfy the condition specified by the intension. The reason why the definition of essence can be made homogeneous is because, given an arbitrary intension, there will always be a corresponding property. For instance, the ι-office 'the tallest woman' will correspond to the prop-

Marie Duží


erty 'being an x such that x is identical to the tallest woman’. Again, for our purposes it is sufficient to define an essence of an office. The polymorphous type of such an Essence is ((ο(οα)τω)βτω): given an arbitrary β-office of type βτω, Essence returns the set of α-properties that are the requisites of this office. Thus, we define: Definition 6 (essence of an office). Let Z →v βτω; q →v (οα)τω; Req/((ο(οα)τω)βτω). Then: [0Essence Z] = λq [0Req q Z]. Remark: It is beyond the capacities of human beings to know an intension such as an individual office or its essence. This would account for knowing actual infinity, i.e. an uncountable infinite mapping. Yet we have capacities to know potential infinity. The meaning of an expression, such as ‘the King of France’ is the construction λwλt [0King_ofwt 0France]. Its encoding in TIL language of constructions can be viewed as an instruction read as follows: In any possible world (λw), at any time (λt) evaluate whether this or that individual satisfies the condition of being the King of France ([0King_ofwt 0France]). Thus, if one understands the meaning of ‘King of France’, it suffices to follow the instruction specified by this meaning, which does not, however, mean that one is always able to execute this instruction and thus to know the holder. Application of the invalid inference rule (*) is a typical fault that can be found in almost all ontological proofs with the exception of St. Anslem’s argument presented in Proslogion III.31

6. Conclusion In this paper I have brought out the semantic, as opposed to the pragmatic, character of the ambivalences stemming from topic-focus For details and analysis of this argument see M. Duží, St. Anselm’s Ontological Arguments, op. cit.


Ambiguities in Natural Langauge and Ontological Proofs


articulation, pivoted on de dicto vs. de re supposition and confusing intensional and extensional level of abstraction. I have shown that these ambiguities have much in common. In particular, ignoring the intensional vs. extensional context yields many problems. The procedural semantics of TIL provided rigorous analyses such that sentences differing only in their topic-focus articulation have been assigned different constructions producing different propositions (truth-conditions) and having different consequences. I showed that a definite description occurring in the topic of a sentence with de re supposition corresponds to the Strawsonian analysis of definite descriptions, while a definite description occurring in the focus with de dicto supposition corresponds to the Russellian analysis. While the clause standing in topic position triggers a presupposition, a focus clause usually entails rather than presupposes another proposition. Thus, both the opponents and proponents of Russell’s quantificational analysis of definite descriptions are partly right and partly wrong. Moreover, the proposed analysis of the Russellian reading does not deprive definite descriptions of their meaning. Just the opposite, ‘the F’ receives a context-invariant meaning. What is dependent on context is the way this (one and the same) meaning is used. Thus I also demonstrated that Donnellan-style referential and attributive uses of an occurrence of ‘the F’ do not bring about a shift of meaning of ‘the F’. Instead, one and the same context-invariant meaning is a constituent of different procedures that behave in different ways. The proposed analysis of topic-focus ambivalence has been then generalized to sentences containing not only singular clauses like ‘the F’ but also general clauses like ‘John’s children’, ‘all students’ in the topic or focus of a sentence. As a result, I have proposed a general analytic schema for sentences equipped with a presupposition. This analysis makes use of the definition of the if-then-else function that complies with the desirable principle of compositionality. This is also my novel contribution to the old problem of the semantic character of the specification for the if-then-else function.


Marie Duží

Finally, I have illustrated the flaw that can be found in ontological proofs. This flaw arises from ignoring the difference between the intensional and extensional reading of a sentence. The moral to be drawn from my contribution is this. Logical analysis disambiguates ambiguous expressions but cannot dictate which disambiguation is the intended one (leaving room for pragmatics here). Yet, our fine-grained method of analysis contributes to language disambiguation by making its hidden features explicit and logically tractable. In case there are more senses of a sentence, we furnish the sentence with different TIL logical forms. Having a formal, fine-grained encoding of linguistic senses at our disposal, we are in a position to automatically infer the relevant consequences.

Acknowledgments This research was funded by Grant Agency of the Czech Republic Project 401/10/0792 Temporal Aspects of Knowledge and Information. Versions of this study were read by the author as an invited talk at the University of Western Australia, Perth, Australia, March 4th, 2011. Portions of this chapter elaborate substantially on points made in Duží, Strawsonian vs. Russellian Definite Descriptions, “Organon F” 2009, vol. XVI, no. 4, p. 587-614 and Topic-focus articulation from the semantic point of view, [in:] Computational Linguistics and Intelligent Text Processing, A. Gelbukh (ed.), SpringerVerlag LNCS, Berlin–Heidelberg, vol. 5449, p. 220-232. I am indebted to Bjørn Jespersen for valuable comments that improved the quality of this study.

Kim Solin Uppsala University

Mathematics and Religion: On a Remark by Simone Weil* 1. Introduction


hen using formal, mathematical methods and results in some area, this is occasionally done without considering more closely how they are actually used. That the application can be done is often taken for granted without questioning, as something so obvious that it need not be discussed further. The success of mathematics in science has shown the way, the saying goes. At times, one also hears that we have to start somewhere if we want to make progress. Progress then often means progress in the formal system, and slowly the actual application area is forgotten or considered secondary to the system. Perhaps there is some sort of sliding going on: what started as a genuine concern with the area under investigation turns into a concern with details of the formal system, or into some form of marketing defence of one’s own particular speciality. This sliding often happens without oneself noticing it. While actually en civil being more concerned with the formal details of the system, one still officially maintains – and maybe truly believes – that one’s principal interest lays with the original questions; this is a sort of self-deception, if you allow me to say so. It can also be that one is so attached to one’s favourite system that one thinks it is applicable * This contribution was made possible through the support of a grant “The Limits of Scientific Explanation” from the John Templeton Foundation.


Kim Solin

everywhere, and one’s concern with some area is the concern to show the superiority of one’s own expertise. And so, it seems to me, it often happens that one is more involved in the formal system than in the questions that the system was supposed to clarify in the first place. This need not be problematic, as long as one is aware of it and admits it, not least to oneself. But how many of us are up to that challenge? There is, of course, also the danger of almost the opposite nature: that one is so estranged to mathematical language that the appearance of a single formula, makes one stop reading, or unable to take an author seriously. This can mean that one unjustly rules out the insights that mathematics can bring. In both cases, the underlying problem is the same: that one has not given the use enough thought. A person that has a high level of understanding of a certain mathematical system might still be hopeless when it comes to a sane judgement of the system’s usefulness, whereas a person with a low or medium level of understanding (but, needless to say, yet some level of understanding) of a certain mathematical system could nonetheless be able to see that it is not applicable to this or that domain. For instance, in contrast to the person with the high level of understanding, acquired through years of work, the latter does not have so much to lose in admitting inapplicability, and his mind is not so caught in the nets of concepts pertaining to that particular system. It is in this and similar ways that a lower level of understanding could actually be an advantage, paradoxical as it may sound. I want to challenge the common idea that only people with a high level of understanding of some mathematical system have something to say about it. Certainly, it can be the case that someone misjudges a system since this person does not have enough knowledge of its details, but a medium level of knowledge by no means necessarily leads to misjudgement. On the contrary, this can often yield a more sober view on the whole enterprise. One of the more difficult situations, in my experience, is when one has a more humble understanding of mathematics and is ashamed to admit it, looking up to the experts as if they were demigods. This,

Mathematics and Religion: On a Remark by Simone Weil


I think, will make a sane judgement of the use of mathematics impossible. Note, this is not the same difficulty that the real expert has, nor is it the problem of the total ignorant who arrogantly waves away the use of mathematics as pointless; it is another sort of difficulty. And if one holds such an undue admiration, what then? Can one just wish it away? It is not as simple as that. In addition to intellectual work, it also requires a lot of work on oneself, on one’s views, and on one’s longings. This sort of work is what I, together with Rush Rhees, take philosophy to be about, although I cannot delve into this issue here.1 Philosophical problems are not only intellectual problems, but akin to moral problems and problems of the will. I shall in this essay consider the use of mathematics in religious questions. This I shall do with the aid of a remark made by the philosopher Simone Weil. The sister of André Weil disagreed with him on how one should understand mathematics, including Greek mathematics in relation to algebra. Witold Gombrowicz, whose Diary is one of the last century’s most interesting literary works, says the following about her: Simone Weil is difficult, dense, loaded with internal experience. One has to return to many of her thoughts over and over again. [...] It is not true that all are equal, and that each person can discuss another. Simone Weil was falling into the hands of those less sophisticated minds, of those probably less mature souls, and in this hapless mode they began to work over a phenomenon that is a lot higher and superior to themselves? They [participants at a meeting that Gombrowicz was attending, my remark] spoke modestly and without airs, but no one could get himself to say that he did not understand and that he had no right to talk about this at all. The strangest thing was that they, who personally were a great deal inferior to Weil, treated her from on high, from the heights of that colBut see, for instance, R. Rhees, Wittgenstein and the Possibility of Discourse, 2nd Edition, Wiley-Blackwell, Oxford 2006, p. 260-265; R. Rhees, Without Answers, Schocken Books, New York 1969, p. 133-134.


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lective wisdom that made them superior. They felt that they were in the possession of Truth. If Socrates himself had shown up at this session they would have treated him like a freshman, because he hadn’t been initiated. They know it all much better. It is exactly this mechanism – which allows the inferior man to avoid a personal confrontation with the superior man – that seemed immoral to me.2

Gombrowicz is not the one to applaud mediocrity or even commonly acknowledged brilliance, so if Gombrowicz of all people says this, it is a sign that Weil ought to be listened to carefully. There is much in this quotation from Gombrowicz that is important, and he also has more to say about Weil. What I, however, want especially to bring out is Gombrowicz’s characterisation of her, namely that she is “dense, loaded with internal experience”. When one fails to understand what Weil says it is often because one has not had similar experiences that Weil’s extraordinary life gave her. This means, as Gombrowicz points out, that we need to return to her texts over and over again, for them to open up and even begin to make sense. While acknowledging her intelligence and wit, it is still tempting to treat her as simply hysterical, half-mad, and many have been unable to resist the song of this siren. This is a reaction that she anticipated herself and the danger of which also Gombrowicz was aware.3 But it is more fruitful to resist this temptation, and instead follow Gombrowicz’s advice, keeping in mind his warning for the iniquity at the heights of collective wisdom. I said above that one need not be an expert in some field of mathematics in order to be able to talk about its use, that it actually can be an advantage to only have average understanding. Why am W. Gombrowicz, The Diary, Volume 1, Northwestern University Press, Evanston 1988, p. 28-29. 3 W. Gombrowicz, The Diary, Volume 1, p. 174-175; letters from Simone Weil to her parents, 18.7.1943 and 4.8.1943, published [in:] Cette guerre est une guerre de religions. Écrits de Londres, Gallimard, Paris 1957. 2

Mathematics and Religion: On a Remark by Simone Weil


I then saying that Weil is difficult to understand and that in order to appreciate her one needs to carefully and patiently enter her world of thought? Could one not argue that it would be advantageous to only have a pedestrian understanding of Weil? Or, alternatively, that by the same line of reasoning, advanced understanding of some field of mathematics is actually pivotal for understanding its use and applicability? But there is an important difference here. The difficulty in learning more mathematics is mostly an intellectual problem, making new definitions, new techniques, proving original theorems, and the alike. The problem with understanding Weil, on the other hand, goes deeper than these intellectual troubles. Weil calls upon us to challenge ways of thinking and ways of being that are so deeply embedded in our lives that merely paying attention to them, just being able to see them, and then taking the challenge of them seriously, is hard enough. Weil is not adjusting some detail in our thinking and so adding yet a brick to the wall of thought, but rather challenging it at its very foundations.4 This is why, it is too easy to disregard Weil from the perspective of our common way of thinking. Certainly, there can also be similar difficulties in mathematics, such as the acceptance of some definition, of some theory, of some proof technique, or the alike. But what I shall be concerned with in this essay is rather what these sorts of definitions and results can tell us on the whole, in particular in connection to their use in religion and theology. This, I aver, is a different and on many occasions more difficult endeavour than mathematical understanding. Taking the risk

Weil is thus not the kind of thinker Ludwig Wittgenstein considered F.P. Ramsey to be: “Ramsey was a bourgeois thinker. I.e. he thought with the aim of clearing up the affairs of some particular community. He did not reflect on the essence of the state – or at least he did not like doing so – but on how this state might reasonably be organized. The idea that this state might not be the only possible one partly disquieted him and partly bored him. He wanted to get down as quickly as possible to reflecting on the foundations – of this state. This was what he was good at & what really interested him; whereas real philosophical reflection disquieted him until he put its result (if it had one) on one side as trivial”. See L. Wittgenstein, Culture and Value, revised edition, Blackwell, Oxford 1998, p. 24e. 4


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that Gombrowicz would find me immoral, I shall also in dialogue with Weil consider what religion can bring to mathematics.

2. The remark by Simone Weil This is the remark from Simone Weil’s Gravity and Grace that I shall take as the point of departure for the remainder of this essay. Pythagoras. Only the mystical conception of geometry could supply the degree of attention necessary for the beginning of such a science. Is it not recognized, moreover, that astronomy issues from astrology and chemistry from alchemy? But we interpret this filiation as an advance, whereas there is a degradation of attention in it. Transcendental astrology and alchemy are the contemplation of eternal truths in the symbols offered by the stars and the combination of substances. Astronomy and chemistry are degradations of them. When astrology and alchemy become forms of magic they are still lower degradations of them. Attention only reaches its true dimensions when it is religious.5

In the remark Weil picks out three different things: (1) astrology and alchemy as forms of magic; (2) astronomy and chemistry; and (3) astrology and alchemy as the contemplation of eternal truths in the symbols offered, which she calls transcendental astrology and alchemy. This is a typical statement of Weil’s that at first glance seems strange and provocative. The common view is that astronomy and chemistry arose from alchemy and astrology as the result of an Entzauberung, but now Weil tells us that this is a mistaken view, and that one must differentiate between magic and contemplation of eternal truths. The latter is, according to Weil, something better than regular astronomy and chemistry, whereas the former, magic, is S. Weil, Gravity and Grace, Routledge, London 1972, p. 120. The translation uses the word ‘transcendental’, but one could also have chosen ‘transcendent’. I shall follow the translation in this essay. 5

Mathematics and Religion: On a Remark by Simone Weil


rightly considered to be laying below the regular branches of science. To get clear about what Weil means, let us now consider these three things in more detail. What Weil has in mind with (1) astrology and alchemy as forms of magic is, I think, for example predictions of how one’s life will turn out based on some constellation of the stars when one was born, or that some placement of the planet Jupiter will affect how this year will turn out for one. Similarly, I suppose that Weil thinks that alchemy would be the attempt of making gold with the aid of secret knowledge of some esoteric reality, or affecting the body and the soul using certain magical objects, metals, or something of that kind. This is, at any rate, how I have read the above passage. What she means by (2) astronomy and chemistry is what we usually understand by these words, for instance some form of stellar cartography or an investigation of the interaction of certain materials, based on scientific method, and possibly some engineering on that basis. In Waiting for God she calls science a theoretical reconstruction of the order of the world.6 All this ought to be close to the common view on (1) and (2). The two are usually easy to keep apart for persons of our time, although our culture nevertheless has a tendency to have undue expectations on scientific results in ways that are reminiscent of magic. Gombrowicz writes that our “intellect has been ‘demystifying’ for such a long time that it has finally become the tool of a monstrous deception”.7 I shall below return to this tendency for the case of mathematics, and now instead turn to Weil’s thought-provoking third option. What does Weil mean by transcendental astrology and alchemy? It is, then, neither of (1) or (2). And, moreover, it is interesting that in the remark one can discern two aspects of this transcendental science, if I may call it that. On the one hand, (3a) that it provides the possibility of contemplation of eternal truths in the symbols offered – so transcendental science is good for understanding these eternal S. Weil, Waiting for God, Harper & Row, New York 1973, p. 169. W. Gombrowicz, The Diary, Volume 3, Northwestern University Press, Evanston1993, p. 45.

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truths – and then, on the other hand, (3b) that transcendental science is good for science itself. Simone Weil’s writings are full of the first aspect (3a). Already the title chosen by her literary executor to her book Gravity and Grace reveals this. In this book, Weil repeatedly uses ideas from science, such as gravity, to clarify how we should understand eternal truths, as for example the grace of God.8 These contemplations are often something more than mere metaphors or comparisons, since they give us concrete objects of contemplation for understanding these eternal truths. In the Christian tradition, one finds similar uses of what one could call natural phenomena, as for example Thomas a Kempis’ passages on the difference between nature and grace.9 The interesting thing with Weil is that she uses her contemporary understanding of the order of the world (physics) for similar aims that a Kempis in his time – a time before our notion of gravity – made with his understanding of the world. In both cases, Weil and a Kempis, the prevailing understanding of nature is used for the contemplation of what Weil calls eternal truths. This is something other than both regular science and magic. For instance, regular science and magic often strive to manipulate, whereas the transcendental is more about receiving and gratitude.10 Today, it is easy to confuse the transcendental perspective with the magical. W.V.O. Quine, just as an example, does not seem to be aware of this difference.11 But dark ages or not, already Plato was acquainted with the difficulty.12 S. Weil, Gravity and Grace, op. cit. See for instance p. 1-4 and p. 122 (Readings). T. à Kempis, The Imitation of Christ, Chapter LIV “Of the different motions of nature and grace”, Dutton, New York 1960, p. 171-173. 10 Cf. R. Rhees, Discussions of Simone Weil, SUNY Press, Albany 2000, p. 92-93. 11 See for instance his discussions [in:] Two Dogmas of Empiricism, “The Philosophical Review” 1951, no. 60, p. 20-43, p. 36-42. In these passages it is clear that Quine views the gods epistemologically on par with physical objects, and makes himself blind to the transcendental role they can play. The same holds when he in: Theories and Things, Harvard University Press, Cambridge 1981, p. 13, discusses Dalai Lama, or when he discusses God in Methods of Logic, Henry Holt and Company, New York 1950, p. 211 (see especially footnote 1 for the attitude with which Quine approaches these questions), and in: Mathematical Logic, W.W. Norton and Company, New York 1940, p. 82-83 and p. 150. 12 See Plato, Apology, especially 35d. 8 9

Mathematics and Religion: On a Remark by Simone Weil


Turning now to the other aspect of transcendental science (3b), Weil suggests that the transcendental perspective is good for science itself. She claims that only the mystical conception of geometry could have provided it with the necessary attention for its begetting. Attention is a key concept in Weil’s thinking, although I cannot go more deeply into that here. But as Mario von der Ruhr notes: [f]ar from being the opposite of absent-mindedness, then, true attention contrasts with attachment, illusion, prejudice, the projection of personal desire, false fascination, obtuseness, pride and vanity.13

It is this sort of attention that is needed in order to truly see the structure of the world even more clearly than before, and so for the inception of a new science, or the next stage in the development of some branch of science. According to Weil, the highest form of attention is religious, transcendental. Weil worries that the 20th century was largely blind to this transcendental aspect of human existence, and that eventually this will mean the end of science. She writes that “science today will either have to seek a source of inspiration higher than itself or perish”.14 The challenge is to regain this transcendental view on existence without falling into the muddy hands of magic and irrationality.

3. Mathematics and religion in the light of Weil’s remark Apart from contemplating the structure of our world, Weil’s writings also contain mathematical elements as objects of contemplation. And considering mathematics, how should one understand the other two categories? Next, I shall deliberate upon this and have thus reached the topic of the essay: mathematics and religion. Let us begin by looking M. von der Ruhr, Simone Weil. An Apprenticeship in Attention, Continuum, London–New York 2006, p. 29. 14 S. Weil, Gravity and Grace, op. cit., p. 119. 13

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at what (1’) mathematics as magic would amount to. I shall only briefly touch upon this issue, to return to it when discussing mathematics as an object of contemplation. Just like in the above case, having a magical view on mathematics means thinking that mathematics can do things for us that it actually cannot. So, when magical astrology purports to predict and control our lives by means of star constellations, magical mathematics purports to predict and control our lives using mathematical methods. What I am thinking of now is the use of certain mathematical methods in the social sciences or in economics, but there is trouble to be found already in, say, the natural sciences and in computer science. This mathematics can be ever so sophisticated, it can demand much work to acquire the skill needed only to understand its technicalities, and yet when attentively and self-critically looking at what this mathematics actually can do for us, one must conclude that it is overrated, and that the belief in it is akin to belief in magic. In contrast to our ability to distinguish between magical and regular science, I think that our current capacity to distinguish between magical and regular mathematics often fails us. In the introduction, I touched upon some issues that lay behind these problems and I shall not repeat this, but only recall that the problem is that one has not considered the use of the methods in enough detail, and moreover, that one has not done this in a self-critical way. Krister Segerberg expresses similar worries as follows: The gap between representation and reality needs to be explained. Modal logicians – but also other formal analysts – habitually fail to discuss the detailed relationship between their abstract models and the reality to which those modellings are meant to apply. How to apply the theory is taken for granted. [...] [A]mong the problems facing me in my own work this one – the connection between the formal and the actual – stands out. [...] “What am I doing?” And the inevitable follow-up: “Why?”15 K. Segerberg [in:] V.F. Hendricks and J. Symons (eds), Formal Philosophy, Automatic Press/VIP, New York 2005, p. 166. See also Melvin Fitting’s passages on the use of formal methods in the same volume, p. 28-30. 15

Mathematics and Religion: On a Remark by Simone Weil


Mathematics as magic thus amounts to some form of hubris, blindness for and disinterest in the limits of mathematical methods, often fuelled by the admittedly great achievement of learning these difficult methods, and at times by the sheer, undeniable beauty of many mathematical theories. How should one then understand (2′) regular mathematics? The unsurveyable literature on the philosophy of mathematics shows us that this question has no straightforward answer, and I shall not even try to sort this out here. But I think that one can be clear enough about this issue in order to see a contrast to the magical use of mathematics and to the transcendental mathematics. Mathematics is often divided into pure and applied mathematics, although this division is not always a happy one. Applied mathematics plays important roles in science and in engineering, and can lead to new developments in mathematics that were not motivated by the initial applications. Here, the value of mathematical theory is beyond doubt. Pure mathematics does not have applications as its main goal, but nevertheless lives up to rigorous standards, is often distantly motivated by applications, and sometimes finds employment in some unforeseen field. Magic often enters, as I said above, when one has exaggerated expectations on the use of the mathematics. But could there be some form of magic involved in mathematics that would not concern the application? That is, magic in pure mathematics? This would probably concern the use of some method within mathematics, or – more likely – the aptness of some definition that is supposed to correspond to an ‘intuitive notion’. As far as I can tell, the difficulties are at least similar. Investigating the difference between (1′) and (2′) is a responsibility that would deserve much more attention, and I shall revisit it below, but let us now turn to (3′) the transcendental use of mathematics, which in the context of this essay is the most interesting of the three. When writing about friendship, Simone Weil uses a fact from elliptic geometry as an object of contemplation. This is an example of how (3a′) mathematics can be good for understanding eternal truths. It is consequently not only a metaphor, a comparison, or a descrip-

Kim Solin


tion, but a symbol that aids our attention when contemplating an eternal truth about friendship. This is what she says: Pure friendship is an image of the original and perfect friendship that belongs to the Trinity and is the very essence of God. It is impossible for two human beings to be one while scrupulously respecting the distance that separates them, unless God is present in each of them. The point at which parallels meet is infinity.16

This fact in elliptic geometry, that parallels meet in infinity, is used here by Weil to contemplate friendship. The mathematical fact has an air of paradox, just like what Weil says about friendship has. It is impossible, Weil says, for two persons to be one without God, and this is what she contemplates with the aid of parallel lines: the friends – and then infinity – God. By improving one’s attention with the aid of this mathematical fact, one can become better aware of this eternal truth. Why am I insisting that this is not only a metaphor in the commonplace sense of the word? The reason is that I fear that Weil’s remark would otherwise be misunderstood as if it were merely some intellectual truth, and that once we have grasped this metaphor, we can use it to quickly move on to a theory of friendship. But Weil wants us to remain with it longer than that, she wants us to use it for developing our attention in order to understand friendship; and this understanding is different from a theoretical, intellectual understanding. It is something similar she is after when she claims that, even though it sounds contradictory, “the transcendent can be known only through contact since our faculties are unable to invent it”.17 When she says that “[t]he desire to discover something new prevents people from allowing their thoughts to dwell on the transcendent, indemonstrable meaning of what has already been discovered”, she

16 17

S. Weil, Waiting for God, op. cit., p. 208. S. Weil, Gravity and Grace, op. cit., p. 110.

Mathematics and Religion: On a Remark by Simone Weil


again makes a similar point.18 Weil does not only want an intellectual understanding of friendship, but rather to better know friendship, and it is here that the object of contemplation enters as an aid for the attention. There are, of course, also other objects of contemplation that would serve the same purpose, that would open up the same eternal truth. The one that Weil uses here, I mention only since it contains mathematics, the topic of this essay. Returning now to mathematics, it should then be obvious that in the quote above, Weil does not want to create a mathematical theory about friendship, although she uses mathematical language. She does not want to say that since friendship can be formalised in elliptic geometry, we can based on some corollary conclude that friendship also satisfies a property that follows from that corollary. This would fall under the magical use of mathematics, it would mean that one has not considered the use of mathematics in enough detail. In the quote, Weil directly addresses mathematics when she says that “only the mystical conception of geometry could supply the degree of attention necessary for the beginning of such a science”. Here Weil considers (3b′) what a transcendental attitude can bring to mathematics itself. As I mentioned earlier, according to Weil the greatest level of attention is achieved when it is religious. This means that the best way to do mathematics attentively is to constantly have the transcendental aspect present, and it was this, if one believes Weil, that led to the advent of geometry. This might be slightly sensational and untrue to the history of mathematics, especially that only this attitude could have given rise to geometry. That notwithstanding, there is still something to Weil’s observation. It is well known that Georg Cantor was largely motivated by the theological and philosophical implications that his work could have. I think that one finds in Cantor a person who often held a genuinely transcendental attitude towards mathematics. Through his theological and philosophical attitude, he could find the attention needed to set in motion much of what led to 18

Ibidem, p. 118.

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contemporary set theory, and Cantor’s work was in many respects a sharp break with earlier mathematics. But, nevertheless, Cantor at times – as is easily done, still today – conflates the transcendental with the magical. For instance, Christian Tapp says that: Cantor adds that the loftiness of God would be the more considerable the more extended the area of things below him is; and the more conceptual resources there are (e.g. Cantor’s mathematical theory), the more we can guess [my emphasis] how far extended it really is.19

This guesswork, I hold, would be akin to magic. Cantor here thinks that we on the basis of his mathematical theory can say something new about God, that we could not know without this theory (assuming, of course, that Tapp reflects Cantor’s views correctly). Simone Weil warns against this when she remarks that “[t]he object of our search should not be the supernatural, but the world. The supernatural is light itself: if we make object of it we lower it.”20 But when Cantor, as quoted by Tapp, says that “the absolutely infinite sequence of numbers thus seems to me to be an appropriate symbol of the absolute” he is – it seems – using the absolutely infinite sequence rather as an object of contemplation in the transcendental sense, as an aid for understanding the absolute.21 The understanding of the absolute that this contemplation brings is not the same as the one Cantor thinks we could achieve by gaining more conceptual resources. Rather, it is an understanding that one could describe as a deepening of what one already, at some level, is familiar with. And this is, from Weil’s perspective, vital to both the development of mathematics, and for an understanding of what Weil calls eternal truths. I want now to return to the difference between the (1′) and (2′), the magical and what I should like to call the regular understandCh. Tapp, Infinity in Mathematics and Theology, “Theology and Science” 2011, no. 9 (1), p. 91-100, p. 97. 20 S. Weil, Gravity and Grace, op. cit., p. 118. 21 Ch. Tapp, Infinity..., op. cit., p. 96. 19

Mathematics and Religion: On a Remark by Simone Weil


ing of mathematics. If transcendental mathematics can be good for regular mathematics, then what about the relationship between magical mathematics and regular mathematics? Can magical mathematics have something to offer mathematics itself, a sort of ‘inspiration’ if you like, or is the magical perspective also harmful to mathematics itself? Weil is of the opinion that the magical science is a further degradation of the transcendental science than the regular science is. Another philosopher who was concerned with these sorts of questions was Ludwig Wittgenstein. Rush Rhees, one of Wittgenstein’s literary executors, describes Wittgenstein’s endeavours as follows: I will repeat only that if someone like Kreisel should say that in suggestions like these Wittgenstein is trying to do what he ought to have left to the mathematicians – then he would show he had not grasped what Wittgenstein was up to. Wittgenstein was not pretending to do mathematics. He was describing [something] which could be mathematics. (In this way it is similar to his construction eines Sprachspiels or of the Lebensweise eines Stammes, whose grammar differed in important ways from ours, for whom ‘intelligibility’ and ‘understanding’ were different from the intelligibility and understanding we seek; and so on. And this could illuminate, or separate certain features of our speech and questions and reasonings.) Wittgenstein is not suggesting that any mathematical problem would be solved through the development of [these] ways of calculating. What he does suggest, I think, is that if one pays attention to the possibility of [conceiving a certain mathematical concept in another way] [...] then one may be less prone to ‘mythological’ or ‘fantastic’ interpretations of the forms of calculation mathematicians do use.22

What Wittgenstein wants to do with his remarks on the foundations of mathematics, for instance with his investigations of Cantor’s

Letter from R. Rhees to G.H. von Wright, 4.4.1973, located at The von Wright and Wittgenstein Archive in Helsinki. 22


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work, is thus not, according to Rhees, to create new mathematics. This is why, Wittgenstein constantly underlines that he is not concerned with mathematical logic and that philosophy of mathematics must leave mathematics as it is (albeit not our understanding of mathematics). With his remarks Wittgenstein wants to bring out the grammar of our mathematical language so that one can avoid understanding mathematics in a mythological or fantastic way, of which what I have called a magical way is one instance. Wittgenstein’s main aim with his objects of comparison is thus a form of clarity; he wants us to see clearly what mathematics is and what it is not, what mathematics can tell us, and what it cannot. This has been largely misunderstood, both by those building on Wittgenstein’s work and those opposing it. Although Wittgenstein’s work on the philosophy of mathematics is not in itself mathematics, it can of course nevertheless influence us as to which kind of mathematics we find reasonable to pursue and which not. If we realize that our main motive for pursuing some field of mathematics was a fantastic interpretation, then we might no longer be interested in that field of mathematics, or at least we will need to find some other motivation for concerning ourselves with it. What is important in the context of my current topic, mathematics and religion, in the light of Simone Weil’s remark, is that Wittgenstein has a different aim with his mathematical remarks than Weil has. When Weil is primarily concerned with the use of mathematics for transcendental goals, mathematics as objects of contemplation, Wittgenstein is primarily concerned with ridding us of magical interpretations of mathematics through objects of comparison. Would Wittgenstein have accepted the transcendental use of mathematics, or would he just have seen it as an instance of a fantastic or mythological interpretation? It is well known that Wittgenstein disliked the mystification of science, and perhaps he would have deemed Weil’s use of mathematics and science as such.23 But

See, for instance, Wittgenstein’s critique of James Jeans’s book The Mysterious Universe in Lectures and Conversations on Aesthetics, Psychology, and Religious Belief,


Mathematics and Religion: On a Remark by Simone Weil


Wittgenstein was without doubt aware of the difference between what I have here called the magical and the transcendental.24 He was also wary of conflating the transcendental endeavour with the philosophical, as is evident when he apropos a perplexing passage of St. John of the Cross says that “it is true, that crooked concepts have done a lot of mischief, but the truth is, that I do not know at all, what [against the religious background] does good and what does mischief”.25 So it is possible that Wittgenstein would have approved of Simone Weil’s methods. It is also important to recognise that in Weil’s writings one finds different kinds of remarks.26 One has the religious contemplations, such as the one about friendship above. But then, one also has remarks that are better viewed as plain philosophy, they are not primarily intended for religious contemplation. The remark of Weil’s that is the pivot of this essay is rather an example of philosophy than of religious meditation. These types of philosophical insights might point to the transcendental, and in Weil’s writings often do. On the other hand, the attention that the transcendental contemplation brings about is vital when making comparisons. So Weil’s and Wittgenstein’s aims are certainly related, for in order to appreciate the transcendental use of mathematics, one must first learn to separate this from the magical use, and then from the regular use, and significant attention is needed for this separation.

4. Conclusion In this essay I have shown that Simone Weil’s distinctions between transcendental science, regular science, and magical science University of California Press: Berkeley–Los Angeles 1966, p. 27. Rhees’ question in the footnote is similar to the question I ask here. 24 L. Wittgenstein, Culture and Value, op. cit., p. 82e. 25 Ibidem, p. 83e. Wittgenstein’s Unheil has been translated into mischief, and his Heil into good. The German words have important connotations that vanish in the translation. 26 This poses a difficulty for Rhees in his Discussions of Simone Weil, op. cit., p. 86ff.


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carry over to mathematics. The same distinctions can be made in mathematics. There is a transcendental use of mathematics, there is regular mathematics, and there is magical mathematics. I have tried to argue that it is easy to conflate these forms of use and, therefore, to ignore one or the other. There is also a difference between philosophical investigation and religious contemplation. And it has been pointed out that, in Weil’s view, the transcendental perspective on science and mathematics is good for science and mathematics themselves and for understanding what she calls eternal truths. Having said this, I’ll end by posing a question: in what way are words like ‘infinite’, ‘eternal’, ‘light’, and so forth, used in various creeds and church documents?

Pavel Materna Brno Masaryk University

Science – Logic – Philosophy. An Old Problem Resuscitated* If we take in our hand any volume; of divinity or school metaphysics, for instance; let us ask, Does it contain any abstract reasoning concerning quantity or number? No. Does it contain any experimental reasoning concerning matter of fact and existence? No. Commit it to the flames: for it can contain nothing but sophistry and illusion. The last paragraph of: David Hume, Enquiry Concerning Human Understanding

1. “Scheinprobleme”


his famous Scheinprobleme in der Philosophie,1 Rudolf Carnap presents a Gedankenexperiment and sends two geographers, one a realist, the other an idealist, on an expedition where the existence of an alleged hill should be confirmed or refuted. The point is that both scientists will agree as concerns empirical questions related to the hill, “In allen empirischen Fragen herrscht Einigkeit”. The disagreement begins as soon as the geographers start to solve philosophical problems like ‘Does the hill really exist?’. R. Carnap’s

The present paper has been supported by the Grant Agency of the Czech Republic, project No. 401/10/0792. I am grateful to the anonymous referee for his valuable comments. This contribution was made possible through the support of a grant “The Limits of Scientific Explanation” from the John Templeton Foundation. 1 R. Carnap, Scheinprobleme in der Philosophie, Suhrkamp Verlag, Frankfurt am Main 1966. *

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conclusion is that the choice of a philosophical standpoint cannot contentually (inhaltlich) influence the (empirical) science. Thus, we have a Realwissenschaft, which contains the claims that are empirical and assert something, then of course mathematics and logic, so that it seems that the remaining – philosophical – writings really should be destroyed by the Humean flames. Because what other fate should they deserve? Carnap (and Carnap’s spirit in the Vienna Circle) knows the solution-it is expressed by the title of his2 paper in “Erkenntnis”: Überwindung der Metaphysik durch logische Analyse der Sprache.

From the viewpoint of non-positivist philosophy, this recipe was a provocation. It was just the Vienna Circle philosophers and some sympathizers (like Bertrand Russell) who respected the role of (modern) logic in philosophy. In the present article, we do not follow the interesting history of neo-positivism after 1932, in particular Carnap’s recognition of the importance of the notion of meaning in his Meaning and Necessity3. What is relevant for our purpose is that Carnap’s notion of the logical analysis of language was infected by nominalism4 and his moving towards conventionalist conception of truth. The principle that the conventionalists did not recognize (against Tractatus) was formulated by J.A. Coffa5 as follows: There is no truth by convention; there is only meaning by convention and then truth in virtue of meaning. R. Carnap, Überwindung der Metaphysik durch logische Analyse der Sprache, “Erkenntnis” 1932, No. 2, p. 219-241. 3 R. Carnap, Meaning and Necessity, Chicago University Press, Chicago 1947. 4 Commenting on Carnap’s ideas in Logische Syntax der Sprache (cf. J.A. Coffa, The Semantic Tradition from Kant to Carnap, Cambridge University Press, Cambridge 1991, p. 293) states: “By embedding his ideas in a Procrustean nominalist mold, he had deprived himself of the possibility of grasping their true nature...”. 5 Ibidem, p. 321. 2

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Another point relevant for our rejection of Carnap’s way of thinking is his well-known proposal of ‘saving’ philosophy from nonsense: the sentences that can be claimed by philosophy (‘time is one-dimensional’, ‘every colour is at a place’, ‘every process is univocally determined by its causes’, ...) are ‘pseudo-object-sentences’. They do not speak about time, numbers, causes, etc., although it seems so because they are formulated in the material mode. This mode has to be replaced by the formal mode, which makes it explicit that such sentences (‘propositions’) speak about forms of language. Carnap’s conception of logical analysis of language reduces philosophy to engaging in linguistic enterprise. In my opinion, Carnap has elaborated on the Humean scepticism concerning philosophical statements and has shown that this scepticism can be formulated with the support of the more sophisticated means offered by modern logic. Carnap’s diagnosis is, however, more interesting than his therapy. Reducing the logical analysis of language to the pursuit of linguistic reformulations of philosophical claims is rather an escape from solving genuine problems. In what follows, I will try to offer a realist formulation of the problem and some thoughts concerning its possible solution.

2. A procedural theory of concepts 2.1. TIL constructions The term ‘concept’ occurs frequently in philosophy as well as in science. Unfortunately, it seems too often that using the term, the author does not know exactly what (s)he talks about. We will set aside the cognitivists’ use of the term (Fodor et alii): concepts in the cognitivists’ sense are some kind of mental representations, so that there arises the problem of sharing, which has not been solved (and cannot be solved ex definitione). We will follow the tradition, which in most cases conceived of concepts as of objective, logically relevant entities.

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The need of using the category of concepts can be traced to Aristotle’s theory of definitions6. Aristotle’s ‘ορισμος, i.e. definiens would most likely correspond to what we call concept. From the very beginning concepts are considered to be structured. Pavel Tichý, defending the structured character of meaning and its logical priority w.r.t. semantic notions like truth, analyticity, etc. appreciated Aristotelian theory in his work7: True, the classical idea of sense being a simple family of features or qualities is inadequate as is the idea that all the simple sentences are of the form S-P. However, the opinion that the notion of intension8 logically precedes the notions of truth, analyticity and synonymy, and not vice versa, is in our opinion quite justified [...].9

P. Tichý’s appreciation of Aristotle is understandable. Already at that time, Tichý began to build up a theory of structured meaning (exploiting the notion of abstract procedure unlike Cresswell in his works Hyperintensional Logic10 and Structured Meanings11). Tichý’s first attempt12 consisted in applying Turing machines. Later, Tichý defined a theory of constructions working in the type-theoretically classified milieu and enabling us to deal with possible-word intensions extensionally, i.e. as with objects sui generis. This theory is

See P. Materna, and J. Petrželka, Definition and Concept. Aristotelian Definition Vindicated, “Studia Neoaristotelica” 2008, vol. 5, no. 1, p. 3-37. 7 P. Tichý, Smysl a procedura, “Filosofický časopis” 1968, no. 16, p. 222-232. Translated as Sense and procedure: P. Tichý, Collected Papers in Logic and Philosophy, [in:] V. Svoboda, B. Jespersen, and C. Cheyne (eds), Filosofia, Czech Academy of Sciences, Prague and University of Otago Press, Dunedin 2004, p. 77-92. 8 Here Tichý uses the term ‘intension’ in the sense of meaning rather than in the sense of possible-world intensions. 9 P. Tichý, Collected Papers in Logic and Philosophy, op. cit., p. 81. 10 M.J. Cresswell, Hyperintensional Logic, “Studia Logica” 1975, no. 34, p. 25-38. 11 M.J. Cresswell, Structured Meanings, MIT Press, Cambridge 1985. 12 P. Tichý, Smysl a procedura, op. cit.; P. Tichý, Intensions in Terms of Turing Machines, “Studia Logica” 1969, no. 26, p. 7-25 (P. Tichý, Collected Papers..., op. cit., p. 93-109). 6

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known presently as Transparent Intensional Logic (TIL), discussed by Tichý13, but also M. Duží, B. Jespersen, and P. Materna.14 We will use TIL to define a procedural theory of concepts. We will, however, not try to define the key objects dealt with by TIL, referring to Duží’s work15 in this respect. Some general principles have to be articulated, though. First of all, let a general characteristic of TIL be given by the following quotation from the above-mentioned publication by Duží and others:16 Transparent Intensional Logic is a logical theory developed with a view to logical analysis of sizeable fragments of primarily natural language. It is an unabashedly Platonist semantics that proceeds top-down from structured meanings to the entities that these meanings are modes of presentation of. It is a theory that, on the one hand, develops syntax and semantics in tandem while, on the other hand, keeping pragmatics and semantics strictly separate. It disowns possibilia and embraces a fixed domain of discourse. It rejects individual essentialism without quarter, yet subscribes wholeheartedly to intensional essentialism. It denies that the actual and present satisfiers of empirical conditions (possibleworld intensions) are ever semantically and logically relevant, and instead replaces the widespread semantic actualism (that the actual of all the possible worlds plays a privileged semantic role) by a thoroughgoing anti-actualism. And most importantly, it unifies unrestricted referential transparency, unrestricted compositionality of sense, and all-out hyperintensional individuation of senses and attitudes in one theory.

The word ‘transparent’ in TIL means that TIL is anti-contextualist: Every expression of the given language expresses its meaning (Frege’s sense) independently of any context. While other logics prefer to say that the expression, say, ‘soldier’ denotes a class in the P. Tichý, The Foundations of Frege’s Logic, De Gruyter, Berlin, New York 1988. M. Duží, B. Jespersen, and P. Materna, Procedural Semantics for Hyperintensional Logic, Springer Verlag, Dordrecht–Heidelberg–London–New York 2010. 15 Ibidem. 16 Ibidem, p. 1. 13 14

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sentence ‘Charles is a soldier’ and a property in ‘Charles wants to be a soldier’, so that its meaning is dependent on a context, for TIL (top-down approach) this expression conveys one and the same construction in both contexts and, moreover, this meaning can be itself denoted as in ‘Charles believes that Peter is a soldier’. The key notion of TIL is thus construction. Due to this notion, TIL becomes a hyperintensional theory, where we can explain why (intuitively): (a) the semantics of an expression A may differ from the semantics of an expression B, although A is logically/analytically equivalent to B (b) there may be more analytically equivalent expressions that denote one and the same object. Ad (a) Observe that, e.g. all mathematical claims that denote the same truth-value differ semantically (no ‘Great Fact’!). Ad (b) Consider the expressions: natural numbers greater than 1 and divisible just by themselves and 1 natural numbers possessing just two factors.

Clearly, both expressions denote the set of primes. If their meanings were definable set-theoretically (as one interpretation of Frege’s definition of Begriff has it17), then the semantic diversity of these expressions would be unexplainable. It is just the notion of construction what makes it possible to jump into hyperintensionality. An informal characteristic will now compensate for the fact that for the technical reasons the exact definitions cannot be reproduced. Let us begin with a most important warning: Constructions are not formal expressions: they are abstract procedures and, therefore, extra-linguistic objective entities. See M. Duží and P. Materna, Can Concepts be Defined in Terms of Sets?, “Logic and Logical Philosophy” 2010, no. 19, p. 195-242, where more details can be found. 17

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This stipulation means that TIL is (logically) a Platonist and (semantically) a realist theory in the following sense of Platonism and realism: [...] [p]latonism, the view that over and above material objects, there are also functions, concepts, truth-values, and thoughts. [...] realism, the idea that thoughts are independent of their expression in any language and that each of them is true or false in its own right.18

Further: To deal with constructions we need, of course, some ‘pseudolanguage’, which will mediate instructions to do particular (abstract) actions. The ‘expressions’ of this ‘language of constructions’ (LC) are no formal expressions (as we have already stated) – they do not admit various interpretations since they unambiguously determine the particular steps to be taken. The formal inspiration is here the typed λ-calculus due to the fact that its founder (A. Church) recognized that practically each operation can be reduced to either the creation of a function by abstraction or application of a function to its arguments. Not by chance, the essentially same philosophy has been accepted by Richard Montague, who has used the typed λ-calculus rather than the predicate logic as the logical tool. Thus, the two constructions corresponding to the creation of a function and, respectively, application of a function to its arguments are in TIL closure and composition, respectively. Further constructions that we will deal with are variables (countably infinitely many for each type), where the usual letters like x, y, ..., k, I, ... are just names of them (because variables are special constructions that construct objects dependently on valuations: they f-construct, v a parameter of valuations, and are therefore also extra-linguistic entities), Trivializations, 0X, which just mention the given object and return it without any change, Double Executions, 2 X, which construct twice over (we will not need it here), Compositions, [XX1 ... Xm], where the construction X (v-) constructs a func18

P. Tichý, The Foundations..., op. cit., p. vii.

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tion and the other constructions (v-construct) the arguments, the result being the value (if any) of that function at those arguments, and, finally, Closures, λх1... хmX, where the construction X is abstracted from so that an m-ary function arises. A simple hierarchy of types is based on some atomic types, mostly: ι ... the universe of individuals o ... the class {T, F} of truth-values τ ... the class of real numbers/time moments ω ... the logical space (possible worlds), in terms of which functional types are defined: (αβ1 ... βm) are sets of partial functions, where α is the type of the value and β1 through βm are types of the arguments. As far as constructions are just used, the simple hierarchy is sufficient. As soon as constructions themselves are mentioned, the ramified hierarchy is defined. Here constructions of order n19 are defined and the set of constructions of order n, denoted as *n (as well as the types of order n) is a type of order n + 1. For n ≥ 1 the types *n are sets of hyperintensions. To adduce some examples, consider the following expressions: (a)

3 + 5 = 6 + 2.

Supposing (as we must) that we already understand the expressions ‘=’, ‘+’, and the numerals, we can write down a construction: (a′)


Comment. This LC-expression is an instruction: Let the function = (type: (οττ)) apply to the results of (i), applying the function + (type (τττ)) to the numbers 3, 5, (ii) applying the function + to the numbers 6, 2. 19

Roughly, they construct objects whose types are of order n .

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Observe. – The construction (a′) is a Composition (see above). – The result will be a truth-value (type o). – The construction itself is the abstract procedure proceeding according to the instruction. – As soon as we agree that this construction is the meaning of the expression (a), we have fulfilled the requirement that the meanings should be structured. – The denotation of the expression (a) is the result of the construction (a′), so the respective truth-value (here T). – Being an abstract procedure, the construction (a′) does not contain brackets, which only encode the instruction connected with (a′). After all, a construction cannot contain any expression. – No object is directly represented in a construction: Objects are only mentioned. Therefore, for example the function + is trivialized. (b)

Charles calculates 3 + 5

This time we have to analyze an empirical expression. In TIL, a thorough argumentation proves that empirical expressions always denote non-trivial intensions, i.e. such functions from possible worlds to chronologies of some type α (type schema of intensions: ((ατ)ω), abbreviated ατω for α any type) that there are at least two possible worlds for which their values differ. In (b) ‘calculate’ is an empirical expression, therefore, (b) is an empirical expression as well (see, however, the example (c)). Let t, w be variables ranging over τ, ω, respectively. Then – supposing again that we know who Charles is and understand the expression ‘calculate’ – we get the following construction: (b′)


λwλt[0Calculatewt20 0Charles0[0+0305]]

This is an abbreviation for [[0Calculate w]t].


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Observe. – Calculating relates individuals to constructions: a calculating individual does not relate him/herself to numbers or expressions: what remains is just the respective construction. Therefore, the construction (a′) is here ‘mentioned’, trivialized. With this trivialization absent, Charles would have to ‘calculate’ the number 8. – The construction (b′) is a Closure (see above). It constructs a function from possible worlds (λw) to chronologies (λt) of truth-values (results of the Composition [...]). Such functions (intensions) are called propositions. – Applications to possible worlds and times (here 0Calculatewt) cannot be realized mathematically/logically: we have to ‘go into the world’, i.e. to make an empirical step. The outcome of such an empirical step will hold for the actual world-time (in the positive case) but it will be a contingent value of the proposition: it is thinkable that Charles in some possible world does and in another possible world does not calculate 3 + 5 at the given time. – The type of Calculate is (οι*n)τω: a function from possible worlds (ω) to chronologies (τ) of the relation between individuals (ι) and constructions of order n(*n) (n is here 1). Notice. Considering together (a′) and (b′), we could wrongly infer (Leibniz’ rule of the substitution of identicals) that their conjunction would imply that Charles calculated 6 + 2. Fortunately, Leibniz holds but cannot be applied: in (a′) the identity holds between results of two constructions, which are used here. Charles is related to one of those constructions rather than to the number that is constructed. The numeric construction is here mentioned.

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Every pianist is a musician.21

To be a pianist as well as to be a musician is a property of individuals, type (οι)τω, i.e. a function that associates every possible world and time with a class (maybe empty) of individuals. Let ‘every’ be a kind of quantifier, type ((ο(οι))(οι)): applied to a class A of individuals ((οι)), then it returns the class ((o(οι))) of such classes b that A is a subclass of b. Then the meaning of (c) is a procedure that behaves according to the following instruction (construction): (c′)

λwλt[[0Every 0pianistwt] [0musicianwt]].

Observe. – As we can easily check, the construction (c′) constructs a proposition (similarly as (b′ )). This time, however, we would not discover any world-times that would return distinct values: the linguistic convention determined a necessary semantic link between the properties ‘being a pianist’ and ‘being a musician’: possessing the latter is a necessary condition of possessing the former. This link is independent of the state of the given world-time. The expression (c) thus denotes a trivial intension, a trivial proposition. The case (c) is the case of non-empirical non-mathematical expression.

2.2. Concepts We have argued that concepts cannot be set-theoretical objects. They should be structured, and one way how to fulfil this requirement is the hyperintensional system offered by TIL. The first idea We assume, of course, as we have to, that the analysis is applied to a disambiguated language. Even such a clear expression as ‘pianist’ has not escaped an ambiguity, as we are told by Oxford Dictionary of Modern Slang, where also radio operators are called pianists. 21

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which might cross our minds after Section 2.1 has been read is probably: let concepts simply be constructions! The problem is, however, not that simple. Consider the following expressions: (a) the highest mountain (b) a mountain higher than x. In the case (a), we are most likely to say: this expression expresses a concept, the respective construction constructs an intension called ‘individual role’, i.e. a criterion, which – given a possible world W and time T – selects such an individual (if any) that is a mountain higher than every other mountain in W at T. This intension is the denotation of the expression (a).22 Thus, we could say that (a) expresses a concept. We will see that our decision, being essentially right, needs some specification. On the other hand, if we hesitate to say something similar about (b), then we are absolutely right. The expression (b) contains an ‘indexical element’, represented by the individual variable x. The expression will denote a property of individuals, but if x stands for the Eiffel Tower, then the resulting property will differ from the property that would result if x were, say, Mont Blanc. Thus (b) does not denote anything, and to claim that it expresses a concept would not correspond to our intuitions. Therefore, the first approximation of the definition of concepts is: Concepts are closed constructions. (Closed constructions do not contain any free variables.) Yet neither this proposal is satisfactory. We will show why. We would like to explicate the notion of concept in such a way that we could claim that every expression expresses just one concept. We have to distinguish (in TIL) the denotation, which is constructed by the meaning, and reference, which is the (contingent!) value (if any) of the denotation in the actual world-time. The reference of the expression (a) is indeed Mt Everest. 22

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Our last preliminary definition does not fulfil this requirement. For consider the following examples: the (real) numbers greater than 2 (α) We have greater than, i.e. >, type (οττ). The concepts that fulfil (α) are: λx1[0> x1 02], λx2[0> x202], ..., λx56[0> x5602], ... λx275[0> x27502], ..., Thus, there are countably infinitely many candidates for the concept expressed by (α). Which criterion could select the ‘right one’? Another example is: to believe


The type of Believe can be23 (οιοτω))τω, i.e. it is an empirical relation. Believe; λw [0Believe w]; λwλt 0Believewt; λwλtλxy [0Believewt xy] ...


Again, there are infinitely many candidates. To solve our problem, we need some auxiliary definitions. Definition 1. Any two equivalent constructions that differ just in the choice of bound variables are called α-equivalent. Definition 2. Any two equivalent constructions one of which is an η-reduction (η-expansion) of the other are called η-equivalent. Definition 3 (procedural isomorphism). Constructions C, C′ are procedurally isomorphous iff there are constructions C1, ..., Cm such

Another relation of believing, denoted also by the English expression ‘believe’, is a relation between individuals and constructions. See M. Duží, et al., Procedural Semantics..., op. cit., Chapter 5. 23


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that C1 = C, Cm = C′, and for any constructions Ci, Ci + 1, it holds that they are α-equivalent or η-equivalent.24 The relation presented in Definition 3 is provably reflexive, symmetric and transitive. Thus, we can see that just one class of constructions that are procedurally isomorphous corresponds to every concept. Aleš Horák from Brno Masaryk University proposed an operation called normalization, which makes it possible to associate every (meaningful) expression with just one concept.25 If this procedure is applied to a closed construction C, the result, NF(C), is the simplest member of the equivalence class generated by C. The simplest member is defined as the alphabetically first, non-ηreducible construction. For every closed construction C, it holds that NF(C) is the concept induced by C, the other members of the same equivalence class point to this concept. This way, A. Horák’s solution makes it possible to define concepts as normalized closed constructions (their type is always *n,n ≥ 1). Remark. Practically, we need not take care of normalizing a closed construction. Any closed construction at least points to a concept, and there is an algorithm that finds the concept proper when applied to any construction that points to it.

3. Three kinds of concepts Returning to the three examples from Subsection 2.1, viz. (a), (b), and (c), we can state that each construction exemplifying the meaning of the respective expression is a concept according to the definition in the preceding Section (still, take into account Remark). Definition 3 from Ibidem, differs only slightly from A. Church’s definition of Alternative 1 of ‘synonymous isomorphism’ presented in A. Church, A Revised Formulation of the Logic of Sense and Denotation. Alternative (1), “Noûs” 1993, no. 27, p. 141-157. 25 See A. Horák, The Normal Translation Algorithm in Transparent ILntensional Logic for Czech, PhD Thesis, Masaryk University, Brno 2002, http://www.fi.muni.cz/ hales/ disert/. 24

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The example (a) exemplifies a mathematical concept. Mathematical concepts construct logical objects (truth functions, quantifiers, modalities, etc.) or objects studied by mathematical disciplines. Concepts expressed by mathematical/logical sentences are concepts of truth-values. The respective sentences are either logically or analytically true.26 (While the sentence ‘2 is a prime or 2 is not a prime’ is logically true, the sentence ‘2 is a prime’ is analytically true.) NB: Mathematical concepts do not contain variables for some possible world as their constituents. They never construct intensions. The example (b) exemplifies an empirical concept. Empirical concepts construct non-trivial intensions. This means that the concepts expressed by empirical sentences are concepts of (non-trivial) propositions. One consequence of our definitions is that every empirical expression denotes something. Empirical concepts construct functions; it is impossible simply that they construct nothing (unlike mathematical concepts, which can construct nothing, as in the concept expressed by the expression the greatest prime). Empirical expressions can indeed miss references.27 For example, take the expression ‘(to be) a man taller than Eiffel tower’: here the denotation is simply the constructed property, while there is no reference.

26 You can say: well, is Continuum Hypothesis (or Goldbach’s conjecture) true? A realist answer: We don’t know. An anti-realist is probably silent and waits for a proof. 27 In TIL, reference is the contingent value (if any) of the denotation in the actual world-time.


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The example (c) exemplifies a non-mathematical non-empirical concept. Non-mathematical non-empirical concepts construct trivial intensions. This means that the concepts expressed by sentences are concepts of trivial propositions. The respective sentences, if true, are analytic or analytically true sentences.28 Combining criteria empirical and mathematical, we get four options: empirical + mathematical empirical + non-mathematical non-empirical + mathematical non-empirical + non-mathematical. Since the first combination is impossible (empirical concepts construct intensions, mathematical concepts do never it, see NB above) and the three remaining options are covered by examples (b), (a), (c), respectively, we can see that our three kinds of concepts make up an exhaustive classification of concepts.

4. Science and logic/mathematics (Empirical) sciences primarily use empirical concepts. As far as they use also logical/mathematical concepts, they need them as tools for building up a consistent system of the given empirical concepts. The purpose of the use of empirical concepts is clear: to get such pieces of information that hold for the real (‘actual’) world. The empirical concepts provide criteria, according to which scientists test the state of the world by means of experiment, observation and the

Also such concepts that are expressed by sentences like ‘3 men + 2 = 5 men’ belong to this kind.


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like. For example, to learn whether the velocity of light is limited, the scientist has to know the empirical concept ‘velocity of’, which means that (s)he has defined, maybe in terms of some other empirical concepts, a function whose values are dependent on the state of the world, and which will be the intended intension constructed by that concept. Some experiments show the probable actual course of the values of this function, etc. The word ‘actual’ is of key importance: empirical concepts construct possible values of the respective functions, while the actual values can be received just empirically: logic and mathematics hold for all possible worlds, but to know which of them is the actual one equals being omniscient. Logic and mathematics use logical/mathematical concepts. They cannot submit information concerning the actual world since the concepts they use construct just such abstract objects that cannot serve as criteria deciding about what is real. This does not mean, of course, that logic cannot study relations between empirical concepts. For example, logic can define relations called requisites and use concepts that define these relations. We have adduced an example of such a relation, which holds between the intensions ‘pianist’ and ‘musician’. Note that we have not used the empirical concepts that construct these intensions in order to learn who the pianists/musicians are in the actual world. This would be a task for empirical science. Summary. – Empirical sciences use empirical concepts to get some knowledge of reality (actual world). – Logic and mathematics study concepts themselves.

5. Philosophy We can see now that the problem which has been tackled by Hume and the Vienna Circle (especially Carnap and, of course, the


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Tractatus), viz. what philosophy does talk about, can be formulated in a much more definite manner: Which kind of concepts does (or can) philosophy use?


In Section 3 we have stated that our classification of concepts as: (i) empirical (ii) non-empirical mathematical (iii) non-empirical non-mathematical is exhaustive. So let us try to answer the question (Ph) in terms of this classification. (i′) Does philosophy use empirical concepts? If so, philosophy would somehow double science. Some attempts include ‘ecophilosophy’, which tries to argue using some facts. The clearly observable danger consists in more or less critical repeating of what a real science (ecology or some particular ecological discipline) claims. But in general: how would a philosopher verify her/his claims if the concepts (s)he uses were empirical? Would (s)he conduct some experiment? Would (s)he argue referring to the results of some observation? And would there be no science which would be able to do the same verification? Finally, should philosophical claims characterize reality? This is the Scylla of the philosophical dilemma. (ii′) Does philosophy use (only) logical/mathematical concepts? This is, of course, the Charybdis. Let us admit that philosophical claims use always non-empirical concepts. Then two options are eligible:

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(iia) (iib)


logical/mathematical concepts or non-mathematical concepts.

Answering (iia), we will answer (ii′). Let us first refer to the option (iib). Implicitly, we have considered mathematical concepts to be mathematical or logical concepts. Thus non-mathematical concepts are concepts that are neither logical nor mathematical. To be such a concept and, at the same time, to be a non-empirical concept means to contain some empirical subconcepts. If philosophy were engaged in using this kind of concept, it would be a collection of banal claims like All bachelors are men, If XY is left from Z then Z is right of XY, If A is stronger than B then B is weaker than A, All mammals are vertebrates, etc. Evidently, such claims are not what we expect to be a philosophical claim. Thus, let us reject the option (iib). What remains is option (iia). Here the situation is not hopeless. True, Kant’s categorical imperative, when analysed, contains empirical concepts (act, handeln) but (based on Encyclopaedia Britannica29): ‘Act only according to that maxim by which you can at the same time will that it should become a UNIVERSAL LAW’ is a purely formal or logical statement.

Indeed, the concept of acting is surely either (a) just mentioned, and if an empirical concept X is mentioned then the result, i.e. 0X, is no more an empirical concept or (b) used, but the constructed intension (acting) is in the de dicto supposition so that its value in the given world is not required. Besides, the categorical imperative is a norm, and the concepts that are analyses of norms are not empirical concepts: they do not help getting a piece of information about reality. 29


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Nonetheless, the concepts expressed by the Categorical Imperative are trivializations of empirical concepts or even empirical concepts in the de dicto supposition. So what does justify the use of such concepts in philosophy? Well, one can say that any concept can occur in a philosophical discourse as soon as it is a trivialization or constructs an intension in de dicto supposition. Then the banalities from (iib) will no longer be the subject matter of philosophy but their following counterparts can become the part of this subject matter: To be a man is a requisite of the property of being a bachelor. X being left of Y is the same as Y being right of X. X being stronger than Y is equivalent to Y being weaker than X. Being a vertebrate is a requisite of being a mammal.

But then do not forget that a philosophy that articulates such claims does what logical analysis of (natural) language does. More examples of this kind can be given. There are, however, examples of another kind. Consider philosophical claims in the spirit of Anselm’s ontological proof, Saint Thomas Aquinas’ Five Ways “all of which end with some claim about how the term ‘God’ is used”.30 Here the situation differs from cases like ‘bachelorhood implies manhood’ and the like, where the linguistic convention is stated. There is no simple linguistic convention connected with the terms like ‘god’. Similarly, no such linguistic convention can justify philosophical claims concerning determinism, causes, free will etc.: linguistic dictionaries may mention such terms but such a reference is not comparable with a philosophical treatise. So what can be called a philosophical concept (or, more cautiously, notion)? There is a phenomenon of vague or homonymous expressions whose semantics consists in expressing pre-conceptual guesses. No definite 30

St. Thomas Aquinas, Stanford Encyclopedia of Philosophy, 10.1.

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unambiguous construction can be associated with such expressions. We would like, however, to get a concept because we guess that there can be some interesting connections between such ‘guesses’, e.g. between causes and freedom, mind and matter, etc. The transition from such a ‘guess’ to a concept has already got a name: it is an explication.31 It seems that we have found a form of ‘therapy’ for Humean scepticism: philosophy explicates interesting pre-conceptual guesses. One can object, though: (1) What do you mean by ‘interesting’? (2) Are there uninteresting pre-conceptual guesses that we should not call ‘philosophical’? Ad (1) A difficult question, indeed. Consider, however, the history of philosophy. You will find a list of problems that have been considered interesting. The history of philosophy can be construed as the history of attempts to explicate pre-conceptual guesses connected with these problems. Ad (2) There is a ‘filter’ that can distinguish, e.g. physical, astronomical and such like guesses from those which we should classify with philosophical tasks. If the guess concerns concepts rather than ‘things themselves’ (like properties, relations of objects) then a philosophical (but maybe a logical) problem starts a process of explication. Some concluding remarks are, however, necessary. First: We went from Carnap to Carnap. This is not surprising, since Carnap changed some of his views. To save philosophy via reformulations of its claims, viz. translating them from the ‘material mode’ to ‘formal mode’ essentially differs from explicating pre-con31

See R. Carnap, Logical Foundations of Probability, The University of Chicago Press, Chicago 1962, p. 3.


Pavel Materna

ceptual guesses: in the latter case philosophy is no longer engaged in linguistics. Second: Explicating can be more or less successful. Some pseudo problems may arise in the process of explicating. Hence, our solution is compatible with a critical attitude to a philosophical explication. Thus, it can be shown that explicating concept as a kind of mental entity is incompatible with the requirement that concepts have to be shared.32 Third: According to our solution, philosophy is not an empirical science and, therefore, philosophical claims do not concern the reality. This means, however, that philosophy is a kind of theory of concepts. But mathematics and logic are theories of concepts as well. Claims of any theory of concepts are necessarily a priori ones. This statement is not at all surprising in the case of logic and mathematics. I am convinced that nearly everybody is surprised in the case of philosophy. Why? In my opinion, the reason is clear: when we inspect the a priori claims in logic/mathematics, we have to state that such claims are a priori true or a priori false (at least when we do not share Dummett’s anti-realism). Nothing of this sort can be stated in the case of philosophy. Why? This is not an easy and simple problem. No very simple answer can be given. Let us try to formulate something like a hypothesis: The way we talk about (the) philosophy is misleading. We know indeed that philosophical schools are legion; each of them represents some attempts to explicate those pre-conceptual guesses which the given school appraises as being philosophically interesting. Evidently, there are always more schools that try to explicate the same guesses. In virtue of the essential indeterminacy of guesses, the explications offered by a school A may be (and mostly are) distinct from the explications given by a school B. Naturally, no such arguments can be adduced


See H.J. Glock, Concepts: Where Subjectivism Goes Wrong, “Philosophy” 2009, vol. 84, no. 1, p. 5-29.

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by the adherents of any school that would use empirical concepts: philosophy is not an empirical science. Thus, the only way how to argue that some philosophical claim is true consists in proving that it is compatible with other claims of the same school. This gives indeed a very unsatisfactory result. Solipsists easily prove that their claims make up a coherent system. Does it mean that it is impossible to find philosophical claims that are surely true? First, let us not forget that many philosophical schools produce norms rather than claims. This is a specific problem of any axiological theory, but in any case the question of truth does not arise in such systems. Second, we need not be too pessimistic. Some development of philosophy can be stated; sometimes some claims proved to be true due to analyses made by philosophical logic, in particular when the respective guesses are not too much indefinite. Indirect proofs are the preferred methods of arguing (see, for example Tichý’s proof that “alternative possible worlds are alternative states of affairs as regards the same domain of objects”).33 We would of course appreciate it if some philosophical theses that seem to be quite absurd could be shown to be false (see solipsism), which seems to be a hopeless task. Carnap from the thirties would have had a simple solution: such theses are just “pseudosentences”, which are sinnlos. A thorough analysis of the solipsist claims from the viewpoint of procedural theory of concepts leads to the same result. Thus, we can hope that similar cases of ‘unsympathetic’ philosophical claims will prove to be cases of meaningless or truthless sentences. Remark. From the viewpoint of TIL, there is an essential difference between being meaningless and being ‘denotationless’ (in the case of truthless sentences). Since meaning is a construction (in 33

P. Tichý, The Foundations..., op. cit., p. 180-182.

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the case of non-indexicals, a concept), to be meaningless means that there is no construction that would be derivable from the grammatical structure of the respective expression. Having no denotation means that the expression is meaningful so that we understand it but the construction does not construct anything. Here are examples: The number 3 is green.34 – This is a meaningless expression: it is impossible to associate this sentence with a concept of a truth-value (and we can see that we do not understand the sentence). The greatest prime is odd. – This is a truthless sentence. We can find a concept that is expressed by it, viz. [0Odd [0Gr 0Prime]], and we indeed to understand the claim but no truth-value is constructed. ⊣ Thus, we can after all admit that the Humean scepticism concerning philosophical claims, as well as the more elaborated scepticism formulated by the Vienna Circle, in particular by Carnap, is partly justified but that Carnap’s ‘therapy’ can be rightly criticized. A procedural theory of concepts based on TIL offers a more sophisticated realist solution of the problem. Philosophy is a specific kind of a theory of concepts and its claims are, therefore, a priori ones. An essential component of philosophy is the logical analysis of language, which cannot be reduced to linguistics.


We understand the expression ‘number’ so as it should be understood. People use to apply expressions in a way that goes against the way the given expression has been introduced into the language. Thus ‘number’ can be certainly used in the sense of ‘being a numeric label’. Such cases of deformed application of semantic rules are, of course, frequent but this is an empirical, pragmatic problem, and if logical analysis should take into account such cases, then it would never begin to work.

Jan D. Szczurek The Pontifical University of John Paul II

The Rationality of Theology* 1. Introduction


human understanding of the depths of the cosmos and the mysteries of the human genome may fill us with admiration and pride. However, the resulting complacency hides the danger of reductionism, which means reducing the meaning of the whole of the existing reality to what is universally understood through science. The criticism of science formulated by E. Husserl seems reasonable. He noted that despite their achievements, the natural sciences cannot answer existential questions concerning man, in particular the question about the meaning of man’s existence. In addition, there is a conflict of values between today’s culture, with its emphasis on the value of the individual (each particular person), and science, where the individual does not constitute a value.1 The existence of values introduces us into those areas of reality which are sometimes called ‘metaphysical’, or beyond-physical. The argument over values is a great challenge for theology as a science, because its rationality is such that it is not only able to demonstrate the universality of certain values in an analogous way to the universality of the laws of nature discovered by the natural sciences, but also to associate the values with that ‘metaphysical’ reality.

* This contribution was made possible through the support of a grant “The Limits of Scientific Explanation” from the John Templeton Foundation. 1 H. Poser, Wissenschaft. Wissenschaftstheorie, [in:] W. Kasper et al. (ed.), Lexikon für Theologie und Kirche, vol. 10, Herder, Freiburg 2001, p. 1248.


Jan D. Szczurek

The task of this paper is to present theology as a human activity subordinate to the rules of rationality, though with limitations. Let us therefore focus firstly on the concept of rationality, and secondly on that of theology, after which we will try to show the function of rationality in Catholic theology and the relationship between Catholic theology and mystery.

2. About the concept of rationality According to the Longman dictionary, the term ‘rationality’ refers to ideas or behaviours that are open to reason and are compatible with reason (sensible, according to reason).2 According to Wikipedia, the term ‘rationality’ in philosophy means “the use of reason”. In other words, it is a question of a conclusion reached through careful consideration. It also means matching one’s beliefs with one’s reason where a conviction is concerned, or one’s actions with one’s reason for such actions. It also indicates that with regards to a conviction, one’s beliefs must be consistent with reason, and that one’s actions must be consistent with one’s reason for such actions. However, the term ‘rationality’ is more often used in specialist discussions of economics, sociology, psychology and political science. A rational decision is one which is not only rationally justified, but it is also optimal to achieve a particular purpose or to solve a given problem. The term ‘rationality’ is used in various disciplines.3 P. Procter, Longman Dictionary of Contemporary English, Longman, Essex 1981, p. 914. 3 Cf. http://en.wikipedia.org/wiki/Rationality (17 November 2011): “In philosophy, rationality is the exercise of reason. It is the manner in which people derive conclusions when considering things deliberately. It also refers to the conformity of one’s beliefs with one’s reasons for belief, or with one’s actions with one’s reasons for action. However, the term ‘rationality’ tends to be used in the specialized discussions of economics, sociology, psychology and political science. A rational decision is one that is not just reasoned, but is also optimal for achieving a goal or solving a problem. The term ‘rationality’ is used differently in different disciplines”. 2

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Looking at business activity, Herbert Simon, a Nobel laureate in economics, distinguished between both real rationality and procedural rationality. “Rationality refers to the actual result of an economic decision. The decision is rational, when it leads to the best possible result. This is the absolute criterion”.4 Naturally, this is rationality defined by the objective of action. This approach seems to disregard the ethical aspect of such a decision. However, we are not interested here in the rationality of management, but the rationality of theology. We are looking for a concept of rationality such that, after relatively slight modification, could be applied to theology. The answer appears to lie with the concepts of rationality of religion, rationality of existence and rationality of the world. According to A. Bronk, the rationality of religion is the non-contradiction of religious truths; rationality as seen from within a religion consists in the cohesion of the truths of faith with one another; whereas rationality looked at from outside a religion indicates the compliance of the doctrinal part of religion with human reason and knowledge as a whole.5

It seems that the philosophical idea of the rationality of existence brings us even closer to a proper understanding of rationality in relation to theology. According to classical philosophy, existence is rational insofar as it is governed by the principles of identity, noncontradiction, excluded middle, determination, and sufficient reason, and insofar as it can be recognized intellectually and understood in the light of the truth to which its knowledge leads. This is the ontological understanding of rationality.6 Cf. http://pl.wikipedia.org/wiki/Racjonalne_gospodarowanie (17 November 2011). A. Bronk, Racjonalność religii, [in:] M. Rusecki et al. (ed.), Leksykon teologii fundamentalnej, Wydawnictwo M, Lublin–Kraków 2002, p. 992ff. 6 Cf. A.B. Stępień, Racjonalizm, [in:] J. Herbut (ed.), Leksykon filozofii klasycznej, TN KUL, Lublin 1997, p. 458. 4 5


Jan D. Szczurek

The rationality of existence is the source of the rationality of the world as a concrete realization of existence as a whole. M. Heller speaks of the rationality of the world. According to him, “by ‘the rationality of the world’ we are meant to understand ‘that feature of the world thanks to which one can examine the world in a rational way’.”7 It is noticeable that the concept of rationality quoted above relates to two areas of reality: one is a set of propositions, the second is the reality to which they relate. Before answering the question of how such a rationality can apply to theology, it is necessary to present the scope of the term ‘theology’.

3. The concept of theology According to the Longman English Dictionary, the term ‘theology’ can have two meanings: (a) it is “the study of religion and religious ideas and beliefs”; (b) “the study of God and God’s relationship with man, especially by studying the origin and development of a particular religion”.8 This is obviously colloquial understanding of theology, quite distant from what theology really is, especially Catholic theology. Generally speaking the concept of ‘theology’ is determined by its two Greek components: θεός and λόγoς. Λόγoς derives from the verb λέγω, which means to choose, to count (to enumerate), to speak (to say), to name, to claim, to think. Λόγoς has many meanings, depending on the philosophical system in which it is used. Basically, it means report, ratio (proportion), base (principle, right), thinking, a tale (story), words (speech), statement (oracle), matter (issue), speech (convincing).9 One must remember that λόγoς can be contrasted with myth (μύθός), which complements the λόγoς by describing the reality inacM. Heller, Filozofia i wszechświat. Wybór pism, Universitas, Kraków 2006, p. 47. P. Procter, Longman..., op. cit., p. 914. 9 Z. Abramowiczówna (ed.), Słownik grecko-polski, vol. 3, PWN, Warsaw 1962, p. 43-46. 7 8

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cessible to λόγoς. Thus, for example, in the religion of ancient Greece, there are many mythical stories about the genealogies of gods. Finally, the Greek term θεός originally meant mysterious elemental powers, personified in time, or intelligent beings directing them, or gods. Deity in the sense of elemental powers is still present in the days of Paul of Tarsus (cf. Gal, 4: 8-9). The term ‘theology’ is used for the first time in Plato, where it denotes myths and legends about gods and their stories.10 Plato criticizes some of the stories about gods, noting their corrupting nature. Plato proposes, therefore, what today we would call the ‘demythologizing’ of contemporary theology, which eliminates the unfavourable elements for the sound education of the youth based on imitating the gods. His initiative starts the movement away from myth towards λόγoς. For Aristotle, the poets Hesiod and Homer are theologians (θεoλόγοι). By theology Aristotle also understands a philosophical and metaphysical treatise on existence as such; theology, therefore, also means the so-called ‘first philosophy’. In ancient Christian literature, the term ‘theology’ was accepted only in the 4th and 5th centuries, when it began to mean ‘knowledge of the true God’. Originally, it appeared in statements on the Trinity and on the nature of God (Athanasius),11 while the doctrine on the Incarnation and Redemption was at that time known as ‘the economy’, or ‘the rule of God’ (οίκονομία). All the biblical authors were at that time called ‘theologians’, among whom St. John the Evangelist was recognized as the greatest.12 Plato, The Republic, II 379a (recent edition: The Republic, translated from the Greek by B. Jowett, Simon and Schuster Paperbacks, New York 2010, p. 524).


St. Athanasius, On the Incarnation of the Word, 10, 1-2; 18, 1 (θεολόγοι – inspired writers); 12, 5; 14, 3 (theología – knowledge of God). Recent edition: St. Athanasius, On the incarnation. Greek original and English translation, transl. by J. Behr, St. Vladimir’s Seminary Press, Yonkers 2011. 12 On the scope of the terms θεός and θεoλόγοι cf. H. Fries, Teologia, [in:] H. Fries (ed.), Dizionario teologico, ed. 2, t. 3, Brescia 1969, p. 471ff. For the analysis of the term theós cf. C. Gallavotti, Morfologia di ‘theos’, Roma 1962 (Studi 11


Jan D. Szczurek

So as to better describe the scientific (rational) nature of theology, one must first distinguish between two basic ways of understanding theology: according to the first one theology is ‘the science of God’, while according to the second understanding, theology is ‘the study of the content of the Catholic (or another) faith’. The first understanding is closer to St. Thomas Aquinas, who says that theology is “a science subordinate to the principles of higher knowledge” (ex principiis notis lumine superioris scientiae), and therefore “subordinate to the knowledge revealed by God”, as opposed to “the science which gains knowledge by itself on the basis of principles learned by the natural light of reason” (ex principiis notis lumine naturali intellectus).13 The first understanding of theology points to the rationality of God Himself, that is, as the One about Whom internally non-contradictory statements can be formulated; in other words, God can be conceived as Being. God as Absolute Being is rational in accordance with the rationality of existence, and by analogy, rationality proper to the world can be attributed to Him, as having its source precisely in the rationality of the Absolute. On the other hand, the second understanding of theology clearly indicates its scientific character, which makes it in this case ‘a thorough knowledge of what has been revealed by God’. Consequently, the rationality of theology can be understood as an internal non-contradiction of a set of theological assertions expressing the above-mentioned knowledge. The rationality of theology makes it possible to apply to theology the basic principles guiding every science. However, to avoid confusion, one must remember that the term ‘science’ can be interpreted differently, depending on the discipline it refers to, or on the language in which the statement about the scientific nature of theology is formulated. The question as to the sense in which theology is a science will be disregarded here. e Materiali di Storia delle Religioni, 33), p. 25-43; cf. J.D. Szczurek, Trójjedyny. Traktat o Bogu w Trójcy Świętej jedynym, WN PAT, Kraków 2003, p. 13ff. 13 St. Thomas Aquinas, Summa Theologiae, I, q. 1, a. 2.

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4. Methodical nature of theology Theology is a complex field of knowledge, which includes various theological disciplines. The Polish regulations covering higher education refer to this area of study as ‘theological sciences’, an area which is divided into disciplines, and further subdivided into treatises. Each discipline within theological sciences differs according to its formal subject and its method. For example, within ‘systematic theology’ there is ‘dogmatic theology’, the formal subject of which is the content of the teaching of the Church on God and on His relations with man, its method being the analysis of the sources of this teaching and theological speculation. However, within practical theology there is pastoral theology, where the formal subject consists in the practical actions of the Church designed to pass on effectively the content of the Catholic faith, and among the methods used are those familiar in the fields of sociology or social communication. Ignoring the complexity of theology and the diversity of the methods used in theology can be a source of many misunderstandings. There is no doubt that the choice of method influences the value of the knowledge gained using that method. This applies to theology in a special way because the process of verifying the value of this knowledge is more difficult than in the case of the natural sciences. Theology must therefore strictly adhere to the general methodology of sciences. However, at the same time theology has its own detailed method, which is generally called ‘the theological method’, and which among other things makes it possible to distinguish what is revealed from what is not revealed. This method consists of three elements: auditus fidei (the equivalent of ‘experience’ in the natural sciences), intellectus fidei (equivalent to a systematic knowledge of the subject covering a given field of knowledge in other sciences), speculatio (the equivalent of theories and hypotheses in other sciences). The first element leads to a statement of the fact of revelation: the second to an

Jan D. Szczurek


understanding of its content; and the third to an expression in logical and ontological terms.14 The claim of theology to be of a scientific character is based upon its methodical nature.15

5. Speculative method 5.1. Concept The rationality of theology can be seen most clearly in the application of the speculative method, often used in scholastic theology, and recommended by the Second Vatican Council (cf. OT, 16). The speculative method in theology has as its purpose the leading of a person to the truth through the use of reasoning based on certain rules of inference defined by formal logic, thus making it possible to extract theological content hidden in the revealed premises. The most frequently used inference is deduction, the classic example of which is the Aristotelian syllogism. Theological speculation makes it possible to verify a host of theological propositions and enables the development of models, hypotheses and theological theories. Theological speculation also uses other forms of inference, such as reductive, inductive, or inference by analogy. The speculative method thus enables us to probe any vague areas within divine revelation and to explore the relationships between them (nexus mysteriorum/veritatum, analogia fidei).16 The dynamic development of the speculative method coincides with the flowering of scholasticism (the 12-13th century). The characteristic feature of the speculative method is its close relationship Cf. A. Lang, Theologische Erkenntnis- und Methodenlehre, [in:] J. Höfer, K. Rahner (eds), Lexikon für Theologie und Kirche, vol. 3, Herder, Freiburg 1986, p. 1010. 15 J.D. Szczurek, Teologia come scienza e il suo significato per la società, “Analecta Cracoviensia” 2008, no. 40, p. 312. 16 Cf. Vatican Council I, Dogmatic constitution on the catholic faith ‘Dei Filius’, Chapter 4 (DS 3016); Leo XIII, Encyclical on the study of Holy Scripture ‘Providentissimus Deus’, no. 14 (DS 3283); J.D. Szczurek, Trójjedyny..., op. cit., p. 47ff. 14

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with contemplation. An example of such speculation-contemplation is the Trinitarian theology of Richard of the Abbey of St. Victor (d. 1173). Richard derives his concept of speculation from the Latin term specula meaning a lofty place, a vantage point, or a watchtower. Speculatio, therefore, means among other things tracking, looking out for, and studying. Thanks to such speculation (looking out for), Richard tries to show why, for example, the number of the Persons of the Trinity can be neither smaller nor larger than three. St. Thomas Aquinas (d. 1274), in his understanding of speculation, relates to the idea of a mirror. According to him, speculation is the act of mind in which man discovers “the things of God in created things, as if in a mirror (divina in rebus creatis quasi in speculo)”.17 Therefore, in his view the subjects of speculation “are universal and necessary truths” (universalia et necessaria).18 Speculation belongs, therefore, to theoretical knowledge (from the Greek theoria) as opposed to practical knowledge (in the Greek téchne), the subjects of which are concrete things. Speculation leads to the knowledge or understanding of reality (scientiam vel intellectus). Therefore, speculation also helps with teaching because it enables one to explain the source of truth and understand the basis for it. According to St. Thomas Aquinas, through speculation man can reach a personal certainty of the given state of things, and does not have to rely solely on the authority of the teacher.19 In contemporary theology, speculation is not popular and is not being developed. Its abstract nature must give way to the existential dimension of theology focused, on the one hand, on the salvation of each individual human being and, on the other hand, on the personal relationship with the divine ‘You’. Thus, referring to speculation may give rise to an objection against returning to scholasticism.

17 18 19

St. Thomas Aquinas, Super Sent, Lib. 3, d. 35 q., 1 a. 2, qc. 3 corp. St. Thomas Aquinas, Summa Theologiae, II-II, q. 49, a. 6, ad 2. Cf. St. Thomas Aquinas, Quodlibet, IV, q. 9, a. 3 corp.

Jan D. Szczurek


5.2. Deduction as a tool The main tool of the speculative method is deduction. By deduction we mean: inference from the premises from which the conclusion follows logically [...]. The rules of logic, governing the inference process, ensure the reliability of the deductive inference: provided the premises in a given case of deductive inference are true, the conclusion must be true and can be deemed certain.20

One of the most common forms of deductive inference is syllogism. In a valid syllogism at least one premise must be general, while the conclusion may be either general or specific.21 A classic example of such a syllogism is the “Aristotelian syllogism consisting of three propositions, as illustrated by the following series: 1) Christ is man, 2) every man has free will (indirect premise), 3) therefore Christ has free will. The number of indirect premises can be disregarded. If every partial conclusion, and then the final conclusion, results in a necessary [that is a certain] path, then the whole inference process leads to a deductive inference”.22 In every science, the propositions which are the premises of deductive inference are called axioms (axiomata, dignitates).23 H.U. von Balthasar pointed to three fundamental theorems which serve as axioms, without which one cannot deal with theology. They are the following propositions: God exists, God has revealed Himself to man, Jesus Christ is the fullness of God’s revelation.24 None of them can be derived from the other, because the second and the third express Z. Ziembiński, Logika praktyczna, Wydawnictwo Naukowe PWN, Warszawa 1997, ed. 20, p. 155ff; cf. I. Różycki, Dogmatyka. Metodologia teologii dogmatycznej, Wydawnictwo Mariackie, Kraków 1947, p. 218. 21 Cf. I. Różycki, Dogmatyka. Metodologia..., op. cit., p. 195ff. 22 J.D. Szczurek, Trójjedyny..., op. cit., p. 48. 23 Cf. I. Różycki, Dogmatyka. Metodologia..., op. cit., p. 197. 24 H.U. von Balthasar, Christian Meditation, Ignatius Press, San Francisco 1989, p. 7. 20

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a free act of God, and this does not follow in a certain manner from His very existence. In addition, axioms are theological statements defined as dogmas of faith, not subject to any doubt, such as for example the existence of the Holy Trinity. Some dogmas of faith, however, may be conclusions arrived at in a certain manner deriving from revealed or rational premises, for example, the possibility of a rational knowledge of God’s existence and of some of the qualities of His nature.

6. Deduction in dogmatic theology 6.1. Theological foundations of deduction Dogmatic theology is the theological discipline which willingly uses deductive inference. The best example is the doctrine of the existence and nature of God. The applicability of deductive inference is inscribed into the very essence and purpose of divine revelation. In His revelations to people God uses their language and their concepts, because He wants to be understood by them (cf. DV, 1-2). As a result, when one wants to understand the content of divine revelation, one must use the same tools as are used for understanding the content of human speech and concepts occurring in it. Without a doubt, man uses deductive inference in this case.25 Although many of the propositions of dogmatic theology can be deduced in a certain manner from axioms, dogmatic theology is not entirely a deductive science because there are propositions that cannot be deduced from these axioms. These are propositions directly revealed by God. Knowing through divine revelation is the essence of theology, and therefore “an argument taken directly from divine revelation takes precedence over theological argument in which there are rational premises”.26 Theologians, therefore, need to examine carefully the content of rev-

25 26

Cf. I. Różycki, Dogmatyka. Metodologia..., op. cit., p. 199. Ibidem, p. 209.


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elation to discern properly what has actually been revealed and in what sense divine revelation uses a given word, that is, what is the scope of the revealed concept which is to constitute a premise for further inference.27

6.2. Deduction as a research method Deduction is a valuable and powerful research tool for theology, especially for dogmatic theology. Using deductive inference, one can organize known propositions in a deductive system, or search for propositions as yet unknown and undefined by the Church, but which will provide answers to questions posed concerning divine revelation. With the help of deduction, from a proposition contained in divine revelation, a theologian can deduce a true statement, which is equally contained in the proposition. From the history of dogmas, it is known that such reasoning is open to serious danger. One crucial example would be the reasoning of Nestorius, which became a source of heresy.28 Nestorius’ line of reasoning can be expressed as follows: It is revealed that Christ is a true man. Every true man has a natural human personality. So Christ has both a human personality and a divine personality. Such a statement is heretical as, by attributing a double personality to Christ, it attributes to Him a mental illness. The error in the inference of Nestorius consisted in the fact that he did not discern properly why Christ is called a true man in divine revelation. The inference calls Him a true man because of His human nature, which He has and which is common to all human beings, whereas personality is an individual non-transferable disposition and does not belong to nature as such, as it expresses individual characteristics. Because Christ has a human nature, one cannot conclude that there was a hu-

27 28

Ibidem, p. 213. Ibidem, p. 211.

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man personality in Christ. Nestorius took as a revealed concept what was not a revealed concept, and by accepting it as a premise in his reasoning, he came to the wrong conclusion.29 Nestorius’ example shows that in the process of deductive thinking there is the possibility of error. Basically, in this reasoning there are two types of errors, material and formal. A material error consists in taking false premises as true, while a formal error consists in reasoning according to a formula, which in reality is not a rule of logic.30 Unfortunately, in modern theology it is difficult to find studies that critically evaluate various theological propositions and hypotheses in terms of their logical correctness.

7. Other forms of theological inference In addition to the certain method of deduction, theology uses uncertain inference, i.e. the one in which the truth of premises does not determine the truth of the conclusion. These include reduction, induction and analogy. Reduction in dogmatic theology is sometimes called interpretation. It involves searching for the right premise for a given proposition. There may be more than one such premise. Interpretation is used when one is looking for evidence for the proposition recognized by the Church as a dogma of faith, as in the case of the dogma of the Immaculate Conception.31 Reduction is used when creating theological hypotheses. Induction may be complete (strong) or incomplete (weak). Complete (or strong) induction means the type of inference which is made on the basis of many individual premises, which determine that concrete objects have certain attributes, and reaches a general conclusion stating that each of these objects has these attributes. If the conclusion 29 30 31

Ibidem, p. 212. Z. Ziembiński, Logika..., op. cit., p. 176ff. Cf. I. Różycki, Dogmatyka. Metodologia..., op. cit., p. 220ff.


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relates to a larger number of objects than the number of individual premises, then incomplete induction takes place.32 Inference by total induction is certain. Induction in theology is used, for example, in formulating generalizing propositions which accept as premises specific individual sentences of revelation. Sentences spoken by Christ about God can be given as an example: “My Father and your Father” (cf. John, 20: 17) and in another place, “Abba, Father” (Mark, 14: 36). From such expressions one can draw the conclusion that Christ is true Son of the God of Israel. Analogy is a concept present in various fields of knowledge. Theological inference by analogy consists in the inference of the existence of hidden properties, directly inaccessible to the mind, on the basis of several properties identified with certainty.33 This inference is based on the existence of similarity. The concept of analogy is present in the Holy Scriptures. In the Book of Wisdom (Wisdom, 13: 5), analogy is used to denote the way leading to the knowledge of God through the knowledge of the power and beauty of creatures. This is the analogy of proportionality. The fact that certain inferences exist in theology alongside the uncertain leads to distinguishing degrees of theological certainty. Apart from theologically certain propositions resulting from certain inference, one can distinguish theologically probable assertions gained through uncertain inferences. In addition to these degrees, there are absolutely certain assertions, that is dogmas of faith.

8. Mystery Theology through its speculative nature enhances reason. The particular nature of theology lies in the fact that, as a science, it does not stop at rational data, but reaches out to ‘faith in revelation’ as its

32 33

Cf. Z. Ziembiński, Logika..., op. cit., p. 184-186. I. Różycki, Dogmatyka. Metodologia..., op. cit., p. 218.

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main source (Fides et Ratio). Faith includes propositions which are not obvious to man’s reason, and these are called in theology ‘mysteries of the faith’. Through contact with mystery, the human mind realizes both its limitations and, at the same time, its dignity, which comes from the ability to recognize what is for it a mystery. Theologians must always be accompanied by the awareness of mystery (sensus mysterii), so as to avoid errors when examining the content of the mysteries of faith by means of the deductive method.34 This refers to the truths of the faith such as the existence of the Holy Trinity, the Incarnation, or the freedom of man endowed with grace. The concept of mystery in theology differs from such a concept in the natural sciences. The mystery in theology does not spring from a current ignorance of the object of knowledge that might be overcome, but from the limitations of human reason. A characteristic feature of the mystery of faith is that the human mind is able to understand the elements of the propositions expressing the mystery, but is not able to understand their connection. In the case of the mystery of the Holy Trinity, it is obvious to the mind that God must be one. Based on experience, it is clear what it means to be a father, and what to be a son, and in what way the spiritual life of man can be manifested. However, the combination of these elements in a statement: the One God is Father, Son and Holy Spirit is not obvious. It expresses the essence of mystery.35 The ultimate source of both the mystery and the rationality of theology is the Incarnation of the Λόγoς. Theology is logical, because the Λόγoς is its centre and principle. Therefore, the actions of God in the world are logical and understandable to man, and theology is seen as a set of propositions internally non-contradictory, meaningful and systematized. Rationality is also present in the mystery itself. If God were within the measure of the human mind, if He were possible to understand,

34 35

Ibidem, p. 213. Ibidem, p. 223ff.


Jan D. Szczurek

He would not be God. Mystery teaches us intellectual humility and thus protects against error. The mystery of faith also calls for an ever deeper studying of the truths of faith and seeking personal contact with the Mystery itself.

Mieszko Tałasiewicz University of Warsaw

Science as Theology*


popular account of the provinces of science and religion today holds that science and theology are incompatible: non-overlapping magisteria, which results in their complete separation, both in terms of the object of their pursuits and the research methods they use. There is an extent to which this account is sometimes questioned: attempts are made to initiate a dialogue between the two sides. Stanisław Wszołek1 believes that science and theology lend themselves to mutual explication, that is, each throws some light on and gains support from the other for its own findings. In this paper, I would like to advance a far more radical claim. I will argue that the relation which exists between science and theology is neither that of separation nor local explication but the complete inclusion of one within the other. From the atheist’s point of view, theology is included in science, for from this perspective theology has essentially no subject matter. It is useful in so far as it provides some insight into the sociology or history of religious beliefs, which are themselves the proper subsets of sociology or history tout court. From the point of view of religion, it is science, all science in fact, that is contained in a broadly understood theology. The air of radicalness of this claim flows from the common view that science and theology are concerned with different ontological orders: the natural and the supernatural. From the Christian point of This contribution was made possible through the support of a grant “The Limits of Scientific Explanation” from the John Templeton Foundation. 1 S. Wszołek, Racjonalność wiary, Wydawnictwo Naukowe PAT, Kraków 2003. *


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view, however, there is no question of there being any different orders. There is one order: God made Heaven and Earth, including all things on Earth and all the laws which govern them. Nature and nonnature can only be distinguished epistemologically: nature is what we can know by scientific methods. Science – the scientific method – is here the definiens of the term ‘nature’. The ontological distinction between nature and non-nature is possible only on the grounds of heretical gnostic doctrines, as a Manichean claim that the spirit comes from God, while nature is that which comes from the demon. A Christian cannot accept this view. To them, everything that exists (created) has the same origin. The claim I am making may also appear radical for reasons of a narrow understanding of theology as a group of disciplines which are actually taught in departments of theology, under the said name, or, as it is also understood, a study based on revelation. Clearly, I do not mean for it to have such narrow understanding. I do not aim to claim that physics or biology should be subjects taught in the departments of theology, nor that the laws of nature were revealed to us. I subscribe to the broad understanding of theology as the study of God and all God’s works. Since physics and biology are both studies of certain God’s works, they represent disciplines of the broadly understood theology. Another reason why my claim may go counter to our intuition is the belief that its acceptance would mean the subservience of natural sciences to theological dictates. Such reasoning is fallacious, too. Not an iota is changed between natural sciences taken as theological disciplines and the same sciences taken outside this perspective: either as regards their object of pursuits or their methods (I leave aside the question of whether some scientific research is morally acceptable; anyway, it does not depend on the substantive connections between science and theology). The treatment of natural sciences as disciplines of a broadly understood theology does not change science, however, it makes a big change to theology; some things take on a different meaning even in parts of theology as taken narrowly.

Science as Theology


Revelation tells us of the meaning of human life, of the task God gave to man, and of how to act and what to eschew. It also says that the world was created by God. It does not explain, however, how the world was created and what mechanisms God furnished it with. If we so wish, we can seek such explanations by ‘the natural light of reason’. And so we do. We have developed sophisticated scientific research methods and made a number of important discoveries which reveal to us how the God-created world works. The scientific methods of studying the world are the product of our efforts, and the proposition that the world was created by God, or that it was not, does not change anything. Sciences, in turn, theoretically have no bearing on what was revealed. They do not tell us of the meaning of life, of what task God gave us, of how to act and what not to do. In practice, however, they are important to our understanding of certain aspects of revelation. The clearest example of this can be found on ‘the front line’ of understanding: at the level of verbal comprehension. Revelation is expressed in our language, and in order to understand it correctly, it is important to know how human languages work (in particular, the languages in which revelation was expressed). The narrowly understood theology (as an academic discipline) took this fact on board a long time ago, making the study of language one of its ancillary disciplines. Similar relations, however, can be observed in other sciences. If we were to follow the teachings of revelation as regards how to act and what not to do, we would need knowledge – in matters of psychology and biology – about the mechanisms which we, as biological organisms, rely on for our natural behaviour, about what our natural tendencies are, and what is likely to cause a special difficulty for us. In doing so, we would essentially be like a teacher of logic who tries to understand what reasoning mechanisms are responsible for certain errors notoriously made by his students and who uses such knowledge to match the educational tools to the students’ needs. Psychology does not define the laws


Mieszko Tałasiewicz

of logic – it can, however, determine the way in which those laws are taught.2 We can say this much: some disciplines of theology, where the latter is taken in its conventional sense (not the theology that deals with all of God’s creation but only with what was revealed) seek above all how to make humans aware, in the most effective way, of what was revealed and how best to help them fulfil this mission. To this extent, thus understood theology should be interested in studying the actual biological, cultural and social determinants of man’s existence so that they could make the most of the favoring circumstances, while guarding against circumstances which conspire against them. Theology asks these questions of science, and if it wants to be rational, it should respect the answers, whatever they may be. The relation holds in one direction: science says how things are; theology takes it on board and tries to work out how to use it to aid in the understanding of revelation. Telling science what theology would like to hear is devoid of purpose. There are no good or bad answers. Actually, whenever science makes a discovery which appears to contradict revelation, the worse of a threat such knowledge represents, the more valuable it should be to theology: an enemy that is known and understood is easier to defeat than that which is unseen and underappreciated. Any attempt to negate and shut out such knowledge is not only ineffective but positively harmful. The correct response would be to adjust the way of explaining revelation to the new evidence.3 The classic example to support our case is of course the theory of evolution and the history of its reception. Some theologians found the theory repulsive, as seeking to undermine man’s belief in creation by These methods, too, can change depending on the development of psychology. Similarly, the methods of teaching foreign languages often change, although the languages – their grammars – change very slowly. 3 “When an apparent conflict arises between a strongly supported scientific theory and some item of Christian doctrine [...] it may well be that [...] the scientific understanding will enable the doctrine to be reformulated in a more adequate way” (E. McMullin, Introduction, [in:] E. McMullin (ed.), Evolution and Creation, University of Notre Dame Press, Notre Dame–Indiana 1985, p. 2). 2

Science as Theology


God. In time, however, the Church came to terms with the evidence and argued that God must have apparently thought it useful to choose evolution as the way of accomplishing His creation. There are no more attempts to abolish the theory of evolution. Instead, more and more often, although not often enough as yet, evolution becomes an important part of the knowledge necessary to interpret and communicate revelation as well as generating new ways of thinking intended to enhance our intuitive understanding of certain revealed truths.4 It is hard to imagine theodicy today without the theory of evolution.5 Evolution also offers an intuitive solution to another problem, raised among others by Jean Paul Sartre: how could God, whose will governs all existence, have created a being who is able to act according to his or her own will? Consideration of this purely theological question in abstracto, without any empirical evidence, strongly suggests that in the act of creation God must have placed some kind of screen between man and Himself, that He must have grounded human personality in some axiologically neutral medium which exceeds human ability to comprehend. The world of nature, which is structurally rich and value-free and which provided a setting for the evolution of a human as an animal, is an ideal answer to this purely theological question. Evolution, as well as creation of a highly complex material

For although “the tradition of faith is not given to us for the satisfying of curiosity” (cf. J. Ratzinger, Eschatology: Death and Eternal Life, The Catholic University of America Press, Washington 1988, p. 161), “it must be recognized that human beings, in their need for an object to contemplate, felt compelled to unfold images once again. There is nothing perverse in such a re-creation of iconic forms. Indeed, it would be foolish to strive for a completely imageless piety, in blatant contradiction of human nature. However, precisely this consideration makes it all the more important to evaluate images in terms of their true measure, [...] and to prevent them from shooting off into the realms of mythology. [...] It is not the business of those entrusted with preaching the faith to expel the images from the Church. But it is very much their business to purify them again and again” (ibidem, p. 132). 5 See M. Murray, Nature Red in Tooth and Claw: Theism and the Problem of Animal Suffering, OUP, Oxford 2008. The main contribution of the theory of evolution to theodicy is the presentation of pain as a generally desirable adaptation and, yet more generally, the death of individual organisms as a necessary condition for the development of higher forms of life. 4


Mieszko Tałasiewicz

world, need not be an unsolved puzzle, ‘God’s whim’, but can be interpreted as the basis of human quasi-independent existence and free will. The distrust felt towards scientific discoveries did not disappear from theology, though, along with the acceptance of the theory of evolution. We regularly witness theologians condemning new discoveries. To illustrate it, let us consider a few issues which have been the focus of debate in the last few years. A good point to start with is the scientifically controversial and hotly disputed (by theologians) discovery, or alleged discovery, made by Dean Hamer of a genetic predisposition towards religion (the socalled ‘God gene’).6 Hamer postulates that there is a correlation between a certain form of gene (VMAT2) and susceptibility to religious belief, measured, among other things, by the subjects’ ‘openness to believe things not literally provable’. Hamer also identifies an ecological mechanism responsible for the transmission of the religionprone allel: the carriers of this allel, thanks to the increased production of neurotransmitters encoded by it, have a more genial personality and are more optimistic, which in turn leads to their having a larger number of children. Hamer’s work has run into serious criticism from other scientists,7 and it is unclear whether the mechanism described by this author really exists. I am not going to debate the issue here. However, the idea itself that the predisposition towards religious belief, towards what is invisible, varies in degree, and that the variability is somehow congenitally determined, does not seem absurd to me. On the contrary, it accords with my evolutionary intuitions and I think it quite credible. Perhaps the genetic background of such a character trait is more complex than simply the mutation within one gene. The ecological mechanism involved here may well be different, too. Intuitively D. Hamer, The God Gene: How Faith Is Hard-Wired into Our Genes, Doubleday, New York 2004. 7 C. Zimmer, Faith-Boosting Genes: A Search for the Genetic Basis of Spirituality, “Scientific American” 2004. 6

Science as Theology


speaking, I am inclined to think that the character trait that is subject to selection pressures is not the predisposition towards belief in the invisible among individuals but the frequency with which this predisposition occurs within a group that is a subject to kin selection.8 Meanwhile, the idea of genetic variation as regards predisposition to religious belief attracts vehement opposition from many theologians, including Polkinghorne, for whom I have high regard and whom I have often affirmatively quoted, but who in response to an interviewer’s question said that, “The idea of a God gene goes against all my personal theological convictions”. The force of these reactions reveals a conviction that, according to such theologians, there is something thoroughly ungodly in accepting that some people are naturally more predisposed to believe in God than others. Everybody is the same distance away from God, they say. I find this opposition baffling, while the claim that everyone is the same distance away from God is, to my way of thinking, patently false. Some people are spared by life’s hardships, others bear the full brunt of them. Some people grow up in peaceful times, others are caught up in the throes of war. Some are able-bodied, others disabled. Some are brought up in a religious environment, others not. In a word, some are a short distance away from God, others far, far

Qualitatively speaking, we can easily imagine a model which favours groups consisting of both individuals who are especially prone to belief in the invisible and those who are especially reluctant, but which eliminates groups which consist only of the “credulous” or only of “skeptics”. A mixed group, when acting as a whole, is capable of a wider range of reactions to a new and unfamiliar situation. The “credulous” are not only quicker to believe in transcendent experiences but are also more willing to believe in old legends told by the elders around evening bonfires and passed down from generation to generation – legends which, besides the obvious exaggerations and fantasies, contain a healthy dose of information about rare but not uncommon unusual situations, once occurring, and the ancestors’ ways of dealing with them. If the new situation which affects the group is only relatively new (outside the living memory of members of the group) and has, in fact, occurred in the past, the “credulous” who have stuck to traditional wisdom can come up with credible solutions, which is preferable to having to work out new solutions by trial and error (where the error may be fatal). If on the other hand the situation is also new to legend, the group will benefit from having skeptics amidst its ranks, who are naturally predisposed to look for ad hoc solutions. 8


Mieszko Tałasiewicz

away. God’s justice does not consist in everyone being the same short distance away from Him, but in God’s knowing the different distances we have to walk and always meeting us at the right spot. With some, He needs only take a step forward to meet them, with others He will not stop short of carrying them off. This is what the Bible says: Jesus approached St. Peter and said: “Come with me”, and St. Peter followed Him. Doubting Thomas, the Apostle, would not be satisfied until Jesus showed him His hands and side. St. Paul had to be struck blind in the desert before he could see the truth. Thus, regardless of whether Hamer’s work is indeed a scientific achievement or not, his hypothesis could at least be true. It does not go against faith’s intuitions. On the contrary, it accords with them and expresses them: it would not be far-fetched to refer to the corresponding genetic profiles as Peter’s profiles and Paul’s profiles. Another example is furnished by the still hotly-debated discoveries of Benjamin Libet. His subjects were asked to press a buzzer at a point of their choosing along a time interval and to remember the position of the experimental clock indicator at the time they made the decision to press the buzzer. At the same time, all subjects were undergoing an EEG test. Libet demonstrated that the so-called ‘readiness potential’ in the part controlling motor-functions, reflected in the EEG test, precedes by about 500 ms the point which the subjects indicated as the time at which they had made a decision. Libet’s research (carried out over the 1970s and 1980s) had a tremendous impact at the time, which, far from declining, has actually gained in force over time (perhaps thanks to the recent publication of Libet’s book9). Many commentators and followers (including for example Daniel Wegner10) hold that the research findings suggest that the sense of agency is an illusion, and that man has no free will after all. It is not hard to guess that, for this reason, theologians have treated the findings as highly suspect and, again, this is completely unjustified. B. Libet, Mind Time: The Temporal Factor in Consciousness, Harvard University Press, Cambridge 2004. 10 D. Wegner, The Illusion of Conscious Will, MIT Press, Cambridge 2002. 9

Science as Theology


The findings, as any findings, should have been treated with suspicion by science itself. Critical analysis of one’s own findings is after all a measure of the rationality of science and a part of scientific methodology. William Klemm11 identifies a number of intra-scientific reasons why a claim whereby the arbitrary pressing of a button is outside the influence of conscious free will should be treated with a great deal of skepticism. Let us list a few by way of an example: – the point which in this research is taken to be the point at which a conscious decision is made is, in fact, the point when we engage in a conscious reflection over the making of the decision and when we place the decision on an experimental time scale; measurements were made of the meta-reflection over the decision, not of the decision itself; – the EEG test focused on selected parts of the brain, while making decisions freely may be a process which extends over time and over large parts of the brain; thus no firm conclusions were reached about whether the potential to act is prior to the point of decision-making. Let us suppose that further research clears up, hypothetically, Klemm’s doubts and finds that humans have, beyond any doubt, no conscious control over their spontaneous pressing of a button, that an impulse is generated in the brain quasi-randomly, while consciousness creates the illusion of a free decision only ex post facto. Again, the finding would contain nothing that the Church or moralists, including secular moralists, have not been proclaiming all along: And what does it mean to be free? It means to know how to use one’s freedom in truth-to be ‘truly’ free. To be truly free does not at all mean doing everything that pleases me, or doing what I want to do.

W. Klemm, Free Will Debates: Simple Experiments Are Not So Simple, “Advances in Cognitive Psychology” 2010, no. 6.



Mieszko Tałasiewicz

Freedom contains in itself the criterion of truth, the discipline of truth. To be truly free means to use one’s own freedom for what is a true good. Continuing therefore: to be truly free means to be a person of upright conscience, to be responsible, to be a person ‘for others’.12

In short, we need to differentiate will from wish. A situation where at a certain interval I am to press a button is not a paradigmatic example of free will, but an example of following random impulses. It is precisely a paradigmatic example of giving in to the illusion of free will. When I do what I wish, I am not truly free; I am a slave to my desires. I am free only when I do what I ought to do. According to the so-called ‘freethinkers’, Christianity stands for enslavement: a set of prohibitions, injunctions and restrictions. Where is the freedom?, they ask. They rebel against faith by invoking freedom. If I feel like eating sausage on a Friday, then that is what I am going to do. If I feel like getting together with a new partner, then I ditch my existing partner and opt for the new one. Meanwhile, the moralist says, since time immemorial – now backed up by Libet’s and his followers’ studies – these are the cases where it may at most 12 John Paul II, Apostolic Letter Dilecti Amici to the Youth of the World, 1985. See also: Catechism of the Catholic Church (CCC, http://www.christusrex.org/www1/CDHN/ ccc.html), article 2339: “Chastity includes an apprenticeship in self-mastery which is a training in human freedom. The alternative is clear: either man governs his passions and finds peace, or he lets himself be dominated by them and becomes unhappy. ‘Man’s dignity therefore requires him to act out of conscious and free choice, as moved and drawn in a personal way from within, and not by blind impulses in himself or by mere external constraint. Man gains such dignity when, ridding himself of all slavery to the passions, he presses forward to his goal by freely choosing what is good and, by his diligence and skill, effectively secures for himself the means suited to this end”. CCC 2223: “The home is well suited for education in the virtues. This requires an apprenticeship in self-denial, sound judgment, and self-mastery – the preconditions of all true freedom”. CCC 908: “That man is rightly called a king who makes his own body an obedient subject and, by governing himself with suitable rigor, refuses to let his passions breed rebellion in his soul, for he exercises a kind of royal power over himself. And because he knows how to rule his own person as king, so too does he sit as its judge. He will not let himself be imprisoned by sin, or thrown headlong into wickedness” (after St. Ambrose, Expositio Psalmi CXVIII, 14, 30: PL 15, 1403 A).

Science as Theology


seem to us that we are making a decision, but in fact we are being driven by desires, which, in order to sweeten the pill, may give us the epiphenomenal feeling of having made a decision. Thus, the message that follows from Libet’s studies fits in with the age-old moral teachings: if we want to demonstrate our free will, we should not fall back on a feeling that attaches itself to some ad hoc ‘decisions’. The only way in which free will, hence true freedom, manifests itself is in our making a prior decision, sufficiently far in advance, about what we are going to do at a given point in time, regardless of the circumstances. The best thing is if the running order of future decisions is determined outside our mind. If we have free will, then we will find it in us to work through this arbitrarily set order. If we cannot find in us what it takes to dispatch the task, then we have neither free will nor freedom. This is what lies at the heart of the liturgical Lent calendar. The Lent calendar aims to impose an arbitrary pattern on our decisions. We are required to decide at certain points in time to refrain from eating. This is always technically doable (the same cannot be said about an actual decision to eat something), yet by refraining we hold off a powerful desire. To this extent, success is a good measure of our being able to fulfil our freedom. The same goes for other desires, and for other Church-ordained prohibitions or prescriptions. Such prohibitions and prescriptions sometimes appear to be unnatural precisely because they are meant to be helping us to detach ourselves from our animal, automatic, desire-driven and slaving nature. They help us make decisions which are not determined by desires, that is, decisions which are an expression of our freedom and fulfilment of our human nature. Libet’s research is thought to put a question mark over free will only because the concept of freedom in post-Libet arguments has been distorted. It is said now (e.g. Seth Lloyd) that we are free when we do not know what we are about to do (although this can be known to a researcher examining our brain waves). However, true free will is precisely the opposite: when I follow my will, I know what I am

Mieszko Tałasiewicz


going to do not in 500 milliseconds but in five months – something the researcher looking into my brain does not know. It is this understanding of free will that finds expression in our language: we say that someone has a ‘strong will’ not when they are actively pursuing their desires but when they are able to resist them. My last example is the question of the existence of an innate moral sense (an area studied, for example, by Marc Hauser13). In this case, the methodology of the research is fairly beyond reproach (pace recent accusations of scientific misconduct), and we are justified in claiming that humans possess a universal moral sense which has evolved over time and thanks to which we perceive doing personally physical harm to another person as evil. Historically, theologians have come down on both sides of the question. On the one hand, they feel vindicated: it is the ‘freethinkers’ who used to say of morality as being a fancy notion, dreamed up by humans, and as something occupying a relative position with respect to culture, while theologians have argued that conscience – the ability to distinguish between good and evil – is an innate disposition, a responsiveness to the universal natural law. On the other hand, provision of universal morality used to be held solely within the domain of religion, so the discovery of a biological moral sense strikes at the heart of this claim. In fact, as with the examples discussed earlier, scientific discoveries are fully in line with age-old moral knowledge: humans have an innate sense of morality, which, in some circumstances, allows them to distinguish between good and evil: For when the Gentiles, which have not the law, do by nature the things contained in the law, these, having not the law, are a law unto themselves. Which shew the work of the law written in their hearts […], wrote St Paul (Rom. 2, 14–15).14 M. Hauser, Moral Minds: How Nature Designed a Universal Sense of Right and Wrong, Harper Collins/Ecco, New York 2006. 14 Quotations from the Holy Scripture based on The Holy Bible, King James’ Version. 13

Science as Theology


At the same time, however, innate moral sense is not enough to produce complete morality, which must come from a rational source and which must be controlled by consciousness. Jesus said: You have heard that it hath been said, Thou shalt love thy neighbour, and hate thine enemy. But I say unto you, love your enemies, bless them that curse you […] For if ye love them which love you, what reward have ye? Do not even the publicans do the same? And if ye salute your brethren only, what do ye more than others? Do not even the publicans so? (Matt. 5, 43-47).

Jesus’ teachings do not invoke natural law. On the contrary, they are explicitly opposed to natural morality. It is natural to love our friends, but Jesus has us love our enemies as well. It is natural to reciprocate kindness with kindness (reciprocal altruism), but Jesus has us reciprocate evil with good, too. It is natural to pay for the work done, but Jesus proposes to pay the same wages to a worker who has worked barely an hour as to the one who has worked the whole day. Anyone today who thinks this is natural should try to agree a similar paying arrangement with his builders… Natural moral sense is necessary for the understanding of morality, for the ability to have the concept of good and evil. But it is not itself a morality. “For the promise, that he should be the heir of the world, was not to Abraham, or to his seed, through the law, but through the righteousness of faith” (Rom. 4, 13). What is so disturbing then in the Harvard Moral Test? *** Summing up, on the understanding of science as a theological discipline, no scientific discoveries are ever likely to pose a threat to revelation; they can at most prompt us to adjust our notions, which Joseph Ratzinger urged us to do (see footnote 4), and to constantly reinvent what we think is the best way in which we communicate re-


Mieszko Tałasiewicz

vealed truths to our contemporaries, in a world which is awash with scientific and technological achievements. Perhaps this is the main reason why so many theologians are reluctant to embrace scientific achievements. The progress of science is marked by relentless change, and so must be the ways in which theological truths are communicated. Perhaps theologians cling too firmly to the status quo, regarding knowledge about the world and its application to communicating truths about faith, which was established once and for all by St. Thomas, and are irked to have to admit that so much in this eternal order needs changing, nay constant changing. Well, I do not think there is an alternative. St. Thomas thought he had solved the problem once and for all, because in his view earthly knowledge, which he possessed, was the ultimate achievement of the human race, requiring but small adjustments in the future. Meanwhile, knowledge and the view of the world it projects have since undergone fundamental changes. And change they will in the future. Just as scientists have come to terms with the idea that they are not laying down theories to last for centuries, but only for a few or a dozen years at most, so theologians should get used to the fact that the work of their generation will be out of date in the next generation. That said, it is vital that that work be continued, because if it is not, then a generation of believers will cease to understand their faith and the world they live in, and the next generation of theologians will have even a bigger job on their hands. Let us not be afraid. He who believes that God created the world has nothing to fear from finding that the world is incompatible with God. It may be incompatible at most with our notions. Another reason for the reluctance of theologians to accept certain scientific discoveries is a desire to use other scientific discoveries in Christian apologetics. All those who wish to bring in science to support a religious tenet try hard to denounce all discoveries that do not fit this particular tenet. At best their effort is useless. Science is too far-ranging to be made to fit any preconceived notions; for each discovery supporting the preferred view, there will be another that goes

Science as Theology


against it. Moreover, a particular discovery will represent evidence in favour, to some, and evidence against, to others (see the case of moral sense). At worst, their effort will have serious negative consequences. It is a psychologically proven fact, however logically unsound, that people think of rejecting some evidence in favour of a theory as if proving opposite theory. Enemies of faith are fond of using this mechanism against their opponents: they show, without much difficulty, weaknesses in such apologia, and announce triumphantly that they have proven religion to be false. And people believe them. That is why, it is not worthwhile to try to prove the existence of God using arguments from science. Theology is not a part of science. Rather, science is a proper part of theology. Science as theology assumes the existence of God but stops short of proving it. It is after all, as I have argued, a methodologically neutral assumption: science about a world created by God is empirically indistinguishable from science about a world which has sprang up from nothing, by accident.

Wojciech P. Grygiel The Pontifical University of John Paul II Copernicus Center for Interdisciplinary Studies

Physics in the Service of Theology: A Methodological Inquiry* 1. Introduction


ontemporary theology is a very broad and diversified discipline. Its main goal, however, is solely based on the conviction that the observable realm of the physical Universe is not the ultimate reality beyond which nothing more exists. If this were indeed the case, natural sciences would define the ultimate knowledge that could be attained by the human mind. The exercise of theology, on the other hand, presupposes that some truths can be communicated by means of the so-called revelation. In other words, these truths are not attainable by purely rational inquiry, but rather the light of reason is capable of yielding explanations of what has been revealed. Thus theology becomes science because it offers the exposition of the revealed material in a systematized form.1 By considering philosophy to be the ‘maid of theology’ (philosophia ancilla theologiae), Saint Thomas Aquinas clearly indicated that revelation, as accepted by the human mind and illuminated by faith, must reach an understanding in accordance with the famous Augustinian dictum fides quaerens intellectum.2 Since understanding is achieved by means of concepts,

* This contribution was made possible through the support of a grant “The Limits of Scientific Explanation” from the John Templeton Foundation. 1 Cf. St. Thomas Aquinas, In Boethii De Trinitate, q. 2, a. 2. 2 St. Thomas Aquinas, Summa Theologiae, I, q. 1, a. 5.


Wojciech P. Grygiel

the role of philosophy in theology consists primarily in supplying the conceptual framework as well as the inferential rules so that the proper theological explanations can be given. It means that, in principle, one could admit a variety of philosophical systems in the service of theology. After the onset of the Humean critique, however, that abolished the basis of medieval epistemology and the direct correspondence between an objectively existing reality and its mental representation, theology had to face the scepticism of the epistemological paradigm of the probable reasoning.3 Two responses can be given to such a state of affairs. The first of them is to regard post-Humean philosophy as entirely unfit for theological discourse and to retain the medieval model by protecting it with the principle of the non-intersecting methodological planes. This attitude is particularly vibrant in the neo-Scholastic tradition.4 Although the classical theological discourse remains intact in such a case, it is entirely sealed off from any interaction with science. In the second response, on the other hand, one recasts the conceptual foundations of theology and the rationality of belief in the framework of the critical realism espoused by the contemporary philosophy of science. Such an attempt has been already carried out by Nancy Murphy, who has taken pains to demonstrate that the tenets of theology yield knowledge in the same way as those of the natural sciences.5 The goal of this article is to propose a methodological approach to theology that strengthens its claims in the perspective of scientific reasoning by incorporating several key elements central to the methodology of the most formalized branch of the natural sciences, namely, physics. It will be shown that the validity of the theological A very detailed study of the impact of the Humean thought on theology and, in particular, on the problem of authority can be found in: J. Stout, Flight from Authority, University of Notre Dame Press, Notre Dame 1981. 4 Cf. M. Heller, Philosophy in Science: A Historical Introduction, Springer, Heidelberg–Dordecht–London–New York 2011, p. 153. 5 N. Murphy, Theology in the Age of Scientific Reasoning, Cornell University Press, Ithaca–London 1990. 3

Physics in the Service of Theology: A Methodological Inquiry


inquiry can thus be enhanced by taking into account the totality of the theological outputs generated with the use of the different conceptual schemes. Since the Divine reality is independent of the existence of the human mind, the truths pertaining to this reality should be invariant in respect to the selected conceptual scheme. Ultimately, this approach brings a new methodological dimension of the relations between science and theology to the fore, thereby setting off the attitude of conflict and opposition between the two disciplines.

2. Why bother? The value of attempting to enhance the theological discourse by drawing from the method of the formalized natural science such as physics is at least three-fold. First of all, the methodology of physics is very well studied and its mode of inquiry directly relies upon the logical consistency of its basic tool, that is, mathematics. Secondly, the methodology of physics has turned out to be extremely effective in revealing the laws governing the Universe, thereby giving an unprecedented insight into the Divine plan of creation. Thirdly, there exists a certain analogy between the theological inquiry and the inquiry of physics (and of the natural sciences in general). It finds its basis in the famous dictum that can be traced back to the times of the early Christianity. The dictum asserts that God has written two books, the Book of Scriptures (Revelation) and the Book of Nature,6 thereby revealing Himself in two ways: by means of a directly communicated statements contained in the Bible, and by means of the mysteries of Nature discovered by scientists in experiments and presented in the form of the mathematical laws. A well-known medieval thinker, Hugo of St. Victor (1096-1141) comments on this issue in the following way: The detailed discussion of this dictum in the historical and philosophical context can be found in: O. Pedersen, The Two Books: Historical Notes on Some Interactions Between Natural Science and Theology, Vatican Observatory Foundation, Vatican 2007. 6

Wojciech P. Grygiel


The whole of the sensible world is like a book, written by the finger of God, being created by His power. And the individual creatures are like symbols or letters that are not invented in an arbitrary way by man, but ordained by the will of God in order to demonstrate His Wisdom.7

The issue appeared in a particularly sharp focus in the works of Galileo as he struggled to clarify his understanding of the relation between revelation and the natural sciences. In his Letter to the Duchess Christina, Galileo gave nature the name of “the executrix of God’s commands”. Consequently, both the scientific laws and the scripturally revealed truths are alike from the methodological point of view insofar as they both yield statements that require rational explanations with the use of conceptual frameworks resident in the human mind. In other words, they are justified within the context of a larger of a physical or a theological theory, respectively. The methodological likeness of theology to the natural sciences has been succinctly observed by John Polkinghorne: Science does not have a privileged route of access to knowledge through some superior ‘scientific method’, uniquely in its own possession; theology does not have access to knowledge through some ineffable source of ‘unquestionable’ revelation, uniquely its own possession. Both are trying to grasp the significance of their encounters with manifold reality. In case of science, the dimension of reality concerned is that of a physical world that we transcend and that can be put to experimental test. In the case of theology, it is the reality of God who transcends us and who can be met only with awe and obedience. Once that distinction is understood, we can perceive the two disciplines to be intellectual cousins under the skin, despite the differences arising from their contrasting subject material.8 Hugo of St. Victor, Didascalion, VII, 3. J. Polkinghorne, Science and Theology: An Introduction, Society For Promoting Christian Knowledge, London 1998, p. 20.

7 8

Physics in the Service of Theology: A Methodological Inquiry


Pointing out this methodological similarity of theology to the natural sciences strengthens the objective character of theology by stressing its primary goal of demonstrating truths pertinent to the Divine reality and not merely surveying its mental representations. Neither theology nor physics proceed from fixed a priori premises but each of them is confronted with the Divine rationality that infinitely transcends the capacities of human mind. Consequently, in physics one must rely entirely on experimentation, while in theology it is ultimately the free will of the revealer that unveils certain otherwise inaccessible truths. The methodological likeness of physics and theology has received further substantiation in the works of Joseph M. Bocheński, especially in his book entitled The Logic of Religion.9 Bocheński points out that in the modes of explanation both disciplines are reductive in the sense that they commence their inquiry from a set of statements that in physics are statements of empirical facts, and in theology they form the so-called ρ-sentences, that is, the elements of the objective faith to be maintained by a believer.

3. Varying conceptual frameworks of theology The origin as well as the ontological status of concepts is a contentious issue in philosophy. The prevalent attitudes in the theological milieu rely on a broadly considered Platonism where concepts are considered to have a fixed content often hypostasized into a separate realm of atemporal ideas or, as it is in the Aristotelian case, resident directly in things thereby constituting their essence. In such cases, the content of concepts does not evolve and the philosophical systems based on them are often treated as perennial. At this point, it seems proper to emphasize that St. Thomas Aquinas, who is often considered as the father of the perennial philosophy, made a very innovative

J.M. Bocheński, The Logic of Religion, New York University Press, New York 1965, p. xii and 180. 9


Wojciech P. Grygiel

step for his time by basing his theology on Aristotle, thus limiting the neo-Platonic component in his inquiry. Thus, he signalled that theology does not have to limit itself to the use of one philosophical system and, moreover, it may greatly benefit from adopting new conceptual frameworks. Two main advantages of such a strategy can be pointed out: (1) changing the conceptual frameworks may permit the use of better logic, thereby securing the certitude of the theological inferences, and (2) some conceptual frameworks may be better suited for the exposition and explanation of the theological truths. The second case is well exemplified by Aquinas’ use of the Aristotelian notions of substance and accidents in the clarification of the doctrine of transubstantiation.10 The history of contemporary theology shows the radical departure from the Thomist tradition in the 20th century with the attempts to base the theological reflection on other philosophical systems such as phenomenology, existentialism or analytical philosophy. For instance, the famous German theologian, Karl Rahner, is known to have incorporated the thought of Immanuel Kant and Martin Heidegger into his works.11 To give another example, the personalism of Pope John Paul II was based on the phenomenology of Edmund Husserl and Max Scheler.12

4. The image of the world and creation It is commonly accepted in the methodology of theology that, although the human intellectual inquiry permits the access to the truths of the Divine reality, these truths (as well as their content) have to be extracted from the raw material provided by an inspired author in the scriptural form. The main task of theology in this context is to sort St. Thomas Aquinas, Summa Theologiae, III, q. 75. Cf. K. Kilby, A Brief Introduction to Karl Rahner, Crossroad, New York 2007. 12 K. Wojtyła (Pope John Paul II), The Acting Person, Reidel Publishing Company, Dordrecht–Boston 1979. 10 11

Physics in the Service of Theology: A Methodological Inquiry


out the proper content of the theological truth to be held as certain from the component of the scriptural account that is conditioned by the historical, cultural and social circumstances of its author.13 One of the central components is the so-called scientific image of the world which, according to Michael Heller, can be defined as “a global picture of the world obligatory for scientists of a given epoch”.14 Heller points out that the scientific image of the world is by no means “a well-defined concept” but it amalgamates a variety of metaphysical preconditions, common-sense representations and religious beliefs. To conclude the list, he adds that “elements somehow enlightening the role and the place of humankind in the world are of the highest importance, because the world image is – as if from its nature – centred on human needs”. The issue of the scientific world image (or in this case pre-scientific) comes to the fore in case of the scriptural account of creation in the Book of Genesis and the Christian doctrine of the creation from nothing (creation ex nihilo). The doctrine obliges a believer to hold that God is the ultimate cause of the Universe that was created from nothing. At the same time, it asserts nothing of how God actually proceeded as He was bringing the Universe into existence. The exposition of the doctrine as it is known today, however, had its long history and it necessitated the development of the conceptual tools for its progressing clarification. This was achieved mainly through the works of St. Augustine and St. Thomas Aquinas. Both these thinkers stressed the problem of the temporal beginning of the Universe in the course of its creation. In particular, in his work De Aeternitate Mundi, St. Thomas demonstrated that there is no conflict between the eternity of the Universe and its dependence on God as its ultimate cause. The creation of the Universe in time, on the other hand, is a matter of belief. In his Summa Theologiae, he asserts that: Cf. D.B. Burrell, Theology and Philosophy, [in:] G. Jones, The Blackwell Companion to Modern Theology, Blackwell Publishing, Oxford 2004, p. 34-46. 14 M. Heller, The Scientific Image of the World, [in:] Creative Tension: Essays On Science and Religion, Templeton Foundation Press, Philadelphia–London 2008, p. 22-28. 13

Wojciech P. Grygiel


I answer that, By faith alone do we hold, and by no demonstration can it be proved, that the world did not always exist, as was said above of the mystery of the Trinity (32, 1). The reason of this is that the newness of the world cannot be demonstrated on the part of the world itself. For the principle of demonstration is the essence of a thing. Now everything according to its species is abstracted from ‘here’ and ‘now’; whence it is said that universals are everywhere and always.15

Following the above, the inference that the temporal beginning of the Universe is the matter of faith and not demonstration depends on the principle of such a demonstration, that is, the essence. If indeed the essences of things are atemporal, there are no grounds that the contingent beings such as the Universe itself, came into existence at a given point in time. This reasoning is consistent with the picture of the static eternal Universe as visualized by the medieval with the Earth at the centre and the stars moving steadily around in the superlunary spheres. Moreover, such a static picture does not permit any dynamics of the Universe in the sense that a time-dependent evolution of its states could be formulated. Inasmuch as such a picture of the Universe is well suited for the explanation of its dependence on the ultimate cause, it does not offer good means for the understanding of the history of the Universe and that it might have had a temporal beginning. The history of the Universe is easily portrayed as one shifts from the description by means of the Aristotelian categories of substance and essence into the description with the use of physical laws formulated in the form of time-dependent mathematical equations. Based on the Friedman-Robertson-Walker models developed with the use of the general theory of relativity as well as a number of experimental results (e.g. Hubble’s law), it is regarded as a fact of science that our Universe began with the Big Bang and is now continuing to ex15

St. Thomas Aquinas, Summa Theologiae, I, q. 46, a. 2.

Physics in the Service of Theology: A Methodological Inquiry


pand.16 The specificity of the theory of relativity, however, precludes the physical description beyond the so-called Planck level, where the density of matter reaches infinity leading to the occurrence of singularities.17 In short, contemporary physics cannot give any account of the nature of the Big Bang itself so that the state of the Universe at time zero remains beyond the reach of any physical description. Since no pre-Big Bang knowledge is available either, it cannot be excluded that our Universe is one of the many in the totality of creation. Although the contemporary cosmological models give clear meaning to the temporal evolution of the Universe, they are nonetheless incapable of explaining why the Universe came into existence and why the laws of physics are as they are.18 Regardless of this deficiency of the classical theory, the main advantage of the cosmological model constructed on its basis is that it allows for the precise understanding of the history of our Universe with the clear indication that it did have a temporal beginning and it may be a part of the larger history of creation, provided that the concept of time is applicable beyond its boundaries. The upshot of this discussion is that the shift of the conceptual framework as applied to the understanding of the nature of the Universe provides better insights into its dynamical properties and its temporal characteristics. The Universe thus conceptualized may be considered to constitute a part of the larger scheme of creation, with time being exclusively suited for its description, especially that recent developments in the area of the non-commutative geometries demonstrate the legitimacy Cf. J. Hartle, Gravity: An Introduction to Einstein’s General Relativity, AddisonWesley, San Francisco 2003, p. 366-399. 17 Cf. S.W. Hawking, G.F.R. Ellis, The Large Scale Structure of Space-Time, Cambridge Monographs on Mathematical Physics, Cambridge University Press, New York 1975. 18 Such an attempt has been made in the context of the famous Hartle-Hawking model where the authors propose the no-boundary condition to assure the self-explanatory character of the laws of the Universe. Although the model does indeed do away with the singularity of the Big Bang, it does not give the explanation of the origin of the laws of physics. Cf. S. Hawking, Quantum Cosmology, [in:] S. Hawking, R. Penrose, The Nature of Space and Time, Princeton University Press, Princeton–New Jersey 1996, p. 75-104. 16


Wojciech P. Grygiel

of the atemporal dynamics.19 The beginning of our Universe in time does not have be reserved as the domain of faith for as the category of time does not apply to the totality of creation, some other modes of explanation may need to be used to account for the creation’s ultimate origin.

5. The link to physics The term ‘invariant’ indicates the link to the main inspiration of this article that originates from the mode of operation of physical theories. In most general formulation, invariance means physical equivalence. This, in turn, ties closely with the notion of symmetry. Invariance occurs as a result of the application of certain transformations to an object under study. If the object is found to be invariant with respect to a given transformation, this transformation is qualified as an element of symmetry. Elements of symmetry form groups that are algebraic structures defined on a set of elements fulfilling the group axioms.20 For instance, the symmetry of a cube is described by the symmetry group denoted as Oh with 48 operations of symmetry as its elements that leave the cube intact in the course of their action. The specificity of this study, however, demands us to turn our attention towards a much more abstract and complex notion of invariance in physics that is proper to the general theory of relativity. Albert Einstein’s main idea in the formulation of the theory was to make the laws of physics independent of the choice of coordinates. In a strict mathematical language, this is accomplished by means of the diffeomorphism invariance. Diffeomorphisms are a special class of maps that

Cf. M. Heller, L. Pysiak, W. Sasin, Noncommutative Dynamics of Random Operators, “International Journal of Theoretical Physics” 2005, no. 44, p. 619-628. 20 Cf. C.J. Isham, Lectures on Groups and Vector Spaces For Physicists, World Scientific, Singapore–New Jersey–London–Hong-Kong 1989, p. 1-56. 19

Physics in the Service of Theology: A Methodological Inquiry


preserve the differential structure of a manifold.21 In the general theory of relativity, for the four dimensional differential manifold M with the Lorentz metric γ, there exists a group of diffeomorphisms f denoted as Diff(M) such that f: M→M making the physical laws invariant with respect to the choice of the coordinate system. In more general terms, what is physically meaningful in terms of representing a distinct physical reality is made independent of the mode of description. This somewhat ideal situation, however, is strictly the domain of a mathematical structure of the general relativity, where the object of the study ‘looks’ exactly the same, regardless of what perspective it is observed from.

6. The physics of black holes and the search for a dogma The analysis presented so far has demonstrated two important points. Firstly, varying the conceptual basis is helpful in the full exposition as well as in the rectification of the meaning of the tenets that pertain to the status of a dogma. Inasmuch as the conceptual framework based on the physical laws expressed in time-based mathematical equations well reflects the dynamic character of the Universe, it lacks the capacity to link the existence of the Universe with its ultimate cause. Secondly, the mathematical structure of the general theory of relativity offers an idealized example that in the context of formalized systems one can achieve a perfect independence of the studied object of the choice of the coordinates, as is the case for Einstein’s field equation. It must be remembered, however, that according to the theorems of Hawking and Penrose, the spacetime structure of general relativity is not free of singularities, and the singularities occur in the application of the theory to the physics of black holes. In the simplest case,

M. Nakahara, Geometry, Topology and Physics, Institute of Physics Publishing, Bristol–Philadelphia 2003. 21

Wojciech P. Grygiel


collapsing stars that lead into the formation of black holes are well approximated by the empty spacetime surrounding a source of curvature with spherical symmetry.22 The corresponding spacetime metric bears the name of the Schwarzschild metric and in the Schwarzschild coordinates the corresponding line element measuring the distance between two spacetime points assumes the following form: ds 2 = 1

dr 2 2M 1 r

2M dt 2 r

r 2d


According to the formula, the metric exhibits two singular points for r = 0 and for r = 2M. It transpires, however, that it does not follow from the above that both singularities are real in the sense that they correspond to the infinite density of mass. The best way to test the problem is to redefine the coordinates and to see whether the singularities survive such an operation. In other words, if the singularity is the invariant of the change of the coordinate system, that is, it does not in effect disappear, it has a real physical sense and represents a black hole. The Schwarzschild coordinates easily transform into the Eddington-Finkelstein coordinates as the time variable is replaced by the following expression:


r = 2 M ln

r 2M


so that the final expression for the linear element ds2 takes up the form:

ds 2 =


2M dv 2 r


r 2d


Cf. J. Hartle, Gravity: An Introduction to Einstein’s General Relativity, op. cit., p. 255-276. 22

Physics in the Service of Theology: A Methodological Inquiry


Since for r = 0, the existence of the singularity does not depend on the choice of coordinates, this singularity corresponds to a real physical situation where the density of mass reaches infinity. The singularity at r = 2M, on the other hand, is an artefact of the Schwarzschild coordinate system. In other words, by varying the coordinates one can identify situations that are coordinate invariant, and thus independent of the conceptual framework selected. Could this situation eventually be transposed into defining a dogma as an invariant of the conceptual framework applied to explain a given theological proposition?

7. Difficulties and hopes Although the very idea of applying the above discussed method of physics in deciding what qualifies as a dogma seems attractive and promising, it faces many severe difficulties. First of all, the notion of a coordinate system in a physical theory has been evidently matched here with the notion of a conceptual framework in a given philosophy. This is a very far-fetched analogy for it is hard to imagine how a single concept could be at all likened to a coordinate. A concept is associated with meaning, while a coordinate in a mathematical notion indicates a direction in a given space. What, if at all, do they have in common? To bring the analogy into better focus, one can fairly safely state that the explanation of a statement in a given conceptual framework is an operation similar to, for instance, defining a vector in a coordinate system. In both cases, the totality of the object defined acquires its meaning from the particular contributions, either concepts or coordinates, respectively. Secondly, no philosophical system as applied in theology functions in an axiomatized form. As is has been already mentioned, even the images of the world latent in the theological discourse are but amalgamates of a variety of views expressed in diverse conceptual frameworks. Inasmuch as one can find mathematical procedures of transforming one set of coordinates into another in a precise way, such transformations do not exist be-


Wojciech P. Grygiel

tween concepts belonging to two different conceptual frameworks. For instance, the philosophical term causality bears distinct meaning in the classical Aristotelian vocabulary that is not applicable, for instance, in the understanding of causality in the theory of relativity. Moreover, the idea of what it means ‘to exist’ varies radically as one moves from the Thomistic metaphysics to W. v. O. Quine’s languagerelated understanding of existence. Moreover, the meanings of common-sense concepts typical to any philosophical system are subject to uncertainties and cannot be encapsulated in a mathematical form. The perfect invariance of Einstein’s field equation, as exemplified in the general theory of relativity by the diffeomorphism group, can never be attained in theology. Does this mean that the invariants of the conceptual frameworks in the explication of the theological propositions can never be found at all? Although the task seems by no means easy, the following reasons speak in favour of retaining it as a viable option. Firstly, as it has been shown in the course of this inquiry, theology does indeed make ample use of the different conceptual frameworks (philosophies), despite some claims to restrict the theological speculation exclusively to the authority of the perennial philosophy. Secondly, in order to demonstrate that the conceptual invariants are at least in principle attainable, it is worthwhile to invoke a contemporary theory developed by George Lakoff and his collaborators.23 In a nutshell, the theory states that all our concepts are metaphorical in their nature and are reducible to a handful of basic metaphors, regardless of whether they are common-sense or sophisticated mathematical notions.24 Consequently, as one sifts the conceptual framework in the theological enterprise, even the conceptually most inconsistent systems with imprecisely defined meanings may share the same conceptual clusters at a deeper level, thus giving rise to appropriate invariants. G. Lakoff, M. Johnson, Metaphors We Live By, University of Chicago Press, Chicago 1980. 24 G. Lakoff, R. Núñez, Where Mathematics Comes From. How the Embodied Mind Brings Mathematics into Being, New York 2000. 23

Physics in the Service of Theology: A Methodological Inquiry


The fundamental advantage of perceiving a dogma as an invariant of the conceptual system would be in not invoking ecclesiastical authority where rational inquiry suffices. It by no means abolishes this authority but – to the contrary – it prevents its misuse in instances that have not undergone sufficient scrutiny of the logical and methodological consistency, similarly to the well-known fallacy of the God of the gaps. A dogma thus substantiated acquires the status similar to that of a law of physics to be universally and inter-denominationally recognized, provided of course that one accepts that the Divine reality exists and reveals itself to us. Interestingly enough, this coincides with Einstein’s tenet that it is also a matter of belief that the object of the study of a physicist – the physical reality – exists and reveals itself in an experiment. Einstein writes: The belief in an external world independent of the perceiving subject is the basis of all natural science. Since, however, sense perception only gives information of this external world or of physical reality’ indirectly, we can only grasp the latter by speculative means. It follows from this that our notions of physical reality can never be final.25

Can theology ever be final? In principle, if the revelation is indeed complete, the theological ‘theory’ that gives full explanation to the truths revealed could be considered final where the conceptual content fully reflects the content of the revelation. However, it would require a further in-depth study to ascertain to what degree this conclusion is subject to constraints imposed by the Duhem-Quine underdetermination thesis. Secondly, by varying the conceptual frameworks in theology there appears the unique chance of sorting out the artefacts projected onto the theological truths by the particular framework, as exemplified above by the apparent singularity in black hole A. Einstein, Maxwell’s Influence on the Evolution of the Idea of Physical Reality, [in:] J.J. Thomson et al. (eds), James Clerk Maxwell: A Commemoration Volume, Cambridge University Press, Cambridge 1931. 25


Wojciech P. Grygiel

physics. As a result, dogmas could be expressed in the most precise way by eliminating the hidden assumptions that otherwise blur their objective content. Ultimately, a new dimension could be given to the following urge expressed by Pope John Paul II: Science can purify religion from error and superstition; religion can purify science from idolatry and false absolutes. Each can draw the other into a wider world, a world in which both can flourish.26

What if the image of God as an old man with the long beard turns out not to be the invariant of the conceptual framework? We may be sorry to let it go but switching to a more realistic representation should only increase our faith.

John Paul II, A Letter to the Reverend George V. Coyne, S.J., Director of the Vatican Observatory, 1 June 1988. 26