Beyond Sets: A Venture in Collection-Theoretic Revisionism 9783110319750, 9783110319286

Table of contents :
FOREWORD
Chapter One - COLLECTIVITIES
Chapter Two - MODES OF INDETERMINISTIC COLLECTIVITY
Chapter Three - INDEXED INDETERMINACY AND VAGRANT COLLECTIVITIES
Chapter Four - COLLECTIVITY THEORY: FIRST STEPS
Chapter Five - PLENA
Chapter Six - LOGICAL OPTIONS
Chapter Seven - NUMBER BEYOND NUMBER
Name Index

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Nicholas Rescher and Patrick Grim Beyond Sets A Venture in Collection-Theoretic Revisionism

Nicholas Rescher and Patrick Grim

Beyond Sets A Venture in Collection-Theoretic Revisionism

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BEYOND SETS A Venture in Collection-Theoretic Revisionism Foreword 1

Chapter 1:

Collectivities

Chapter 2:

Modes of Indeterministic Collectivity

13

Chapter 3:

Indexed Indeterminacy and Vagrant Collectivities

27

Chapter 4:

Collectivity Theory: First Steps

43

Chapter 5:

Plena

59

Chapter 6:

Logical Options

73

Chapter 7:

Number Beyond Number

93

Name Index

111

FOREWORD This book is the product of a collaboration, stretching over the years 2007-10, whose initial fruit was a paper on “Plenum Theory” published in Nous, vol. 42 (2008), pp. 422-39. The work grew out of the authors’ conviction that standard set theory, which had evolved to meet the needs of mathematics, is not fully adequate to the less abstractly geared and rigidly determine needs of less finalized ranges of inquiry and deliberation. Had set theory been devised to meet the needs of less formalized thinking than is requisite for mathematics, collectivity theory—or something much like it—would have emerged. Each author acknowledges indebtedness to the other for his collaborative geniality. Nicholas Rescher Patrick Grim

Chapter One COLLECTIVITIES 1. WHAT COLLECTIVITIES ARE

C

ollectivities are collections that can have members under all modalities: actual and potential members, definite and indefinite members, past and future members, members identifiable or unknown. The null collectivity aside, collectivities will indeed have members, but their membership need not be enumerable individual by individual or identifiable with precision. Collectivities are pluralities we generally access in terms of qualifying features and modalities rather than lists of identifiable members. Membership in a collectivity standardly corresponds to possession of some specified features encompassed in a qualifying condition: x ∈ C iff Fx This defining feature, however, may be messy. In the case of a given item a, it may be undetermined whether or not F applies, so that neither Fa nor ~Fa determinately holds as a matter of fact. With an indeterminate collectivity C, neither x ∈ C nor x ∉ C need hold for some x. Worse yet, we should allow for the possibility that claims to F-possession and its lack may be equally in order. Both Fa and ~Fa may apply; for an overdetermined collectivity C, both x ∈ C and x ∉ C might hold for some x. Collectivities can thus be significantly ill-behaved by the standards of familiar set theory. Consider setting up a bibliography of all books about cats. Where is the boundary line between in and out here? How often must cats be mentioned as such? What if they are mentioned not as cats per se, but more specifically as angoras or tabbies? Do lions and tigers count? Saber-toothed tigers? What if there is no explicit mention of cats at all, but merely descriptions which in fact fit them? What of the Cheshire cat? In the end, we must recognize that there is just no clear-cut boundary here. In standard treatment, sets are supposed to have well-defined boundaries:

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Nicholas Rescher and Patrick Grim • A Venture in Collection-Theoretic Revisionism

(∀x) (x ∈ S v ~x ∈ S) and ~(∃x) (x ∈ S & ~x ∈ S) Collectivities, including that of books about cats, come with no such suppositions. Collectivities can have members of all modal sorts and conditions, including members undetermined, undecidable, or problematic. What holds for collectivities with regard to modalities of indefiniteness also holds for collectivities with regard to modalities of actuality and possibility, of necessity and contingency, of present and past, of known and unknown. Sets are defined in terms of explicit membership. In that sense, all members of sets are actual members, none merely possible. All members are necessary members, in the sense that we would be dealing with a different set were any member omitted. Set members may be present individuals, past, or future, but the membership of a set cannot change over time. None of these need hold for collectivities, for which some members may be merely possible, some contingent and some necessary, some permanently but others merely temporarily.1 Collectivities, just like sets, have a being of their own—they are items in their own right, over and above the items that constitute their membership. In principle, therefore, they can exhibit features that their members cannot. A collectivity may consist entirely of finite sets, but contain more than a finite number of elements, for example; none of its elements can do so. The members of a collectivity can each be a middle-aged human, for example, but the collectivity itself cannot. Quite generally, collectivities can be introduced for consideration in three basic ways: 1. By explicit specification. Unlike sets, however, this specification can include modalities. For example, one can specify that the collectivity in question has a and b determinably as for-sure members, with c indeterminately so. 2. By descriptive qualification. For example, one can specify that all competent plumbers [NOTE: not licensed plumbers] belong. 3. By logical construction from other collectivities. For example, one can specify a collectivity as the intersection of two given collectivities.

COLLECTIVITIES

3

The second of these is the most familiar way of introducing collectivities, and descriptive indeterminacy is the prime doorway for membershipindeterminacy. It lies in the nature of indeterminate collectivities that they can fail to be presentable extensionally through an explicit inventory of their membership, requiring presentation by way of descriptive intentionality instead. The following items are cases in point: • Those residents of Hiroshima in 1945 who would still be alive today had the atom bomb not been dropped. • The world’s foothills • Napoleon’s great-great-great-grandparents • The next President’s cabinet • Persons who have passed into total obscurity • The determinate collectivities These are all indeterminate collectivities. The nature of such collectivities, the source of their indeterminacy, and the character of our reasoning regarding them is significantly less well understood than is the case with the crisp counterparts of set theory. Our goal is to understand them better. 2. COLLECTIVITIES AND SETS In view of the Axiom of Extension (∀x)( x ∈ S ≡ x ∈ S′) ⊃ S = S′ a set is defined by its membership, without modalities. Sets cannot have merely possible members, borderline members, contingent members, temporary or transitory members. A set cannot come to include a new member, lose an old one, gain definiteness, crystallize or change over time. For these reasons alone collectivities cannot be simply identified with sets. With collectivities membership comes with the shades of all modalities; for collectivities the issue of determinacy, for example, is often crucial. Be-

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Nicholas Rescher and Patrick Grim • A Venture in Collection-Theoretic Revisionism

cause set theory does not—cannot—address issues of determinacy and indeterminacy, collectivities remain outside its scope. There is some irony in the fact that sets have been called on to play a philosophical role for which they are essentially unsuited. Any philosophical introduction to set theory is bound to use familiar collections as introductory examples: collections of teaspoons, packs of dogs, and all the red things, for example. But the pure sets of set theory have no elements other than sets. Once the theory is complete, it becomes clear that the theory is not really about the things used to motivate it. Precisely because the collections used in introduction contain things other than sets, they fail to qualify as sets at all. In the unusual case that sets of ur-elements are considered at the base of a hierarchy of sets, moreover, set theory itself asks no questions and answers none regarding the ontological status of the ur-elements or the modality of their specification. This too is clear from introductions to set theory. When the instructor starts with a ‘set of teaspoons’, students inevitably want to know whether past teaspoons and future teaspoons are included (what are future teaspoons, actually?), and whether the set of teaspoons is something that gets larger over time. Those are questions perfectly appropriate for collectivities, but decidedly foreign to set theory, even a set theory broadened to include ur-elements. Most centrally, allied with classical logic, sets have binary all-ornothing conditions: any given item is either a member of any given set or is not, fully and determinately included as a member or fully and determinately excluded. Here too philosophical introductions to set theory are ironically misleading. Because ‘red’ is vague—with various things neither clearly red nor non-red—‘red things’ is an indeterminate collectivity, and any ‘set of all red things’ would fail to qualify as a set at all. Even were we to consider sets as things with members other than sets, the definitional sharpness of the concept of set would prohibit the red things—and indeed the F things for most familiar Fs—from forming a set. 3. THE DIRECT APPROACH Collectivities, with members under modalities, are common conceptual coin in human reasoning. We know that no list of the worlds’ mountains can mention only foothills. We reason that those residents of Hiroshima in 1945 who would still be alive today had the atom bomb not been dropped would very likely number in the hundreds or thousands, but that some who survived the bomb might well have murdered others. We ponder the com-

COLLECTIVITIES

5

position of the next President’s cabinet, wonder how many major philosophical works disappeared with the Library at Alexandria, and take seriously our responsibilities to future generations. In all of these cases we are dealing with groups membership in which carries the multifaceted modal subtleties and inherent indefiniteness of collectivities. Our attempt in what follows will be to take a direct approach to collectivities: to treat them at face value in terms of a philosophical theory designed to do them justice. Our attempt is philosophical through and through, formal or quasi-formal only in first steps or rough outline. Our intention is to leave various alternative prospects for logical development open, despite outlining basic elements of structure to which any further logical development must do justice. Collectivity theory is set theory as it would have been had set theory been developed to suit the breadth of human language, conception, and reasoning regarding pluralities, rather than the specific and restricted needs of mathematics. We take ours to be the most direct way of approaching collectivities. We don’t intend to argue, however, that it is the only way. Perhaps collectivities could also be approached via the alternative of a set-theoretical extension of intentional logics.2 Perhaps they could also be treated as extensional shadows of property theory.3 Perhaps something like collectivities could even appear within a set theory embedded in an appropriately alternative logic.4 Perhaps. Our attempt here is to take the philosophically most direct route, closest to intuitions, in the conviction that any elaborated alternative would at any rate have to offer a structure for collectivities isomorphic to that developed in the following pages. As a hint of things to come, it should be noted that there can indeed be a Russell collectivity comprised of all collectivities that are not elements of themselves. Its membership boundaries are logically indeterminate or oscillatory, of course, in that it can be a member of itself if and only it is not. That is simply the nature of some collectivities to be considered. Although it is clear that there can be no set of all sets, there will be a collectivity of them all. There is moreover no problem of self-membership with ‘the collectivity of all collectivities.’ For collectivities, unlike sets, need not ‘make up their mind’ in such matters. A similar story can be told regarding the collectivity of all facts, a story regarding all propositions, and another regarding all states of affairs. Most specifiable conditions F admit the modal shading and indefiniteness that typify collectivities, and for this very reason lie beyond the reach of mere sets. We commonly reason regarding possible candidates in the

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Nicholas Rescher and Patrick Grim • A Venture in Collection-Theoretic Revisionism

next election, partisan members of Congress, Presidential cabinets over time, technological innovations over the next century, future generations, our unknown ancestors, things too heavy to lift easily or much larger than a breadbox. Philosophers are often concerned with extensions of concepts, totalities of consequences, alternative actions, those things which satisfy specific constraints, even possible worlds and totalities of propositions. None of these can qualify as the sets of set theory because they include things both indefinitely and under a range of other modalities. Where sets are pressed to serve as placeholders for these richer entities, sets are pressed into work for which they are distinctly underqualified. For a more adequate philosophical treatment of ordinary reasoning regarding ordinary pluralities, what is required is a theory not merely of sets but of collectivities. Collectivities make room for the subtleties and complexities of modality. Here it is indeterminacy rather than determinacy that will be characteristic, with binary conditions the rare exception rather than the general rule. There are also further difficulties with sets for which collectivities provide a corrective and a welcome relief. Set theory was born in paradox, was shaped by paradox, and continues to carry the threat of paradox into its current adolescence. Properly understood, we will argue, the threat of contradiction is not merely formal and is not to be evaded by merely formal techniques. The fact that there can be no set of all non-self-membered sets might be shrugged aside as a minor logical surprise. Beyond Russell’s paradoxical set, however, there lie the serious philosophical difficulties of coherently conceptualizing a set of all things, the realm of unrestricted quantification (or even the sense of restricted quantification), the totality of all events, all facts, all propositions, or all that is true. Sets are structurally incapable of handling any of these.5 As argued in later chapters, all of these are properly conceptualized as collectivities rather than sets, though even that conceptualization carries some surprising lessons regarding indeterminacy, the nature of such totalities, and the universe as a whole. 4. INDETERMINATE COLLECTIVITIES One factor which critically distinguishes collectivities from classical sets is that a set represents the collection of all items that exhibit some binary (on/off) condition F: S = {x│Fx}

COLLECTIVITIES

7

A set is nothing but the extension of its membership-defining condition: x ∈ S ≡ Fx With classical sets it is required that F be sharp-edged, effecting a decisive bifurcation. Every item belongs either to F or to its complement F¯. No item belongs both to F and to its not-F complement F¯. With collectivities these conditions are abrogated. Here neither Excluded Middle nor Bivalence need apply. It is crucial to note that a collectivity C may have two distinct complements, one weak and the other strong. Using ∆ for ‘determinately’ and collectivity brackets [ ] as opposed to set brackets { }, it is possible for one selfsame item to belong both to a collectivity C = [x│Fx] and to its weak compliment PC¯ = [x│~∆Fx]. For weak complements the standard set theoretic principle corresponding to Non-Contradiction ~(∃x) (x∈S & x∈S¯) need not apply in the case of collectivities. For strong complements CC¯ = [x│∆~Fx] the standard set theoretic principle of Excluded Middle (∀x) (x∈S v x∈S¯) correspondingly breaks down. A structure common in collectivities can be illustrated by a three-ring diagram of the following sort: C

C C

P

C

Here CC is the inner core of the collectivity C, and PC its outer periphery. The inner (strong) ring encompasses the determinably for-sure members and the outer (weak) ring the problematic ones—those not determinably non-members. Membership in CC is a sufficient condition for membership in C, which in turn is a sufficient condition for membership in PC. We have (∀x)(x∈CC → x∈C → x∈PC).

Nicholas Rescher and Patrick Grim • A Venture in Collection-Theoretic Revisionism

8

While collectivities are not sets, and cannot be adequately represented as sets, one can develop an approximate first model for collectivities using the familiar tools of set theory. This can be done by means of an approximate model built by associating the collectivity C(F) = [x│Fx] with a complex of several sets, specifically: C

S = {x│∆Fx}

P

S = {x│~∆~Fx}

P

S¯ = {x│~∆Fx}

C

S¯ = {x│∆~Fx}

Here ∆p abbreviates that p is determinable or demonstratable in the informational circumstances at issue. Such an operator ∆ can be further articulated in terms of logical demonstrability├ relative to a given information base B: ∆p iff B ├ p Of course ∆~p entails ~∆p, though not conversely. In this first approximation, a characteristic collectivity can be modeled as a pair of sets {S1 / S2}. Here S1, reflecting CC, marks the determinable (for-sure) members of the collectivity, while S2, representing PC, marks their enlargement by additionally possible members (i.e., those which do not determinately belong to C’s complement). In terms of even this approximation it is readily seen that the union of two indefinite collectivities can be definite. Using ≈ for our approximation relation, we have it that if both C′ ≈ {{a, b} / {c, d}} C″ ≈ {{c, d} / {a, b}} Then C′ ∪ C″ ≈ {{a, b, c, d }/∅}, modeling a collectivity in which all members are determinate members. And moreover the intersection of two indefinite collectivities can be definite. For if

COLLECTIVITIES

9

C′ ≈ {{a, b} / {c, d}} C″ ≈ {{a} / {f, g}} then we have: C′ ∩ C″ ≈ {{a} / ∅} There are therefore features of collectivities that can be captured by a set-theoretic approximation. The shortfall of any such model, of course, is that it sets precise boundaries throughout—to ‘additional possible members,’ for example, whereas this too may be a matter of indefiniteness. Any characterization of the content of collectivities in terms of the standard amalgam of set theory and classical logic will be a characterization in terms of sharp edges. Because collectivities characteristically do not come with sharp edges, but with shades and subtleties of modality and indeterminacy instead, no attempt to do the conceptual work of collectivities in terms of sets alone can prove adequate. A collectivity need not correspond to any single set or set of sets. Membership in a collectivity C is an indeterminate intermediation between the membership of CC and PC. When we depict the matter by a dual ringdiagram there is therefore no exact circle we can draw to represent the collectivity’s membership; any circle drawn would have to be smudged or blurred. On this picture, collectivity theory is a relaxed analogue of set theory which envisions collections and complements with merely penumbral boundaries. A collectivity [x│Fx] defined in terms of condition F will characteristically parallel a certain albeit otherwise indeterminate set that is intermediate between {x│∆Fx} and {x│~∆~Fx} and whose complement is intermediate between {x│∆~Fx} and {x│~∆Fx}. The key point here, however, is that such an approach offers merely an approximation rather than an equivalence. It must accordingly be stressed that there need be no welldefined set S(F) correlative with a collectivity C(F) such that: x ∈ S(F) iff x ∈ C(F) A collectivity may be indeterminate not merely in the sense that we are incapable of conceptualizing, specifying, or identifying its members but in

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Nicholas Rescher and Patrick Grim • A Venture in Collection-Theoretic Revisionism

the sense that there sometimes is simply is no fact of the matter as to its membership. Accordingly, the indefiniteness or indeterminacy at issue may be ontological rather than epistemological, as, for example, with a collectivity whose content depends on the truth of an undecidable proposition. Collectivities occupy logical territory that sets cannot. It should for example be noted that S(F) = {x│Fx} can prove to be devoid of meaning or undefined even when the corresponding collectivity [x│ Fx] is a well-defined object. Membership in a collectivity x ∈ C(F) and equivalently x ∈ [x│Fx] can be perfectly well defined even when there is no corresponding well-defined F-set {x│Fx}. In general neither F itself not any other predicate Z defined essentially in terms of it would allow us to maintain: C(F) = {x│Zx} with x ∈ C ≡ Zx Collectivities can have members under all modalities; it is quite generally the modality of membership that marks the difference between collectivities and sets. Primary among those modalities is indeterminacy; collectivities and collectivity membership remain intractable if issues of determinacy are left unaddressed. Indeterminacy, however, comes in many forms. A first approximation of those forms is available using the derivability/deducibility model for the determinacy operator ∆ above. A claim is determined relative to some information-base B if it can be derived therefrom. When that basis is empty we have logical determination; when it consists of the characterizing principles of the language at work we have semantical determination; when it consists of the manifold of what is accepted as known we have epistemic determination, and when B represents reality itself we have ontological determination. Collectivities that are indeterminate in each of these ways will be examined in chapters to come. NOTES 1

For particular purposes, it may prove useful to restrict attention to particular modalities and collectivities individuated in terms of those alone. It may be sufficient for the identity of ‘actuality collectivities’, for example, that they share the same actual members, perhaps with varying shades of indeterminacy. ‘Possibility collectivities’ would demand more; that identical collectivities contain the same possible

COLLECTIVITIES

11

NOTES

members as well, again with varying shades of indeterminacy. It may be sufficient for the identity of ‘temporally narrow t-collectivities’ that they share the same members at a specified time t. By contrast, ‘temporally broad collectivities’ will demand coordinate membership at all times. 2

See for example Edward N. Zalta, Intentional Logic and the Metaphysics of Intentionality (Cambridge, MA: MIT Press, 1988), and C. Anthony Anderson, “General Intensional Logic,” in D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic (Dordrecht: D. Reidel, 1984), pp. 355-385.

3

See George Bealer, Quality and Concept (Oxford: Clarendon Press, 1982), and Uwe Monnich, “Property Theories,” in D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, 2nd Edition, Vol. 10, (Dordrecht: Kluwer, 2003), pp. 143-248.

4

As possibilities see John Myhill, “Constructive Set Theory,” Journal of Symbolic Logic, 40 (1975), 347-382, and J. Barwise and L. Moss, Vicious Circles (Stanford: CSLI, 1996).

5

See Patrick Grim, The Incomplete Universe: Totality, Knowledge, and Truth (Cambridge, MA: MIT Press, 1991).

Chapter Two MODES OF INDETERMINISTIC COLLECTIVITY 1. A BUDGET OF INDETERMINACIES

T

he indeterminacy of a collectivity’s membership can arise in various ways. It is instructive to start by surveying some of the principal possibilities: (a) Semantic indeterminacy. The collectivities of heaps of sand grains, of tall philosophers, and of the foothills of the world are indeterminate in membership because of the inherent vagueness of the membershipspecifying terminology of “heaps,” “tall,” and “foothills.” The absence of a precise boundary in these cases makes membership in these collectivities indeterminate. (b) Radical contingency. Just who among Hiroshima’s residents in 1945 would be alive today had the bomb not been dropped? That membership would seem to depend on whether the United States would have used more extensive firebombing instead, whether any of those who survived would have murdered others who survived, and who the victims of later epidemics would have been. This imponderable web of radical contingencies renders this collectivity’s membership indeterminate. (c) Epistemic inaccessibility. There were real and specific people who were Napoleon’s great-great-great-grandparents. Their identity, however, is lost in time, perhaps irrecoverably; that collectivity is now epistemically indeterminate. An information vacuum leaves membership in the collectivity indeterminate. (d) Vagrant predication. The unknown victims of the 1938 hurricane, like those all record of whom was lost because of the Krakatoa explosion of 1883, are inaccessible to us by those very characterizations. There are certainly truths no-one will ever know, but as truths no-one will know they clearly cannot be identified as such. These are collectivities membership of which is inaccessible by specification in terms of such “vagrant predicates”—predicates whose instances have no known address.1 (e) Logical indeterminacy. As the concepts of indeterminate collectivity are developed in the following chapters, it will become clear that logic

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alone forces us to acknowledge certain collectivities as indeterminate. We will be able to establish on logical grounds alone that there are issues of indeterminacy inevitably beyond our reach. A first promising distinction among the types of indeterminacy characteristic of different collectivities is the distinction between epistemic and ontological indeterminacy. This first distinction is a rough one, drawn on the lines of whether the indeterminacy at issue is a matter dependent on our capabilities or independent of us. We class as epistemological those forms of indeterminacy due to our inability to establish membership of a given collectivity, including inabilities of conception, specification, and knowledge. We class as ontological those forms of indeterminacy that are not merely dependent on us but are in the things themselves: those issues of membership for which there is not or can be no fact of the matter. Closer examination in later sections will make clear both the value and the limitations of this first rough distinction. The indefiniteness of collectivities in category (d) above is explicitly epistemological. Primary examples of the vagrant predication collectivities in category (e) will be epistemological as well. The authors’ tendency is to treat both the semantically indeterminate collectivities of (a) and the logically indeterminate collectivities of (c) as ontologically indeterminate. Here indeterminacy appears to us to be independent of us, deeper than merely a matter of what we can or cannot ascertain or establish. There are, however, philosophers who argue vehemently that the semantic indeterminacy of (a) is epistemic as well.2 The radical contingency of category (b) is a further case, perhaps more widely disputable, that calls for further attention. Our effort in this chapter is a first attempt at a general taxonomy of the indeterminacies characteristic of indeterminate collectivities. The disputable status of (a) and (b), together with a more complete exploration of (e), will be left to chapter 3. With regard to both our first rough distinction and those to follow, it should be noted that the present explorations do not depend for their validity on the specific examples at issue. Our aim is a treatment of collectivities and their characteristic indeterminacy that will stand even if specific categorizations used and the specific examples given remain open to dispute. In some cases collectivities may be indeterminate in more than one way, for example, or from more than one perspective. The collectivity consisting of the very first and very last members of homo sapiens to exist on earth is of course indeterminate epistemically. But it is criteriologically and in that sense semantically and hence ontologically indeterminate as well, due to

MODES OF INDETERMINISTIC COLLECTIVITY

15

the inherent indefiniteness of whether some potentially qualified creature does or does not belong to the species. In other cases, as noted, the specific type of indeterminacy characteristic of vague or counterfactual collectivities may remain open to dispute. In some cases, indeed, the specific character of indeterminacy at issue may itself remain indeterminate. 2. THE EXAMPLE OF SEMANTIC INDETERMINACY We shall begin with collectivities that are indeterminate because of the vagueness of our specifications. These are perhaps the most familiar cases, and thus offer a convenient entry to the considerations of indeterminate collectivities in general. How tall does one have to be in order to be tall? It is clear that “tall” is vague, and because of this the membership of collectivities specified in terms of such a category will be indeterminate, as is the case, for example, with: • The tall citizens of Pittsburgh. Is Mike Fink a member, at 5’11”? Is Jessica Wainwright, at 5’10”? Because of the vagueness of ‘tall’ it remains indeterminate who counts as a member and thus what the membership of the collectivity consists in. The sources of semantic vagueness can also be more remote: • Registered members of large Pittsburgh fraternal organizations. Even if ‘registered members’ is used with a specific criterion, and even if we take Pittsburgh’s fraternal organizations as those listed as such on the tax roles, the vagueness of the term ‘large’ will render this collectivity semantically indeterminate. Although fuzzy sets have traditionally been introduced as a tool appropriate to semantic vagueness, they fail to capture the true indeterminacy of the collectivities at issue. For fuzzy sets are not collectivities of indeterminate membership but rather determinate collections of quasi-members, each of which lays claim to membership to a precise degree. There is good reason to think that semantically indeterminate collectivities, whose membership is genuinely indeterminate rather than a matter of degree, reflect vagueness much more directly.

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Another common philosophical approach to vagueness is in terms of supervaluations.3 The core idea is that we use terms that do not have precise definitional constraints—there is no specific number of hairs required to quality as ‘bald’, for example—but that are therefore capable of any of various admissible ‘precisifications’ (a precisification in which 1000 hairs is stipulated as the cut-off, perhaps, 999 hairs, or 998…). On supervaluation theories, ‘Bill is bald’ will be true if he would fall in the category ‘bald’ on any acceptable precisification, false if he will fall in that category on no acceptable precisifications, and neither true nor false if he would count as ‘bald’ on some acceptable precisifications and not others. It has been put forward as a selling point for supervaluations that ‘Bill is bald or not bald’ and all other classical tautologies, because true on any ultimate precisification, will qualify as true despite the fact that ‘Bill is bald’ may be neither true or false. For our purposes, however, supervaluations seem as remote as fuzzy logic. It can be argued that supervaluation theories have reversed the order of evaluation—that we understand ‘acceptable precisifications’ in terms of our prior understanding of vague terms such as ‘bald’, for example, rather than the other way around. It can also be argued that supervaluation theories capture vagueness, to the extent that they do, only by smuggling it in via the inherent vagueness of ‘acceptable precisifications.’ It is, moreover, far from clear that we use vague terms in ways that do satisfy classical tautologies. Is Bill good looking or not? Often the proper answer seems to be ‘Well, he’s not really one or the other,’ or even ‘Well, he is and he isn’t.’ Both of these violate core principles of classical logic; to the extent that our reasoning with vague predicates is distinctly nonclassical, therefore, a touted virtue of supervaluation theories may in fact be a vice. For all these reasons, we will put supervaluations aside in attempting a more direct approach to even semantic indeterminacy in collectivities. There is one respect, however, in which supervaluations are an appropriate topic of study here: ‘acceptable precisifications’ constitute an indeterminate collectivity of precisely the sort at issue.

MODES OF INDETERMINISTIC COLLECTIVITY

3. INDETERMINACY APPROXIMATION

AND

CONTINGENCY:

17

A

FIRST

Collectivities are of various types, with correspondingly various types of indeterminacy. Just as it proved helpful in chapter 1 to approach collectivities themselves using a first approximation in terms of sets, it can be helpful to approach distinctions of indeterminacy with some more familiar terms in hand. Here as there it should be emphasized, however, that what these tools can give us is merely a first approximation, to be supplemented immediately with a number of crucial qualifications. In the first approximations of chapter 1 we employed a determinacy operator ∆ articulated in terms of logical demonstrability├ relative to a given information base B. The information base at issue might be any of the following: • All that we know at a particular time • All that has occurred to a particular point in time, together with all natural laws • All information contained in a particular description Using the first of these as our information base B, indeterminacy takes an extremely wide scope. In many cases what we know will not entail p and will not entail not-p: whether p or not p is indeterminate relative to that information base. Because our information base is here an epistemic one, it is of course natural to think of the indeterminacy at issue as epistemic indeterminacy. But there are different things known at different times, and different things known to different people: epistemic indeterminacy is a class as large and as various as the possible bodies of information to which it contextually tied. For you, or for you at a particular time, things may be epistemically indeterminate that are not such to others at other times. Things may be epistemically indeterminate for us now that will not be so tomorrow, next year, or next century. All gambling is based on epistemic indeterminacy. Interestingly, it requires indeterminacy in no further ontological sense. As long as I have or can have no information base sufficient to decide between red or black on the roulette table, the prospects are equally balanced and the bets are even. As long as I have or can have no information base sufficient to decide even

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a question of the historical past—whether there were an odd or even number of Confederate soldiers at Shiloh, for example, the matter is sufficiently indeterminate to make bets even. There are also cases of indeterminacy, however, in which the information base at issue is the second above: all that has occurred to a particular point in time, together with all natural laws. Contemporary physical theory, on standard interpretation, informs us that quantum phenomena are in the category of indeterminacies relative to this information base: whether a particular atom of uranium will decay at a particular instance is a matter governed by statistical laws alone, in place of which no finer-tuned natural laws are possible. The entire history of the universe up to a particular point, together with natural laws representing all deterministic mechanisms, are insufficient to entail either that a particle will decay within a specific time interval or that it will not.4 Here B represents information in the abstract, which need be neither ours nor anyone’s: a body of complete information to which we might aspire but which we do not in fact have, and perhaps could never have. Relative to even that information base some items will be indeterminate. Because that information base is a matter of reality beyond us, it is natural to think of this second as a form of ontological indeterminacy. This example of ontological indeterminacy was relative to an information base wide in scope: the saturated reality of the universe up to a particular point in time. That same notion, however, might well be used relative to narrower information bases, though still information bases independent of us. An event might be indeterminate relative to an information base written in terms of one set of parameters, for example, though not in terms of an information base written in terms of a wider set. Both information bases might be ontological in the sense of being independent of us and beyond our epistemic abilities to trace, but an event determinate relative to the larger information base might nonetheless be indeterminate relative to the smaller. What information base is salient is often a matter left to context. Indeed the relevant information base is sometimes determined by the pragmatics of charity in conversation; we take as implied an information base sufficient to make the claim at issue true. This is often the case with rival counterfactuals: • Had Caesar fought in Vietnam, he would have used machine guns. • Had Caesar fought in Vietnam, he would have used catapults.

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Each of these is plausible in terms of a salient information base; were they to arise singly in conversation, we would assume as background that information base that supported them. Some insist that there are other sources of indeterminacy in the universe, ontological in the sense that quantum phenomena are ontologically indeterminate but significantly closer and more familiar. Here free human choice is a prime contender.5 Libertarians argue that the full history of the universe up to a particular point, together with all governing physical principles, are insufficient to entail either that an agent will choose chocolate or will not. Our effort here is not to settle that debate, but we can indicate its logical position. It is clear that human decisions are in some sense indeterminate. What is at issue in the philosophical debate is the character of that indeterminacy. Is free human choice a matter of indeterminacy as deep as quantum phenomena, a matter of indetermancy relative to a smaller information base—though still one epistemically beyond us—or a matter of ‘merely’ epistemic indeterminacy? An important observation regarding indeterminacy is that its type may not be written on its sleeve: it may be indeterminate what mode of indeterminacy is at issue. In the two classes of cases considered, the character of indeterminacy at issue—epistemological or ontological—is tied to the epistemological or ontological construal of the relevant information base. For a third important class of cases, a further characteristic of the information base is crucial. Had I flipped the coin in my hand at precisely noon yesterday, would it have come up heads or tails? That, we want to say, is indeterminate: the coin might have come up heads, and might have come up tails, but it remains indeterminate which. Certainly the indeterminacy in this case is at least epistemic: there is nothing we know that would tell us that it would have come up one way rather than another. Coin flips are not as radically chancy as is the decay of uranium atoms, however: given the physics of a coin, of the angle of flip, of the vector of gravity, and of the vector of force, it is in fact predictable how a single flip will come out.6 Indeterminacy also has another source in this example, however. ‘Had I flipped this coin at precisely noon yesterday’ is ambiguous across a range of possible coin flips. I might have flipped this coin yesterday using my right hand, using my left, starting with an angle of 20 degrees, or 30, or 45, and with a force anywhere from gentle to forceful. Different combinations of these parameters would have resulted in heads or

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tails, but ‘had I flipped this coin at precisely noon yesterday…’ specifies no set of relevant parameters in particular. Whatever the ontological determinacy of coin flips given a full set of input factors, the description at issue is inadequate to pick out any set of parameters sufficient for either result. Here indeterminacy is not merely epistemic, but is not ontological in precisely the sense outlined above, either. The source of the indeterminacy is not lack of entailment one way or the other from a specified information base but the fact that no information base sufficient for entailment one way or the other has been specified. In some of the cases considered above the term ‘contingency’ seems appropriate; such a term certainly applies in the case of ontological indeterminacy, for example. In cases of gambling we also speak of contingency, though it may be unclear even to us whether we mean an epistemic ‘indeterminate for all I know’ or an ontological ‘indeterminate given facts F,’ where facts F happen to be those that I know. Not all matters of ignorance count as ‘contingencies’, however; I do not think of it as ‘contingent’ whether a friend is in Boston or Barcelona at the moment, though I might admit that I do not know which. We do tend to think of yesterday’s coin flip as contingent, though when it becomes clear that there is a range of possible coin flips at issue rather than one we may no longer be so sure whether ‘it’ was contingent precisely because we are no longer sure what ‘it’ is. The upshot is that the structure of indeterminacy relative to information bases is somewhat broader than, though inclusive of, what we think of as ‘contingency’ proper. Indeed the extent of contingency within indeterminacy seems itself to be semantically indeterminate. This first approximation to kinds of indeterminacy at issue is helpful, we think, in offering a sketch both of what they have in common and what distinguishes them. What they have in common is relativity to an information base. What distinguishes them is the character of that information base, conceived semantically, epistemically, or ontologically. Just as the model of collectivities in terms of sets and a ‘determinately’ operator ∆ in chapter 1 was merely a first approximation, however, it must be recognized that deduction from relative information-bases is merely a first approximation for varieties of indeterminacy. Deducibility is all or nothing: something either follows from an information base or does not. Indeterminacy as it appears in our first approximation, therefore, will always have a sharp edge. Some things will be determinate (they or their negation entailed by the relevant information base), others indeterminate (with no entailment either way), but nothing in be-

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tween. It is because this does not hold for the indeterminacy of collectivities that this model must remain merely a first approximation. Some are determinate members of the collectivity of bald men. Some are indeterminate ‘more or less’ members. But there is no sharp boundary between either those determinately ‘out’ and the indeterminates nor those determinately ‘in’ and the indeterminates. Deducibility, though helpful in thinking of relativity to information bases, is ultimately too sharp for a full understanding of the indeterminacy of collectivities. It might be tempting to replace deducibility with an evidence measure. This need not be as exclusively epistemological as it sounds: we might know or hypothesize that a particular body of information would offer evidence for or against a hypothesis even if that is not a body of information we do or even could obtain. Whatever such an evidence measure might be, however, it cannot be a quantitative measure in the sense that numerical probabilities or values in fuzzy logic are. The indeterminacy of collectivities is not only not all or nothing in the sense of 0 or 1; it has no specific intermediate value of .69 or .70, either. And it is not that such values are merely unknown. Such values simply do not exist; the indeterminacy at issue is essentially and inherently cloudlike, inchoate and unformed, not merely blurred through the dark glass of perception. 4. INDETERMINATE COLLECTIVITIES There are collectivities characterized by each of the types of indeterminacy outlined above, and collectivities for which the specific type of indeterminacy may be debated and may remain unclear. Many collectivities are such that their membership is a matter of imponderable contingency. Consider for example the following: • People living in the year 2050 who have never heard of Harry Truman. • Atoms in certain mass of unstable transuranic material (Californinum252 Cf, for example) that will still be there 30 days hence. While it is effectively certain that there are indeed items in each group, many of them now in existence, full specification of membership is impossible, not merely in practice but in theory.

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These are indeterminate collectivities, but the basic reason for their indeterminacy remains debatable—and indeed debated. Ontological determinists think that the situation is settled in such cases as a matter of actual fact, even though that fact may be essentially out of reach. Others think that in such matters of contingent futurity there simply is (as yet) no antecedent fact of the matter. However, the net effect of these positions is the same: we have no choice either way but to grant that while the objects of the sort at issue do indeed exist they cannot possibly be identified under the descriptions given in their specification. There will be collectivities characterized by all the modes of indeterminacy and contingency outlined above, clear or debated, specifically including the following: • Epistemically indeterminate collectivities, where membership is indeterminate for all that we know, or even can possibly come to know, because the requisite information is totally inaccessible. (Example: Who was enroute on the Appian way on the day of Julius Caesar’s assassination?) • Semantically indeterminate collectivities, where terms used in specification are not precise enough to make a sharp-edged determination of membership possible. (Examples: hats or bald men, to revert to the examples of the Greeks. Modern examples would include ‘presently living person’ and ‘female athlete.’) • Stochastically indeterminate collectivities, where chance and wholly fortuitous eventuations come into play. (Examples: The winners of next year’s state lotteries, or next year’s accidental fatalities.) • Ontologically or physically indeterminate collectivities, determined in principle by no antecedent events. (Examples: the particles that will pass through the left slit; those among yonder California atoms that will still be in existence at this time tomorrow.) • Counterfactually indeterminate collectivities, where membership is as imponderable as the tangled skein of contingencies involved. (Example: People who would be alive today had the World Trade Towers not been felled on 9/11/01. This group would include some victims, but not all, since some of those might by now have died of oth-

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er causes. It would also include children unborn and unconceived at the time, but would exclude people now alive who victims would have in killed automobile accidents had they survived.) • Logically indeterminate collectivities, as for example those numbers larger than the smallest number of steps that can demonstrate a certain Diophantine equation in a system of arithmetic. We will argue in following chapters that the notion of indeterminate collectivities itself logically entails the existence of some surprising indeterminate collectivities. In all these cases we are confronted with collectivities which unquestionably have members but whose membership is nonetheless indefinite thanks to the machinations of one or another form of indeterminacy. We have noted that the type of indeterminacy characteristic of a collectivity may sometimes be debatable. It may also sometimes be mixed; a collectivity may be indeterminate in more than one way. Consider what might first appear as a purely epistemic example: the collectivity of presidents of the United States, past present and future. Some individuals—George Washington and John Adams, for example—are clearly members. Some individuals are possibly in, including the candidates in the current presidential election. But some members are in principle unidentifiable, including “The winner of the presidential election in 3,000 A.D.”—who, after all, knows whether the country will even exist at that stage? The indeterminacy does not appear to simply be one of ignorance; it is not just that we cannot now identify those individuals existing offstage in the wings of the long-range future. As of now it is indeterminate whether there are or will ever be such individuals. Here it is certainly arguable that indeterminacy shifts from epistemic to ontological: that there is now simply no fact of the matter as to what this collectivity's membership is. One embarks on the classical problem of future contingents—an issue that leads beyond the boundaries of these deliberations. Seen in this light, stochastically indeterminate collectivities—the outcomes of the today’s coin tosses, for example—may also be have an indeterminacy epistemic in some aspects and ontological in others. The indeterminacy of a collectivity as a whole, it should be noted, is in principle distinct from any indeterminacy regarding its members individually. Here a telling example is the counterfactual collectivity of people now in a crowded room who would have remained had a fire alarm

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sounded at exactly 10:00 P.M. The membership of that collectivity is counterfactually indeterminate, though everyone who might be a member is someone clearly identifiable as a person now in this room. In this case the identity of potential members is fully determinate, though the membership of the collectivity is not. In other and more troublesome cases the ontological status of even potential members may be in question. Difficulties multiply beyond limit for collectivities that seem to introduce possibilia, conceived of as non-actual and merely possible individuals. Here even candidacy for membership seems to lose its grip. Consider, for example, a proposed collectivity of people who would now exist had Lincoln not been elected President. Had Lincoln not been elected, it is clear that the Civil War would not have occurred in at least precisely the way it did. Some would have survived who did not, and would have survived to have children and descendants living today. Unlike the case of people in this room who would have remained had the fire alarm sounded, however, we are in this case unable to identify even potential members of such a collectivity. Who are these people—who even would these people be—who would now be alive had Lincoln not been elected? In such a case it is not merely membership that is indeterminate, but the things that would be members themselves. Here indeterminacy infects even candidacy for membership, approaching dubious intelligibility. Here as elsewhere in thinking about indeterminate collectivities, philosophical positions in related areas will play an important role—positions regarding individuation across possible worlds, for example. The authors are in agreement, though others may not be, that the issues of indeterminacy on which we are trying to focus are difficult enough without the additional complications of possibilia. For present purposes, therefore, we choose to put possibilia aside. The authors are also in agreement that whatever view is to be taken of possibilia, it cannot be one in which we are forced to envisage definite and distinct individuals, like actual individuals in all other respects, waiting just off stage in the wings of mere possibility7 Our attempt in this chapter has been a first rough taxonomy of indeterminate collectivities in terms of their indeterminacies. We turn to a closer examination of specific categories within that taxonomy, and to related extensions, in chapter 3.

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NOTES 1

See Nicholas Rescher, “Vagrant Predicates and Noninstantiability,” chapter 15 of Epistemic Logic (Pittsburgh, PA: University of Pittsburgh Press, 2005).

2

See Roy A. Sorensen, Blindspots (Oxford: Clarendon Press, 1988) and Timothy Williamson, Vagueness (New York: Routledge, 1994).

3

See Bas van Fraassen, “Singular Terms, Truth-value Gaps, and Free Logic,” Journal of Philosophy 63 (1966), 481-95; Kit Fine, “Vagueness, Truth, and Logic,” Synthese 30 (1975), 365-300; and J. A. Kamp, “Two Theories about Adjectives,” in E. L. Kennan, (ed.) Formal Semantics of Natural Language (Cambridge: Cambridge University Press, 1975), pp. 123-155.

4

Here as elsewhere, in ways not always noted, the problematic notion of ‘natural law’ is crucial. Quantum indeterminism for an event e is clearly not indeterminism with regard to all generalizations, since things will in fact come out some particular way and there will be a generalization, with or without gerrymandered predicates, that entails the actual result in case e. To the extent that a notion of natural law is vague or incompletely understood, common claims regarding the indeterminacy of quantum events are similarly vague and incompletely understood, even by those who make them.

5

See Nicholas Rescher, Free Will (New Brunswick: Transaction Publishers, 2009) as well as the contextual literature cited in his Free Will: An Extensive Bibliography (Frankfurt: Ontos Verlag, 2010).

6

As an illustration of the point, Persi Diaconis has both trained himself to flip a coin so as to come up on a desired side and has constructed a ‘coin-flipping machine’ with predictable consequences. See Persi Diaconis, Susan Holmes, and Richard Montgomery, “Dynamical Bias in the Coin Toss,” SIAM Review, 49 (2007), 211235.

7

One of us has dealt with issues of possible and fictional existence in depth elsewhere. See Nicholas Rescher, Imagining Irreality: A Study of Unreal Possibilities (Chicago: Open Court, 2003). The other of us, though in sympathy with the general lines of that approach, reserves the right to return to indeterminate possibilia at a later date.

Chapter Three INDEXED INDETERMINACY VAGRANT COLLECTIVITIES

AND

1. ISSUES OF INDEPENDENCE

W

ith some collectivities aspects of indeterminacy are apparently independent of us. Take, for example, a specific chunk of Californium 252 Cf, and consider: • Those atoms of that chunk that will decay over the next thirty days.

On standard physical interpretations, the indeterminacy of this collectivity is an ontological one, written into the character of the universe itself. The membership of that collectivity is relative to no epistemic or cognitive agent, and would remain even if there were no epistemic or cognitive agents. In other cases the indeterminacy of collectivities, far from being independent of us, is explicitly grounded in our own cognitive limitations. Consider, for example, • Napoleon’s great-great-great grandparents These were a very specific set of very specific individuals—no vagueness here. But for us that set remains epistemically indeterminate—we cannot possibly say who those people were. We do know, of course, that the set contains at least two people and fewer than 65. Because of prospects for overlap, however, the cardinality of that set remains indeterminate as well; one cannot be sure that its cardinality is exactly 64, nor indeed that it is greater than two. Our cognitive limitations run deeper than merely limitations of what we know or can know. They include limitations in what we can imagine, what we can conceive, and what we can cognitively process. What are at issue are a family of cases similarly dependent on our cognitive limitations but distinguished by the particular cognitive limitations they may depend on.

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2. THE PUZZLING CASE OF SEMANTICALLY INDETERMINATE COLLECTIVITIES The examples considered above are clear cases, encouraging the distinction between epistemologically and ontologically indeterminate collectivities with which we began in the previous chapter. But cases are not always so clear. Consider for example semantically indeterminate collectivities: • The world’s foothills Here indeterminacy does not appear epistemic: it does not seem that we are simply ignorant of some quantitative fact or world geography or some hidden aspect of foothills that would make membership in the collectivity clear and precise.1 The indeterminacy of this collectivity is not merely a matter of what we know. This collectivity, like that of tall men and distinguished philosophers, appears therefore to be an ontologically indeterminate collectivity. Its indeterminacy, like that of the collectivity of Californium atoms that will decay over the next thirty days, is independent of us. The indeterminacy of the collectivity of the world’s foothills lies not in us but in the character of foothills. Are semantic collectivities ontological and independent of us? The argument to the contrary is that it is the concept or linguistic category of ‘foothills’ that is the source of indeterminacy in such a case—an indeterminacy essential to the concept or linguistic category at issue. Because the concept and the language at issue are our concept and our language— because the semantics of semantic indeterminacy is our semantics—the indeterminacy at issue lies in ourselves after all. On this argument, semantic indeterminacy is not epistemic—it is not a matter of knowledge in the sense that the indeterminacy of Napoleon’s great-great-great grandparents is indeterminate—but it is not ontological in the sense of being independent of us either. Here it is important to defuse a basic confusion that has bedeviled philosophy at least since Berkeley. Conceptual categories target aspects of a world; aspects of a world, it must be admitted, that is conceptualized in a certain way. A world conceptualized in terms of those categories, however, is a world nonetheless, independent of us as the world always is. It is a genuine reality that is conceptually categorized. ‘Foothills’ is a category of our language, for example, but foothills are not; they are large, fully physi-

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cal, sometimes scrub-covered and difficult to climb. Had forms of life never evolved on earth with the capacity to form conceptual or linguistic categories, the foothills would have been here nonetheless. The things our conceptual categories target are independent of us, even if our conceptual categories are not. In speaking of foothills we are speaking of those things. Speaking within those categories—and there is no other way to speak of foothills—both the physical characteristics of foothills and the indeterminacy of their collectivity are ontologically independent of us. An optical analogy, though simple, is perfectly appropriate. There are things we can see only through a microscope, but they are things nonetheless. Had we no microscope, we could not see them. Had we no microscope, we would not even be able to refer to those things, but they would have been there all the same. Here it important that reference is fixed in terms of the optical access we now have given a microscope. When we suppose what would have been the case had we not had that device, reference does not go along for the ride—our reference is not then limited in the way it would have been had we had no microscope. In the same way, there are aspects of the world that we recognize in terms of our conceptual categories. Had we not existed, or had we not approached the world in terms of those categories, those things we access in terms of our current conceptual categories would have been conceptually inaccessible to us. But those things would have been there all the same. Here it is important that reference is fixed in terms of the categories in use. When we suppose a world without us or without our conceptual categories, that reference does not go along for the ride either—it remains fixed by the categories in use. The following is Berkeley’s ‘master argument,’ emphasized in both Of the Principles of Human Knowledge and Three Dialogues between Hylas and Philonous:2 . . . I am content to put the whole upon this issue; if you can but conceive it possible for one extended moveable substance, or in general, for any one idea or any thing like an idea, to exist otherwise than in a mind perceiving it, I shall readily give up the cause…. But say you, surely there is nothing easier than to imagine trees, for instance, in a park, or books existing in a closet, and no body by to perceive them. I answer, you may so, there is no difficulty in it: but what is all this, I beseech you, more than framing in your mind certain ideas which you call books and trees, and at the same time omitting to frame the idea of any one that may perceive them? But do not you yourself perceive or think of them all the while? This therefore is nothing to the purpose: it only shows you have the power of imagining or forming ideas in

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your mind; but it doth not shew that you can conceive it possible, the objects of your thought may exist without the mind: to make out this, it is necessary that you conceive them existing unconceived or unthought of, which is a manifest repugnancy. When we do our utmost to conceive the existence of external bodies, we are all the while only contemplating our own ideas. But the mind taking no notice of itself, is deluded to think it can and doth conceive bodies existing unthought of or without the mind; though at the same time they are apprehended by or exist in itself. (Of the Principles of Human Knowledge, sections 22-23).

The argument that esse est percipii is grounded in the fact that we cannot perceive anything that is unperceived and cannot conceive of something that is not conceived of—perhaps even conceived of in terms appropriate to perception. It is equally true that we cannot conceptualize anything except in terms of conceptual categories. None of that entails, however, that something cannot exist unperceived, unconceived of, or independent of conceptual categories. What Berkeley needs to argue is not merely a matter of actuality—that all perceived things are perceived. What he needs to argue is not merely a matter of imagination—that all imagined things are imagined, or even imagined as perceived. What Berkeley needs to argue is rather a matter of alethic possibility: that these things, now perceived and capable of being referred to because of that perception, would not have existed had they not been perceived. Neither his argument nor those that mirror it are capable of reaching that far.3 On the view we are taking here, foothills are real and independent of us just as microbes are real and independent of us. Given a microscope, we can see the latter. Given certain conceptual categories, we can conceptually access and refer to the former. What we thereby refer to are real objects, ontologically independent of us; their characteristics, and the characteristics of the collectivities they form, are ontologically independent of us as well. The focus of the present investigation is one of those ontological characteristics: an essential indeterminacy. Accessed via a set of concepts or linguistic categories, on such a view, the world itself is itself indeterminate. Conceived in terms of foothills, for example, the world is itself indeterminate in certain respects. It is a sign that the indeterminacy at issue is ontological that this fact is itself contingent. Had the geography of the world been different, with altitudes and proximities in sharply distinct intervals, the world’s foothills might have formed a clear and distinct set. The fact that they do not is a fact about the

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world: an ontological fact regarding the indefiniteness of a specific collectivity. The argument that semantic indeterminacy gives us a form of genuinely ontological indeterminacy, however, need not blind us to important differences between that form and the ontological indeterminacy of decaying uranium atoms with which we began. It is clearly a sign of semantic indeterminacy that different categories in the same general referential neighborhood need not be similarly indeterminate. The points on the earth’s surface exactly 2,000 feet above sea level do not form a collectivity indeterminate in the same way that the world’s foothills do. Indeed that may be what it means to say that an issue is semantically indeterminate: that semantic alternatives related in extension need not be similarly indeterminate. Were we to speak of points on the earth’s surface 2,000 feet above sea level, the indeterminacy characteristic of foothills would disappear. No matter how we might choose to speak of Californium atoms, on the other hand, the indeterminacy of their decay would remain. Though issues of semantics give us genuinely ontological indeterminacy, therefore, it is not precisely the ontological indeterminacy with which we began. Like collectivities characterized by ‘epistemic indeterminacy’, collectivities characterized by ‘ontological indeterminacy’ come in importantly different types. Ontological indeterminacy grounded in the vagueness of semantics is clear in other cases as well. Consider, for example, the collectivity of my human ancestors. What are at issue are real beings, not merely concepts or figments of imagination—hence the insistence on ontology. Nonetheless we know that it cannot be true that ‘humans are only and always born of humans’; the history of the species, after all, is not infinite. The indeterminacy of the collectivity of my human ancestors owes essentially to the indeterminacy of that finitude, and ultimately to the indeterminacy of what is to be counted as a human ancestor. Here again the ontological indeterminacy at issue, unlike that of Californium, is essentially conceptual at base. Our attempt to distinguish between even roughly ontological and epistemic classes of indeterminacy, it should be noted, has been in terms of matters ‘independent of us’. Even that phrase, however, can prove troublesome. There is certainly a sense in which the world’s current population is an ontological fact, independent of us in the sense at issue here. There is also a sense, of course, in which that is a fact very much dependent on us—dependent on our particular numbers. There is a sense in which facts regarding the current membership of Congress are facts independent of us; believing the membership were otherwise would not make it so. But Con-

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gressional membership is a social fact affected by social action, and in that sense it is clearly not independent of our actions as voters. It is tempting to try to clarify the issue by replacing ‘independent of us’ with ‘independent of our cognitive abilities.’ But even there the issue will be troublesome: the fact that our cognitive abilities are what they are is precisely the kind of thing we want to classify as an ontological matter, ‘independent of us.’ 3. THE COMPLEXITIES OF COUNTERFACTUAL COLLECTIVITIES Semantically indeterminate collectivities offer one test case for a distinction between matters independent of us and those that are not. Counterfactual collectivities offer another. Consider for example: • Those residents of Hiroshima in 1945 who would still be alive today had the atom bomb not been dropped. At first glance such a case falls squarely in the ontological camp. The flow and eddies of historical contingency—of whether later firebombing would have taken some of the same lives, for example, or whether some who were not killed in that contingency might nonetheless have murdered others—are matters generally independent of us in the core sense at issue. But what of the indeterminacy of this collectivity? Is that indeterminacy an ontological matter or something else? The indeterminacy at issue is clearly ontological if the world’s events follow each other with something like the indeterminacy of quantum phenomena: if a saturated historical reality up to a specific point fails to entail one specific chain of later events rather than another. In that case the residents of Hiroshima who would still be alive are perfectly parallel to the uranium atoms that will decay over the next three years. Nothing now obtaining determines the membership of the latter, and nothing obtaining in the counterfactual contingency determined the membership of the former. If historical events at this scale are not like that, however—if a realitysaturated historical universe at one point fully entails one and only one course of future events—the matter is less clear. In that case it is tempting to say that what is envisaged is a particular event in the past—the Enola Gay’s failure to drop its payload, perhaps—which would have triggered a precise cascade of specific further events. On that picture, that particular event would have cast a very specific and sharply defined shadow into the future, detailed down to the specific people who would now be alive. The

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residents of Hiroshima who would still be alive today would then constitute an ontologically crisp set. Because we could not expect to know the precise series of subsequent events, that set would be one we would be unable to delineate. In that case, however, the indeterminacy of the collectivity defined by a counterfactual contingency would be merely epistemological rather than ontological. As outlined in terms of a ‘first approximation’ in the previous chapter, however, there is a further characteristic of such cases worthy of note. There are a number of large and detailed series of events which satisfy the minimal specification ‘had the atom bomb not been dropped.’ Even assuming a fully determinate universe, those different events would have had different consequences, resulting in different people living and dead. In some respects, therefore, the indeterminacy of counterfactual contingencies resembles not ontological, not epistemological, but semantic indeterminacy. ‘Foothills’ is a conceptual category unspecific enough to determine precise membership in its collectivity. The antecedent clause of this counterfactual, like that of many others, is similarly unspecific enough to determine precise membership in the collectivity evoked in its consequent.4 The realm of collectivities and their characteristic indeterminacies is a tangled and complex one. A first glance suggests a rough division between epistemically and ontologically indeterminate collectivities, or collectivities the indeterminacy of which is dependent on us or independent of us. A more careful second examination makes it clear that those initial categories are only very rough ones: that what is at issue is a variety of both different levels of independence and different types of dependency, keyed to a range of different cognitive abilities. It also becomes clear that the character of some indeterminacies is destined to remain controversial. In some cases that indeterminacy may itself be indeterminate: counterfactual collectivities, for example, can exhibit an indeterminacy that seems epistemic from one perspective, ontological from another, but akin to semantic indeterminacy from a third. 4. INDEXED INDETERMINACIES Cognitive access varies over time—we know things now we did not once know, and can calculate things now that were once beyond our reach. Indeterminacies dependent on cognitive access will therefore sometimes be indexed, either explicitly or contextually, in terms of time.

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We are quite generally ignorant of the future. A range of collectivities members of which lie in the future will therefore be indeterminate for us. Thus consider: • The initial cabinet of the President elected in 2016. • Humans who will be born between 3,000 and 4,000 A.D. The members of the first collectivity are alive now—we can individuate candidate members, though actual membership remains epistemically indeterminate. The members of the second collectivity do not now exist, and the collectivity thus seems indeterminate in a deeper sense—the conceptually tenuous indeterminacy of possibilia alluded to at the end of the previous chapter. The first collectivity is epistemically indeterminate for us now, but will not always be: its indeterminacy is indexed. The fact that it is a future cabinet that is at issue is not in fact essential, however. In some cases we might not know something about the past—who it is who is buried in an Egyptian tomb, for example, or which craters of the moon were formed by impacts older than 1 million years—though future investigation might tell us precisely that. It is the time of knowing, past or future, not the time of what is known, that indexes epistemic indeterminacy. Cognitive access varies across people much as it varies across time. There are things you know that I do not—the membership of your family across three generations, for example. That is a collectivity epistemically indeterminate for me at the moment, though not for you. There can be person-relative and context-indexed indeterminacy of a similar form: • Events at the party which no-one who hasn’t spoken to Jacob yet knows. For those who have not yet spoken to Jacob, the membership of this collectivity remains epistemically indeterminate. For those who have, however, the membership may be clear. Indexes are not limited to indeterminacies dependent on us, however: ontologically indeterminate collectivities can be indexed as well. Consider a clear example of ontological indeterminacy much like that with which we began:

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• Those atoms of that chunk that will lose their electrons between 2016 and 2019. On standard interpretations of quantum mechanics, this is a collectivity indeterminate in the deepest ontological sense: there is nothing in the course of the universe before 2016 that dictates precisely which atoms fit the bill. Between 2016 and 2019, however, there will be a particular set of atoms that decay. Once 2020 rolls around, whether through determinate mechanism or pure chance, the universe will have taken a particular course, and could not possibly take more than one. Some things will have happened rather than other things, and some atoms rather than others will in fact have decayed; in 2020, therefore, the collectivity will no longer be even ontologically indeterminate. Ontologically indeterminate collectivities, although independent of what we know or do not know, can therefore carry a time index as well. In this example there are in fact two factors in play regarding determinacy and indeterminacy. One, which we have emphasized throughout, is quantum indeterminacy: the fact that the course of the universe prior to 2016, together with all natural laws, is insufficient to determine membership in that collectivity. In that sense, membership of the collectivity, even when known in 2020, and even though determinate at that date, must be recognized to have been contingent. It remains true, even in 2020, that precisely which atoms would decay was not determined by events prior to 2016. The other factor in play in the example is the necessity of the past. At any point the events that have occurred are precisely those that have occurred; no matter how indeterminate they might be before the event—no matter how essentially random—they are fully determinate after. In 2016 the collectivity at issue is indeterminate because of quantum indeterminacy. By 2020 it has become determinate by the necessity of the past. Whatever the appropriate verdict regarding determinism as a philosophical doctrine, one must recognize a common sense of contingency in which some events are chancier than others. Excluding edges, the probability of a flipped coin landing either face up or tails up is one. The probability of it coming up heads is significantly smaller. The probability of 100 heads in a row is miniscule. A bet on 100 heads in a row is thus radically chancier than a bet on a head in a single flip. The fact that some events are chancier than others is also made clear by conditionalized sequences in which one must clear hurdle 1 even to reach hurdle 2. Having one’s name put in for

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nomination in either party is chancy; securing that nomination is chancier, and actually winning the final election is chancier still.5 5. VAGRANCY All of the issues we have addressed—reference, characteristic indeterminacies, and indexing—appear in a particularly interesting form in the phenomenon of vagrancy. An important albeit eccentric mode of reference occurs when an item is referred to obliquely in such a way that, as a matter of principle, any and all prospect of a certain form of access or identification is precluded. This phenomenon is illustrated by claims to the existence of: • a thing whose identity will never be known. • an idea that never has or will occur to anybody. • a person whom everyone has utterly forgotten. • an occurrence that no-one ever mentions. • an integer that is never individually specified. Given the form of reference used, these items must all be inaccessible in a particular way. In order to be referred to as such, the thing in the first example cannot possibly be identified. The ideas referred to in the second cannot be entertained. We cannot name the person utterly forgotten, recount the occurrence never mentioned, or produce the integer never individually specified. To produce instances of these in such a way would be to straightaway unravel the characterization at issue. Yet one cannot but acknowledge that there are such items—that they have an identity— notwithstanding the infeasibility of access or identification built into the phrases we use to refer to them. The concept of an applicable predicate instances of which of necessity cannot be identified comes into view at this point. Realizations of such a predicate F will be unavoidably unspecifiable. For while it holds in the abstract that this property at issue is indeed exemplified⎯so that (∃x)Fx will be true⎯nevertheless the very manner of its specification renders it impossible to specify any particular individual x0 such that Fx0 obtains. Such

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predicates are ‘vagrant’ in the sense of having no known address or fixed abode. Despite their having applications, instances cannot be specifically identified—they cannot be pinned down and located in a particular spot. So on this basis we may define: • F is a vagrant predicate iff (∃x)Fx is true while nevertheless • Fx0 is false for each and every specifically identified x0. Predicates of this sort will be such that, while general principles show that there indeed are items to which they apply, nevertheless it lies in their very nature that such instances could never be individually identified. It lies in the very make-up of their specification that when F is vagrant, then Fx0 is a contradiction in terms where x0 is a specifically identified item— an incoherent, meaningless contention. And this is a very real phenomenon, seeing that such predicates are illustrated by: • being a person who has passed into total oblivion. • being a never-formulated question. • being an idea no-one any longer mentions. Throughout such cases, specifically identified instantiation stands in direct logical conflict with the characterization at issue. To identify an item instantiating such a predicate is thereby to contradict its very characterization. And so, while the existence of exemplifications may be an ontological fact, this is offset by the no less firm epistemological fact that the identification of such exemplifying instance is simply impossible. The impossibility lies not in ‘being an F’ as such, but in ‘being a concretely identified F’. The problem is not with the indefinite ‘something is an F’ but with the specific ‘this is an F’. Difficulty lies not with F-hood as such, but with its specific application⎯not with the ontology of there being an F but with the epistemology of its apprehension in individual cases. Accordingly, vagrant predicates mark a cognitive divide between reality and our knowledge of it. Berkeley’s argument that no “extended moveable substance” can “exist otherwise than in a mind perceiving it,” considered above, turns on reject-

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ing any counterexample of trees existing unperceived in a park, or books existing unperceived in a closet, on the grounds that any counterexample put forward must thereby be perceived in imagination. Berkeley incorrectly construes the fact that no specific instantiation can be produced for the vagrant predicate ‘something existing unimagined’ as indication that such a predicate is a contradiction in terms, a “manifest repugnancy.” Such is a misconstrual of vagrant predicates in general. In thinking of a neverimagined star or unthought-of integer I am thinking in general of things that fall under that vagrant predicate, with no need—and clearly no possibility—of identifying any particular instance. There is no inconsistency in employing vagrant predicates or in speaking in general of their instances; the inconsistency would be in claiming we could identify any specific instance in the particular. In the abstract and formalistic reasonings of logic or mathematics— where predicates are cast in the language of abstraction—epistemic modalities and cognitive operators of the sort at issue in predicative vagrancy simply have no place. Here one will never encounter vagrant predicates, because matters of cognition are never invoked: we affirm what we know but never claim that we know. However, with matters of empirical fact the situation can be very different. The closest one comes to a cognitive operator in the context of logic and mathematics is perhaps a provability predicate such as that employed in the Gödel theorems. A provability predicate represents purely numerical properties true of particular numbers, but which given the particular encodings of a Gödel numbering hold of a number just in case that number encodes a formula derivable from the axioms of a system. Though only a rough approximation of a cognitive operator, it is interesting to note that this mathematical approximation begins to exhibit some of the characteristics noted above. Such a predicate is always indexed to a particular system. It also allows the encoding of a numerical parallel to ‘is not a theorem,’ for which the diagonal lemma gives us an instantiation which cannot be both a theorem and true on interpretation. That is of course the core of Gödel’s first theorem: that there will be formulae representable in any system of a certain strength which, though true, cannot be captured as theorem. ‘Seen from inside the system’, as it were, ‘is not a theorem’ therefore has some of the character of a vagrant predicate. In those matters of vagrancy that now concern us, cognitive inaccessibility is built into the specification at issue. Here being instantiated stands in direct logical conflict with the characterization at issue, just as with:

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• being a sand grain of which no-one ever took note • being a person who has passed into total oblivion • being a never-formulated question • being an idea no-one any longer mentions To identify an item of this sort is thereby to unravel its specifying characterization. Predicative vagrancy affords a clear route to epistemically and conceptually indeterminate collectivities. Thus in the case of the aforementioned examples, we realize full well that there are items of the sort at issue, while it is nevertheless impossible—not just in practice but in principle—ever to provide even a single example. As indicated at several points, we do not find a purely epistemic account of vagueness or semantic indeterminacy plausible. In the present context it is interesting to note, however, that the phenomenon of vagrancy might be used in fleshing out such an account. Consider the sorites paradox, in the form of the classic puzzle of how many sand grains it takes to make a heap. Clearly 2 is not enough. The following induction premise seems a plausible one: • There is no n such that n grains do not constitute a heap but n+1 grains do. Starting from fact that 2 grains are insufficient to constitute a heap, however, repeated applications of such a premise would entail that no number of grains would be sufficient: there could be no heaps of sand at all. One might however deny the induction premise above while maintaining a close epistemic variant, much as Williamson does:6 • There is no identifiable n such that n grains do not constitute a heap but n+1 grains do. Given such a position, the following integer characterization would qualify as vagrant predicate as we have outlined it:

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• is just sufficient to constitute a heap. While the position entails that there is such an integer, it also entails it to be forever unidentifiable. Although we remain skeptical of epistemic accounts of semantic indeterminacy and will not pursue this specific line of thought, we will devote a later chapter to the related prospect of using collectivities to introduce new forms of indefinite number theory. 6. INDEXED VAGRANT COLLECTIVITIES As indicated above, both the epistemic and ontological indeterminacy of collectivities may be indexed to a particular time. The vagrancy of vagrant predicates can also come with an index. Consider, for example: • Ancient Egyptian Pharaohs whose existence was unknown before 1890. • Matters of common knowledge after 2016 unknown to anyone before 2016. • Theorems unprovable with merely twenty-first-century techniques. • Events unpredictable before their occurrence. The membership of the first collectivity was by stipulation unknowable before 1890, though at least some of its membership (excluding future discoveries) is epistemically determinate for us now. The membership of the second collectivity is by stipulation unavailable to us before 2016. At that point in time its vagrancy will no longer be guaranteed, though issues of vagueness may certainly remain. Since identification of a theorem demands establishing its provability, the third example will remain a vagrant collectivity until at least the twenty-second century. The last example is one some membership of which will become determinate over time—we will witness events that we can establish were unpredictable in advance. Which future events will quality as members, however, will forever not merely be indeterminate but vagrant: were we able to identify such a member we would have successfully predicted it, unraveling its claim to qualification as a member after all.

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Vagrancy may be indexed not merely to particular times but to particular places or people:7 • Ideas never entertained by anyone within this room. • People of whose existence Bruce Springsteen is completely unaware. • Individual people of whose existence the authors are completely unaware. However hard anyone in this room works at it, the phenomenon of vagrancy—here indexed to people in this room—will prevent anyone in this room from determining the membership of the first collectivity. Any idea introduced as a candidate by a member of this room would automatically fail to fit the referential description. Indexed vagrancy will similarly prevent Bruce Springsteen from identifying any member of the second collectivity. Because the vagrancy at issue is indexed, however, it does not extend beyond Bruce—you and I can certainly identify people of whose existence Springsteen is unaware. Issues of vagueness aside, there is nothing to prevent you and me from identifying the membership of the collectivity as a whole. In the third case it is we the authors who are in Bruce’s position. We know this collectivity certainly does contain members. For us, however, its membership is epistemically indeterminate not only in part but in whole, allowing us the determinate identification of no member at all. Its pervasive epistemic indeterminacy for us is guaranteed by the characterization of the collectivity itself and the fact that we are the ones indexed in that characterization. Those limitations need not make it epistemically accessible to others, however. In the next chapter we turn from specific forms of indeterminacy to some of the basics and some of the surprises in trying to pin down its logic in general. NOTES 1

That indeterminacy in these cases does not appear epistemic is granted even by those who, like Sorensen and Williamson, argue that it ultimately is. See Roy A. Sorensen, Blindspots (Oxford: Clarendon Press, 1988) and Timothy Williamson, Vagueness (New York: Routledge, 1994).

2

George Berkeley, Philosophical Works; Including the Works on Vision. M. Ayers, ed. (London: Dent, 1975). Characterization of this as Berkeley’s ‘master argument’

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NOTES

is based on the central emphasis he puts on it, and comes from Evariste Gallois, “Berkeley's Master Argument.” The Philosophical Review 83 (1974) 55-69. 3

As George Pitcher and many other commentators have observed, Berkeley’s argument relies on conflating mode of representation with what is represented, or what we conceive with with what we conceive of. See G. Pitcher, Berkeley (London: Routledge, 1977) and Lisa Downing, George Berkeley, Stanford Encyclopedia of Philosophy 2004.

4

David Barnett also calls attention to the fact of multiple possibilities under an incomplete specification, though he argues for a conclusion radically different from that here. See “Indeterminacy and Incomplete Definitions,” The Journal of Philosophy 105 (2008) 167-91.

5

In understanding history, it is crucial to understand which events are nigh inevitable and which are highly contingent; which events would have occurred regardless of variance in a wide range of variables, and which demanded a precise coincidence of very specific variables. It is the necessity of the past that tends to wash out this crucial distinction in reading or teaching history.

6

Timothy Williamson, “Inexact Knowledge,” Mind 101 (1992) 217-242; “Vagueness and Ignorance,” Proceedings of the Aristotelian Society, Supplementary Volume 66 (1992) 145-162; Vagueness (New York: Routledge, 1994).

7

This is a phenomenon related to ‘personal paradoxes’, introduced to one of the authors (PG) years ago by David Boyer.

Chapter Four COLLECTIVITY THEORY: FIRST STEPS 1. COLLECTIVITY THEORY

T

he work of previous chapters has been to call attention to non-set collectivities, not to introduce them as something new. There is a sense in which indeterminate collectivities require no introduction; they are common coin in common reasoning and common thought. The members of the next President’s cabinet, those New Yorkers who would be alive had 911 not occurred, and the worlds’ foothills all constitute collectivities, indefinite in intriguingly different ways. Sets are crisp rather than indefinite: for precisely that reason they are not coextensive with collectivities, and cannot do the full work of collectivities. They have been a tempting philosophical stand-in for collectivities, however, precisely because of that crispness. Even given their convenient crispness, logical development of set theory was not without quite fundamental surprises—here Russell’s and Cantor’s paradoxes are prime examples. Similar surprises will appear in the logical development of collectivity theory, and indeterminacy carries surprises of its own. 2. COMBINATORIAL PRINCIPLES OF INDETERMINACY How do indeterminate collectivities form intersections and unions? The intersections of determinate sets are determinate throughout. If it is determinate which individuals have characteristic P and therefore qualify as members of set S, and determinate which individuals have characteristic P′ and therefore qualify as members of set S′, it will in all cases be determinate which individuals have both P and P′ and therefore qualify as members of the intersection of S and S′. Such a principle will hold as long as the sense of ‘determinate’—epistemic, ontological, or semantic, for example—is kept constant throughout. Such a principle could fail only if an individual were determinately P, determinately P′, but not determinately P and P′. The union of any two determinate sets will also be determinate. If it is determinate which individuals have characteristic P and therefore qualify

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as members of set S, and determinate which individuals have characteristic P′ and therefore qualify as members of set S′, it will in all cases be determinate which individuals have either P or P′ and therefore qualify as members of the union of S and S′. As before, such a principle will hold as long as the sense of ‘determinate’ is kept constant throughout. It is clear that the intersection of two indeterminate collectivities may be indeterminate: the collectivity of unwilling combatants at Antietam is indeterminate, the collectivity of those seriously wounded is indeterminate, and the collectivity of unwilling combatants seriously wounded at Antietam is indeterminate as well. Unlike inherited determinacy, however, inherited indeterminacy is by no means guaranteed; the intersection of two indeterminate collectivities need not itself be indeterminate. For consider an indeterminate collectivity C1 and its strong indeterminate complement C2: those things determinately excluded from C1. Consider further the collectivity C3 composed of C1 and the addition of a definite x, and the collectivity C4 composed of C2 with the addition of a definite x. The intersection of indeterminate C3 and C4 will be the determinate {x}. Intersecting indeterminates need not perpetuate indeterminacy; they can in fact lead out of it. It is clear that the union of indeterminate collectivities may be correspondingly indeterminate. Those seriously wounded at Antietam form an indeterminate collectivity, as do those psychologically traumatized by the battle; the union of those seriously wounded or psychologically traumatized forms an indefinite collectivity as well. Here again, however, indeterminacy need not be inherited: it is not the case that the union of indeterminate collectivities need be indeterminate. Thus consider those members of the Friar’s club who are very funny and those members who are not.1 Though each of these may be semantically indeterminate, their union is the fully determinate membership of the Friar’s Club. Union within the indeterminate can also provide an exit from it. Are determinacy and indeterminacy inherited ‘up’ and ‘down’?—by sub-collectivities or super-collectivities? C is a sub-collectivity of C′, and C′ a super-collectivity of C, just in case all members of C are members of C′. Because this definition comes with no restriction on collectivity specification, a set or collectivity that is determinate may nonetheless have indeterminate sub-collectivities. The membership of the Friar’s club is determinate, for example, but the subcollectivity of genuinely funny members of the Friar’s club is not. A determinate collectivity may of course also have indeterminate super-

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collectivities: all members of the determinate Friar’s club are at least minimally intelligent mammals, but the latter is an indeterminate collectivity. Unlike union and intersection, sub-collectivity and super-collectivity are not necessarily determinacy-preserving. The same examples suffice to show that sub-collectivity and supercollectivity need not be indeterminacy-preserving. The indeterminate collectivity of minimally intelligent mammals has the membership of the Friar’s club as a determinate sub-collectivity; indeterminate collectivities can have determinate sub-collectivities. The indeterminate collectivity of genuinely funny members of the Friar’s club has the determinate membership of the Friar’s club as a determinate super-collectivity. A very different argument shows that every indeterminate collectivity will have an indeterminate proper sub-collectivity. For suppose it did not. In that case every sub-collectivity at issue would be determinate. Since the union of determinate collectivities must be determinate, however, our supposedly indeterminate collectivity would then be determinate as well. The reductio establishes that there must be an indeterminate proper subcollectivity for every indeterminate collectivity. Does every indeterminate collectivity have a determinate proper subcollectivity? Clearly not; there will be collectivities for which it is indeterminate in every case whether something qualifies as a member or not. Epistemically vagrant collectivities, such as persons who have passed into total obscurity, offer a clear example: there are no identifiable members, and thus there can be no epistemically determinate sub-collectivity. Every indeterminate collectivity has an indeterminate super-collectivity, since we can always add another member, determinate or not. Does every indeterminate collectivity have a determinate super-collectivity? If there can be a universal collectivity, this question reduces to the question of whether such a collectivity is itself determinate or not. If there is a determinate universal collectivity, every collectivity has a determinate supercollectivity. If there is a universal collectivity, but it is indeterminate, there will be indeterminate collectivities for which there is no determinate supercollectivity—the universal collectivity itself, or the various collectivities obtained by removing some distinct member from it. It is clear from this discussion that a common phenomenon will be collectivities some members of which are determinate members—a determinate core—and some of which are not. Washington, Madison, and Lincoln are epistemically determinate members of the collectivity P of past and future presidents of the United States; each is a member of the determinate

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core P*. Indexed to the present, any future presidents are epistemically indeterminate members. The distinction at issue here is not merely that between determinate and indeterminate sub-collectivities of P: though past presidents do constitute a determinate sub-collectivity and future presidents do constitute an indeterminate sub-collectivity, there are other subcollectivities, determinate and indeterminate, that do not cut the collectivity on precisely the same line. What we need to distinguish is the subcollectivity C* of members that are determinate members of a collectivity C from the sub-collectivity of members that are not. Because determinacy and indeterminacy come in types—epistemic, ontological, and semantic, for example—the concept of a determinate core C* must be indexed to type as well. For collectivities and senses of determinacy in which there is a clear distinction of this type, at least one of the combinatorial principles above can be refined. Although the intersection of two indeterminate collectivities can produce a determinate collectivity, as noted above, this can occur only where all members of the intersection are members of the determinate membership of each collectivity. Should the intersection include any indeterminate member of either collectivity, membership in that intersection will be indeterminate as well. Such a distinction offers no similar refinement for the principle of union: the union of two indeterminate collectivities may or may not be determinate. Such a distinction is applicable, moreover, only where it is clear which are determinate and which indeterminate members of a collectivity. That, as we will see, is itself an issue that is often indeterminate. With indefinite collectivities, as we have seen, the idea of both union and intersection becomes complicated. Because collectivities can have both determinate and indeterminate members, it is tempting to think in terms of not one but three forms of membership: • x ∈d C: classical (determinate) membership • x ∈p C: penumbral membership • x ∈ C: full indeterminate membership, where x ∈ C iff x ∈d C v x ∈p C Such a distinction would give us two main modes of union and intersection. In place of a single notion of intersection, for example, we might think of two:

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• classical (determinate) intersection C1 ∩ d C2 = [x | x∈d C1 & x∈d C2] • full (indeterminate) intersection C1 ∩ C2 = [x | x∈ C1 & x∈ C2] In place of a single notion of union, we might think in terms of: • classical (determinate) union: C1 ∪ d C2 = [x | x∈d C1 v x∈d C2] • full (indeterminate) union: C1 ∪ C2 = [x | x∈ C1 v x∈ C2] Many of the examples given above can be thought of in these terms. We avoided using such a distinction, however, because what it in fact gives us is once again merely a useful first approximation. In the end, we will see, the line between determinate and indeterminate membership can itself be far from definite, vitiating any crisp distinction between types of intersection and union that assume a definite line. The fact that determinacy and indeterminacy can themselves be either determinate or indeterminate leads us into issues of higher-order indeterminacy and some of the more surprising conclusions in the logic of indeterminate collectivities. 3. RUSSELIAN RAMIFICATIONS The goal in developing a philosophical theory for any concept, however common, is philosophical insight. That insight often comes in the form of conceptual surprise, and sometimes in the form of conceptual paradox. Here our goal is a theory parallel to standard set theory’s treatment of crisp collectivities composed of sharply defined items, but appropriate to the wider class of everyday collectivities that clearly do not fit the restrictions of set theory. The development of even standard set theory led to major and important surprises, however, and the same will be true in development of a theory of indeterminate collectivities. In Chapter 2 we surveyed collectivities in terms of a range of types of indeterminacy. In addition to epistemically indeterminate, ontologically indeterminate, and semantically indeterminate collectivities, we there added a category of logically indeterminate collectivities. Some collectivities will be indeterminate, it turns out, for logical reasons alone; in particular, for logical reasons grounded in the concept of indeterminacy itself.

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Russellian collectivities offer an appropriate starting place. We begin with the following specification: • The collectivity CR of all and only those collectivities that do not contain themselves as members. There are many collectivities that clearly are members of CR: the empty collectivity does not contain itself as a member, nor does the collectivity of teaspoons. But for at least one item—the collectivity CR itself—questions of membership raise logical complications. If CR is a member of itself, it cannot be. If CR is not a member of itself, it must be. CR is therefore a logically indeterminate collectivity. There are several ways to envision the character of logical indeterminacy. One is simply to say that both the hypothesis that CR ∈ CR and the hypothesis that CR ∉ CR lead to contradiction. It must be that CR neither is nor is not a member of CR —just as the middle-sized man is neither tall nor not tall, for example, but here for logical reasons turning on contradiction rather than semantic reasons turning on vagueness. When one first approaches phenomena like CR, the thing that is immediately striking is that questions of membership become logically oscillatory. Is CR ∈ CR? If so, it must not be: it must be that CR ∉ CR. Given the characterization of CR, however, this in turn forces to conclude that CR ∈ CR. But that leads to the conclusion that CR ∉ CR. One might generalize from this personal and psychological experience to an abstract logical characterization of CR’s self-membership as oscillating. Abstractly considered, either hypothesis leads to an endless chain of deductions that alternate in their conclusions between CR ∈ CR and CR ∉ CR.2 The truth-situation oscillates; one form of logical indeterminacy is oscillatory indeterminacy. For indeterminate collectivities, Russelian structures of this type form a wide extended family. As noted above, it is often the case that collectivities C contain a determinate core C*—the sub-collectivity of those elements for which membership in the collectivity is determinate in a specific sense. In the case of semantical or ontological determinacy, at least, it is tempting at first to identify that core C* as a classical set: it is indeed tempting to identify the classical sets with those collectivities for which C = C*. The purely indeterminate collectivities, on the other hand, would be those for which C* = Ø. Like the outline of collectivity union and intersection above, this treatment of determinate cores in terms of sets is a good first approximation, though we will note below that it too must be qualified.

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Consider two further forms of collectivity • The collectivity C∈C* of all those collectivities C that are members of their own determinate core C*, and correspondingly • The collectivity of collectivities C∉C* that are not members of their determinate cores. Here different senses of determinacy will give us different forms of determinate cores. For any of those, one question is whether C∉C* ∈ C∉C*; whether the collectivity at issue is self-membered. Another question is whether C∉C* ∈ C∉C**; whether such a collectivity is an element of its own core. For any form of determinacy, if it is a member of its own core, any C∉C* is determinately a member of itself. But C∉C* is in each case the collectivity of collectivities that are not members of their determinate cores, and thus C∉C* must not be a member of its core. Regardless of the form of determinacy at issue, the hypothesis that C∉C* is a member of its own determinate core leads to the conclusion that it is not. If C∉C* is not a member of its own core, however, it fits its own specification—and determinately so. In that case we are forced to conclude that C∉C* is a member of its own determinate core. Whatever sense of determinacy is at issue, for collectivities of this form we again have a logical oscillation, again with a surprising conclusion. Since either hypothesis leads to oscillation between it and its contrary, it must be that C∉C* neither is a definite member of its determinate core nor is a definite non-member of its determinate core. Its core membership is logically indeterminate. The surprise is that we are therefore forced to conclude, regardless of the sense of determinacy at issue, that membership in even a determinate core can be indeterminate: that even determinate cores can have penumbral members. This line of thought leads us to a further conclusion regarding higherorder indeterminacy, to be developed in greater detail below. In this case at least we are forced to conclude that even determinate cores—subcollectivities of determinate members of a collectivity C—may have a membership that is not entirely determinate. Something may be an indeterminate member of a determinate core without thereby being determinately a non-member of that core. Because of that, the approximation to

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determinate cores in terms of sets offered above, therefore—the characterization of the determinate core C* of a collectivity C as a classical set, and of sets in general as those collectivities for which C = C*—must be taken as merely a first approximation. Because even determinate cores can have indeterminate membership, they need not be classical sets. Truly classical sets would have be something more like super-determinate cores, where all membership is determinate membership, all determinate membership is determinate determinate membership, and so on. 4. DIALECTICAL RAMIFICATIONS The logic of indeterminacy brings with it not only some surprising conclusions but some surprising forms of argument. In the previous section we concentrated on the collectivity form C∉C*; for various senses of determinacy, the collectivity of all collectivities not members of their determinate core. Consider also, however, the collectivity form C∈C*; for various sense of determinacy, the collectivity of all collectivities C that are elements of their own determinate core. Will collectivities of that form be self-membered, or not? Equivalently, will they be members of their cores C∈C**, or not? Regardless of the form of determinacy at issue, we get neither contradiction nor oscillation in assuming that C∈C* is determinately self-membered—a member of its own determinate core. The notion that the collectivity of all collectivities that are members of their epistemically determinate core, for example, is itself a member of its epistemically determinate core is clearly no contradiction. Nor is it a contradiction to assume that the collectivity of all collectivities that are members of their ontologically determinate core is itself a member of its ontologically determinate core. On the other hand, we get neither contradiction nor oscillation in assuming the contrary in each case: that the collectivity of all collectivities that are members of their epistemic or ontological core are themselves merely indeterminate members of that core. Whatever the sense of determinacy at issue, contradiction and oscillation force us to neither a positive nor a negative decision regarding determinate core membership for collectivities of the form C∈C*. Far from being an endpoint, however, this leads to a strange logical dialectic with a somewhat surprising conclusion. In the case of epistemic determinacy, how could we know whether C∈C* was itself a member of its own determinate core or not? Logical grounds, in the form of a contradiction from an assumption on either side, surely

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would give us the relevant knowledge. But in this case those appear to be the only route to knowledge: the only considerations which would determine either answer appear to be purely logical ones. A similar case can be made regarding semantic or ontological indeterminacy. Where semantics or ontology are at issue, what would make it the case that C∈C* was itself a member of its own determinate core or not? Were the concept that it was a member inherently contradictory, the result would be determined of logical necessity. Were the concept that it was not a member inherently contradictory, logical necessity would again decide the issue. Absent logical necessity on either side, however, there seems to be nothing that would force one result rather than the other: the only considerations which would determine either answer appear to be purely logical ones. Let us sum up. Whatever the sense of determinacy at issue, contradiction and oscillation force us to neither a positive nor a negative decision regarding determinate core membership for collectivities of the form C∈C*. Fairly obviously in the case of epistemic determinacy, however, and at least arguably in the case of semantic and ontological indeterminacy as well, whether C∈C* is a member of its determinate core or is not is something that only logical considerations of that sort could decide. Because both disjuncts prove consistent, there is no logical determinacy of core self-membership. The dialectical surprise is that such a conclusion does in fact settle the question. If there is no determinacy of core self-membership, C∈C* is emphatically not a member of its own determinate core. Is C∈C* a member of its own determinate core or not? If logic eliminates neither disjunct, and only logic could, the issue remains indeterminate—a conclusion, surprisingly, that eliminates the first disjunct after all. C∈C* is not a member of its own determinate core; equivalently, C∈C* is not determinately selfmembered. The logical dialectic, which we will see again in the logic of indeterminacy, is this: given the content of some pairs of disjuncts, indeterminacy between the two can itself decide in favor of one of the disjuncts in particular. Consider also: • The collectivity CD of all determinate collectivities. Is this collectivity a member of itself, or not? For specificity, we would again have to specify a particular form of determinacy: the collectivity CDE

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of all epistemically determinate collectivities, for example, or the collectivity CDO of all ontologically determinate collectivities. Regardless of the form of determinacy at issue, however, the supposition that a collectivity of form CD is self-membered leads to no contradiction: we would simply have a determinate collectivity of all determinate collectivities. The supposition that CD is not self-membered leads to no contradiction either: the edges of determinacy may themselves be indeterminate. So which is it? And what determines it? Here as before—clearly, in the case of epistemic determinacy, and at least arguably, in the case of semantic or ontological determinacy—it appears that the only considerations which would determine either answer are purely logical ones. Here as before, those do not suffice to determine either answer as opposed to the other. In the same way that CR’s self-membership or non-self-membership was indeterminate because logically over-determined, the self-membership of CD, like that of C∈C*, appears to be indeterminate because logically underdetermined. Here again, however, there is a dialectical surprise. We are led to the conclusion that the self-membership of collectivities CD is determinate after all: that collectivities CD of all determinate collectivities will themselves be indeterminate. The reasoning proceeds as follows. Where ‘P’ plays the role of ‘CD is determinately self-membered’: 1. There is no reason to think that P (by hypothesis) 2. There is no reason to think that not P (by hypothesis) 3. Therefore not P (by 1, 2, and the content of P) Because of the content of P in this case, the conjunction ‘there is no reason to think that P and there is no reason to think that not P’ gives us not-P. The clear oddity of the dialectical argument is that our conclusion is something that one of our premises claims ‘there is no reason to think’. To rub in the oddity, we might also conclude: 4. There is therefore reason to think that not P (by reflection on the argument 1 through 3) In that complete argument, our conclusion is the explicit denial of one of the premises.

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In the logic of indeterminate collectivities, where we are dealing not only with questions of self-membership of characterized sets but determinate or indeterminate self-membership, we can expect to find other arguments of this character. If a question-begging argument is one that is dialectically insufficient because its conclusion appears as a premise, we might call this a question-offering argument, dialectically effective despite the fact that the negation of its conclusion appears as a premise. 3 5. INCONSISTENCIES OF INDETERMINACY There are clearly collectivities that determinately exclude themselves as members: the set of singleton collectivities, for example, is certainly not a singleton collectivity. There will also be collectivities for which selfmembership is indeterminate: we have argued that both the Russell Collectivity CR and the collectivity CD of all determinate collectivities fall in this category. But consider then the complement CND to the collectivity CD of determinately self-membered collectivities: • CND: The collectivity of all and only collectivities which are either determinately non-self-membered or indeterminately self-membered. Will this collectivity be a member of itself, will it not be a member, or will its self-membership remain indeterminate? Any of these options, it turns out, leads us to contradiction. Here we have a three-way web of oscillation: • If CND is determinately non-self-membered, it is a clear and determinate member of itself. • If CND is determinately self-membered, it is clearly and determinately something other than a determinate member of itself. • If CND is indeterminately self-membered, there seems to be nothing indeterminate a about it: it then qualifies under its specification and so is determinately self-membered. It is noteworthy that this logical pattern again arises irrespective of the single or multiple forms of indeterminacy at issue. The pattern could, for example, be phrased purely for cases of epistemic indeterminacy. Consider, for example, an epistemically specified collectivity CNE:

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• The collectivity CNE of all those collectivities for which selfmembership is either determinately excluded epistemically or for which the status of self-membership is epistemically inaccessible. CNE is the collectivity of all those collectivities which we can either know that they do not contain themselves or for which we cannot know whether they do or not. Such a collectivity might well include collectivities which do include themselves as a matter of ontology or logic, for example, though that is something it is epistemically impossible to establish. Does CNE contain itself with epistemic determinacy, exclude itself with epistemic determinacy, or does it remain epistemically indeterminate whether it contains itself or not? If it contains itself with epistemic determinacy, then it cannot be epistemically determinate that it contains itself, on pain of contradiction. If it excludes itself with epistemic determinacy, then it determinately contains itself. The reasoning to this point might seem to offer a dialectical argument like those above: since the alternatives of determinate self-membership and determinate non-self-membership lead to oscillatory contradiction, we are forced to conclude that it is epistemically indeterminate whether this collectivity contains itself or not. But since its membership conditions for CNE specify that it includes all collectivities whose self-membership is epistemically indeterminate, such a conclusion would entail that it determinately includes itself after all. Here again we face a triangular web of oscillatory contradiction. There are two connected lessons here. One should be familiar by this point: the lesson that many logical issues regarding indeterminate collectivities do not depend on the form of indeterminacy at issue. Analogous patterns and analogous problematic can be expected for epistemological, semantic, and ontological indeterminacy. Another lesson is that appeal to the concept of indeterminacy alone will not free us from the difficulties of analogous contexts—the difficulties crystallized in Russell’s paradox and attempted of resolution in various forms of axiomatized set theory, for example. The promise of indeterminate collectivities is a wider logic, but is not guaranteed to give us a logic that can easily or instantly solve the old problems. A more complete discussion of a range of logical options for indeterminate collectivities in general is left to chapter 6.

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Questions of self-membership regarding CD and CNE lead to oscillatory contradiction on any of three alternatives, much as does the strengthened Liar. Here we will sketch one appealing map of the logical territory, tracking one plausible option. Is the Russell collectivity determinately selfmembered or determinately non-self-membered? One plausible response is to reject these two alternatives as exhaustive, embracing indeterminate self-membership as a third option. Are CD and CNE determinately non-selfmembered, determinately self-membered, or indeterminately selfmembered? Here again one plausible response is to reject the alternatives as exhaustive, forcing a category of indeterminate indeterminate membership. For further questions regarding further collectivities such a strategy will lead us ever higher, revealing a hierarchy of non-collapsing indeterminacies motivated by other considerations as well. 6. HIGHER-ORDER INDETERMINACY The indeterminacy of collectivities is not a single and simple category. On the current approach, there are levels of indeterminacy which form a hierarchy quite naturally and without arbitrary imposition. Thus consider a collectivity which has some definite members, but for which it is indefinite whether other items belong. This will of course qualify as an indeterminate collectivity in our sense. But consider the collectivity of those things for which membership is indeterminate. In some cases this will be a very definite set: The original collectivity will be indeterminate, but it will be determinate with respect to what things it is indeterminate. Such a collectivity will come with a determinate set of things in, a clear class of things out, and a determinate set of things for which membership is indeterminate in one or more of the senses we have outlined. In other cases, however, the collectivity of things for which membership is indeterminate will itself be indeterminate. This is a form of higher-order indeterminacy. A collectivity with a distinct and definite set of ‘indeterminates’ can be thought of as indeterminate to degree 1. A collectivity for which the collectivity of indeterminates is itself indeterminate will be indeterminate to degree 2. And the hierarchy will continue. Thus collectivities for which it is indeterminate whether the collectivity of indeterminates is or is not indeterminate will constitute an indeterminacy of degree 3. In work on the Liar’s paradox, hierarchy is standardly rejected as ad hoc or artificial: our language recognizes no hierarchy of content in the sense required by Russell’s theory of types, for example, and no hierarchy of

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truth-predicates or meta-languages of the form required by Tarski and related approaches.4 That objection to hierarchy, however, is largely an objection to descriptive and/or normative claims regarding our language: claims that English comes with some implicit indexing of truth-predicates, for example, or normative prohibitions on certain applications of truthpredicates. The hierarchy outlined here is emphatically not a linguistic hierarchy, and is not offered normatively in any sense. It is a hierarchy inherent in types of indeterminacy—a hierarchy in the subject matter of indeterminacy itself, not merely a hierarchy in the language we normally do or (supposedly) should use regarding it. The pattern outlined for higher-order indeterminacy is thus reminiscent of the pattern of higher-order vagueness. That is not surprising for the case of semantic indeterminacy, since our primary case there is vagueness. Higher-order indeterminacy seems to be a wider phenomenon, however, extending to epistemic, logical, and ontological indeterminacy, including radical contingency. Vagueness is merely a specific case. The indeterminacy of vague terms is often illustrated against an objective scale of some kind—‘tall’ against a scale of height in feet and inches, for example. Those over 6’8” are clearly tall, those under 4’6” clearly not tall, but what of those in between? With such a scale in mind, it is natural to think of the hierarchy of indeterminacies in terms of border regions—a middle region between tall and not-tall, for example—border regions of border regions, and border regions of border regions of border regions beyond . . . At each step the number of categories branches by powers of two: from ‘indeterminate between x and ~x’ at a first step to ‘indeterminate between x and indeterminate between x and ~x’ and ‘indeterminate between ~x and indeterminate between x and ~x’ at the second step, and so on. One might expect runaway or even broadening areas of indeterminacy as a result. But this need not be so. Consider for example: X0 = X lies in the region .5 ± X1 X1 lies in the region .05 ± X2 X2 lies in the region .005 ± X3 • •

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• 1 Xn lies in the region .5 (10 )n ± Xn+1 Such infinite series of higher-order indeterminacies—like those along each of our branches—may be convergent, approaching 0 in the limit.5 The example of vague predicates against an objective scale makes it clear that an infinite hierarchy need not entail a chaos of runaway indeterminacy. The standard assumption in the vagueness literature is that higher-order vagueness generates an infinite hierarchy. There have been counterproposals, however: that iterated vagueness ‘peters out’ after a few generations.6 One argument for this view is the difficulty of understanding “it is vague whether it is vague whether it is vague whether it is vague whether . . .” beyond a certain number of repetitions of ‘vague.’ A parallel argument could be made that the same is true regarding “it is indeterminate whether it is indeterminate whether it is indeterminate . . .” Though we will not here rule on this issue for determinacy, a few comments are perhaps appropriate. To the extent, or in the varieties, that indeterminacy is dependent on our conceptual abilities—conceivability and knowability, perhaps—the argument against an infinite hierarchy has some bite. To the extent that indeterminacy is independent of us, however—as it is in ontological and logical cases, for example—the difficulty of conceiving of repeated iterations loses its bite, failing to show that there is less than an infinite hierarchy of indeterminacies. The possibility remains, therefore, that higher-order indeterminacies of some varieties ‘peter out’ in the sense that has been proposed for vagueness, whereas other forms of indeterminacy obey a strict hierarchy of iterations all the way up. Even these first steps toward a logic of indeterminate collectivities reveal a deep complexity that skirts with paradox. That fact comes into graphic realization when we carry over these considerations to the particularly interesting class of large-scale collectivities we call plena. NOTES 1

Depending, of course, on the sense of ‘not’ at issue; on this point see the discussion of alternative negations in chapter 6.

2

Abstract oscillations of this type are used to introduce an exploration of the wider and more complex semantic dynamics of paradox in the first few chapters of Grim,

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NOTES

Mar, & St. Denis, The Philosophical Computer (Cambridge, Mass: MIT Press, 1998). 3

There is much more to be explored regarding dialectical arguments of this sort, which seem to parallel the destructive force of contradictions in reductios with constructive conclusions despite contradiction in certain cases. That further exploration is left to further work.

4

As one case among many, see Patrick Grim, The Incomplete Universe (Cambridge: MIT Press, 1991).

5

In the limit, on this picture, higher order indeterminacies approach a Cantor set of points without measure distributed across the interval. See for example H.-O. Peitgen, H. Jürgens, and D. Saupe, Chaos and Fractals: New Frontiers of Science 2nd ed. (N.Y., N.Y.: Springer Verlag, 2004), page 65.

6

A topic of general discussion during the conference on paradox and vagueness at Storrs, Connecticut whose papers are collected in J.C. Beall, Liars and Heap (New York: Oxford University Press 2003).

Chapter Five PLENA 1. THE NATURE OF PLENA

A

s indicated in Chapter 1, collectivities are non-set pluralities that we generally access in terms of qualifying features rather than inventories of identifiable members. In this chapter we focus on a particularly intriguing category of collectivities: that we term plena. Plena are collectivities of a decidedly extraordinary kind: large-scale macro-totalities on the order of all facts, all truths, and all things. Here we seek to take some first technical steps toward a more adequate conception of these massive collectivities. It seems clear on first thought that there must be such collectivities. There are facts, and there must be a totality of all facts. There are truths, and there must be a totality of all the truths. There must be a totality of propositions, of things, of sets of things, and of collectivities. But there are clearly difficulties here. Within standard set theory the notion of a set of all sets is well known to lead to contradiction; to all appearances, there can therefore be no such thing. The notion of a totality of truths, of facts, of all states of affairs, or of all propositions leads to contradiction in much the same way. Suppose, for example, that there were a set of all facts F. That set will have a power set PF— the set of all sets of facts. But for each set of facts F in PF, there will be a distinct and specific fact—that F contains precisely the facts that it does, for example. But by Cantor’s theorem, there must be more elements of the power set of any set than there are elements in the set itself. By the reasoning set out above, there will then be more facts than in F. But F was designated as the set of all facts. Contradiction.1 Despite their intuitive plausibility, mega-totalities such as these prove decidedly problematic. It will take some doing to create a secure place for them in the realism of cogent deliberation. And just this is the task of plenum theory, which addresses oversize collectivities that are as large as their own power sets, including:

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• things/entities/individuals/items/objects • ordered sets2 • structured inventories • segmented agendas (for discussion or deliberation) • actualities and possibilities3 • truths/facts/states of affairs Attempts have been made to deal with these intuitive totalities of sets by means of alternative class theories, but often the result is philosophically unsatisfactory. A place may be made for a class of all sets, for example, but at the cost of having no place for an equally intuitive class of all classes. Application of class theories to totalities of truths, propositions, or facts are no more promising.4 Following suggestions of George Boolos, further developed in the work of Agustín Rayo and Timothy Williamson, attempts have also been made to provide a place for broad and universal quantification without the broad and universal collections or totalities that would supply a classical model-theoretic domain for that quantification. Against this background it should be stressed that the concerns of plenum theory are, by contrast, primarily ontological rather than semantic. Our proposal here is that—just as with collectivities generally— one should take such totalities at face value, as intransitively accessible objects. It is these, which the Boolos approach carefully does without, that are precisely the objects to be explored.5 But why should one care about such mega-objects: what are they good for? The answer lies in the very aim and ambition of abstract thought. Theorizing aspires to universality—to a transcendence of ordinary experience’s episodic particularization of this and that. It strives to generalization about totalities of different kinds and—in the end—to the totality of everything-at-large. What plena offer is the prospect of giving substance and structure to this line of thought. Tra-

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ditional approaches to ontology and universalization exhibit a range of theoretical deficiencies which, it is hoped, this further and decidedly different conception may serve to overcome. 2. PLENA, TOTALITIES, AND PLENARY TOTALITIES A plenum as here understood is a collectivity that contains distinct entities corresponding to each of its sub-collectivities, where subcollectivities follow the same pattern as subsets: (∀s)(s ⊂ P ↔ (∀x) (x ∈ s → x ∈ P)) The mark of plena is that every sub-collectivity s of a plenum P is such that there is a member of P that exists in unique correlation with s. Using ∃s to stand for ‘exists in unique correlation with s,’ the definitional condition for plena P is (∀s)(s ⊂ P → (∃sx) x ∈ P) Given this condition, there will be a mapping of all sub-collectivities of a plenum into its members, that is, a mapping m such that (∀s)(s ⊂ P → (∃x)(x ∈ P & x = m(s) & (∀s′)(s′ ≠ s → m(s′) ≠ m(s))))

There may be various ways in which sub-collectivities of a plenum exist in unique correlation with a member. Each sub-collectivity of a plenum may be represented by a member of that plenum, or may correspond to the content of a unique member. In the case of a plenum of all truths, for example, truths to the effect that its sub-collectivities have precisely the members they do will be elements of the plenum of truths. In the case of a plenum of all states of affairs, the state of affairs of a subset of states of affairs obtaining, or for that matter being thought about by a particular agent at a particular time, will themselves constitute member states of affairs. Often, however, the relation will be simpler: the sub-collectivities of a plenum will themselves be direct members:

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(∀s)(s ⊂ P → s ∈ P) Such a plenum will be characterized specifically as a membership plenum, so that P is a membership plenum when [x⏐x⊂P]⊂P A totality is a collectivity defined by the stipulation that it contains any and all items that satisfy a certain descriptive condition, such as belonging to a certain specific natural kind. It is a collectivity that stipulates a natural completeness. And as with other collectivities, it need not necessarily be a set because it need not be sharp edged in the sense of effecting a surgically clean cut between the things that do and those that do not belong. Georg Cantor argued in 1895 that “the absolutely infinite totality of cardinal numbers” is not a set.6 For if it were then it would have a cardinality which could not be exceeded by that of any other set. But the set of all of its subsets would indeed have to have a larger cardinality which (ex hypothesis) it cannot. From the angle of set theory totalities can prove to be decidedly problematic. But other angles are available. A plenary totality or totalistic plenum is a totality that meets the definition of plena above. Among the intuitive totalities we have mentioned are things, ordered sets, structured inventories, segmented agendas, actualities and possibilities, propositions, truths, facts, and states of affairs. In each case we have aggregative closure in that that every collectivity of items of this kind will itself define—and indeed often constitute—a distinct item of that kind. All of these collectivities are accordingly totalistic plena. All of these are also kind totalities, in that there is a natural kind K such that the totality in question is: = [ x⏐Kx ] = [ x│ x ∈ K] It follows immediately that there must be infinitely many plena. If the totality of all items/objects is a plenum, then so is the totality of all such items except for those of some specific sort y (dogs, stars, numbers). If the totality of all truths is a plenum, then so is the totality of all truths that do not mention y’s (dogs, stars, numbers).

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It warrants note that kinds are determined intentionally in terms of meaning while totalities are determined extensionally in terms of their membership. Accordingly, we do not have it that K ≠ K′ → ≠ or equivalently = → K = K′ As kind specifications, triangles ≠ trilaterals (in a Euclidean context) thanks to the difference in a conceptual articulation. But we do indeed have = On the other hand, we cannot have different totalities of the same kind. Thus K = K′ → = is perfectly true. If had a member that lacked, or conversely, then K = K′ could not be maintained. As outlined, a kind totality qualifies as a totalistic plenum if there is a distinct element for each of its sub-collectivities. Truths form a totalistic plenum because there is a distinct truth for each sub-collectivity of truths—a truth, for example, about precisely those truths— including a truth about precisely all the truths. Membership in the totality of truths may be amorphous in other regards, however. It may remain an open question, for example, whether the collectivity of all truths itself constitutes a truth. In previous chapters we have tracked the logic of epistemically, semantically, logically and ontologically indeterminate collectivities. Indeterminacy is not definitional of plena. Because plena are too big for surveyability, however, epistemic indeterminacy of various sorts may turn out to be a common characteristic of plenary totalities. The definitional specification of plena as containing distinct entities corresponding to each of their sub-collectivities, moreover, marks them as inherently amplificatory as well: incomplete in any determinate conception.

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3. IS TOTALIZATION LEGITIMATE? The logical difficulties of a set of all sets are familiar. The logical difficulties for a set of all truths, of all propositions, and of all states of affairs follow a similar problematic. The promise of non-set plena is an alternative approach to such totalities. But there have also been other objections to such totalities that should be acknowledged. Immanuel Kant deserves note as the philosopher who, in his Critique of Pure Reason, first complained about illicit totalization. He rejected any and all totalities that are not closed, as it were, and therefore impossible to survey in toto. Kant held that experience is our only pathway to knowledge about existence, and some totalities are impossible to experientially survey as a whole; at best we can survey only particular instances. As Kant saw it, there is a fundamental fallacy is involved in such totalitarian conceptions: The concept of the totality is in this case [of the-world-as-a-whole] simply the representation of the completed synthesis of its parts; for we cannot obtain the concept from the apprehension of the whole—that being in this case impossible. . . (Critique of Pure Reason, B456.)

For Kant, such closure-defying, unsurveyable conceptions as that of the-world-as-a-whole, whose content goes beyond the range of that which could ever be given in experience, are something ill-defined and thereby inappropriate. Only experiential interaction can ever ensure actual existence—description alone can never do the job: [It is inappropriate to suppose] an absolute totality of a series that has no beginning or end [such as would be at issue with ‘the terminus of all successive divisions of a region’ or ‘the initiation of all the causes of an event’]. In its empirical meaning, the term ‘whole’ is always only comparative. The absolute whole of quantity (the universe), the whole division [of a line segment], or of [causal] origination or of the condition of existence in general . . . along with all questions as to whether this whole is brought about through finite synthesis or through a synthesis requiring infinite extension . . . [are something altogether inappropriate]. (CPuR, A483-84 = B511-12.)

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Kant held that because experiential totalization is unachievable we can never appropriately reify such a totalistic conception into that of an object that has a well-defined identity of its own. Accordingly, such totalizations as ‘the-physical-world-as-a-whole’ represented for Kant an inherently fallacious conception whose acceptance as real leads to inconsistency: [As a sum-total of existence] the world does not exist in itself, independently of the regressive series of my representations, it exists in itself neither as an infinite whole nor as a finite whole. It exists only in the empirical regress of the series of appearances, and is not to be met with as something in itself. If, then, this series is always conditioned, and therefore can never be given as complete, ‘the world’ is not an unconditioned whole and does not exist as such a whole, either of infinite or of finite magnitude. (CPuR A503-05 = B531-33.)

Kant’s objections to totalizations often have a distinctly epistemological flavor; his objections are rooted in the fact that the possible experience of certain totalities is inherently impossible. In outlining plenary totalities as metaphysical or logical constructs, on the other hand, we reject a commitment to Kant’s epistemic restraints. Kant repeatedly claims, on the other hand, that endorsement of unsurveyable unbounded totalizations must result in logical selfcontradiction. This is certainly debatable; it remains open to question whether his four classic ‘Antinomies’ in the Critique of Pure Reason genuinely establish any such fact. Here we follow Kant, however, in assuming that outright logical contradiction is something one must prima facie be at pains to avoid.7 Because Kant held these inaccessible entities to generate inconsistencies, he rejected large-scale totalities such as “all real things” and “all human thoughts” as inherently inappropriate. In this regard modern logicians walk in his footsteps. They reject totalities such as • all sets having the feature F • all propositions having the feature F

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as inappropriate on the grounds that factor-specifications of this sort can lead to contradiction in set-theoretical development. The object of the theory of collectivities and plena is to expand our ontological horizons by recovering such inconsistency-generating collections for the domain of rational deliberation. 4. SOME LOGICAL ASPECTS OF PLENA When viewed from the perspective of anything like sets, plena are self-amplifying and indeed explosive in content. Every time one looks to yet another sub-collectivity one automatically finds further members of the plenum itself. Suppose one starts with a set-like estimate of a plenum’s membership—an estimate of all the things it contains as members. Because a plenum contains a unique member for each of its sub-collectivities, each collectivity of those things will betoken a further member of the plenum. Plena are so large that we can never bring them into view as a whole. Whatever one’s first view, the plenum will contain more—ever adding elements corresponding to all elements of the power set of what is in view. Once one has included those in the plenum as well, it is clear that each sub-collectivity of that larger compendium will represent a further member of the plenum. The process continues and expands one’s view ad infinitum. It also follows immediately that there can be no merely finite plena. Were we to start by contemplating a finite membership, we would have to recognize every sub-collectivity of that membership as generating a further member, and then every sub-collectivity of that larger group, and so on. The explosiveness of plena puts them not only beyond any finite number but beyond any number at all. Any estimate of a given cardinality c for the members of a plenum forces us to recognize a greater cardinality for the collectivity of its subcollectivities. Because the plenum will contain a unique member corresponding to each of its subcollectivities, however, we will have to abandon our original estimate in favor of a greater. That process continues divergently and without limit. Self-amplification marks a particular form of indeterminacy characteristic of plena in general and plenary totalities in particular. Membership of plena is of necessity conceptually incomplete, in the sense

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that no conception on the model of a completed set or list can possibly be adequate. For plena any notion of determinate membership, finished and complete, will prove inadequate.8 Plena are quite literally mind-expanding, growing beyond any determinate conception at any time. This kind of amplificatory indeterminacy need not imply that plenary totalities are “fuzzy” in the usual, degree-admitting sense of the term.9 As surveyed in earlier chapters, indeterminacy often appears in the guise of indecision. It is epistemically indeterminate whether this person would be here had the alarm bell sounded, for example, semantically indeterminate whether this counts as a ‘foothill,’ or ontologically indeterminate whether this Californium atom will decay in the next thirty days. In such a case available information, our semantic specifications, or the world itself haven’t made up their minds as between in or out, so to speak. In the case of plenary totalities, however, indeterminacy can appear not as ambivalence but as the ‘not yet’ of incomplete development. Plenary totalitites are explosive, conceptualizable only in part at any stage, since any conception of their membership forces a revised and more expansive conception. Full membership in plenary totalities remains indeterminate not merely epistemically but conceptually. That necessary conceptual incompleteness renders membership in plenary totalities not so much fuzzy as embryonic: forever possessed of growth potential, but forever failing to be fully formed. Totalities incomplete with regard to membership will also be problematic with regard to universal quantification. Specifically, what are we to mean in saying that all members of such a totality have a certain feature—when it may not even be fully and decisively determinate what that membership consists in? The indeterminacy of x ∈ P will carry that of (∀z)(z ∈ P → Fz) in its wake. Of course whenever z ∈ P entails Fz, so that z ∈ P → Fz obtains on logical grounds alone, then (∀z)(z ∈ P → Fz) must be true. A generalization can be claimed to hold for the entire membership of a totalistic plenum when it holds as a matter of logico-conceptual necessity. In other cases, where x ∈ P slips into indeterminacy, the status of this generalization can be expected to slip into indeterminacy as well.

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5. PLENA AND INDETERMINATE COLLECTIVITIES: DISTINCTIONS AND LINKS

Just what is the relation between plena and indeterminate collectivities? Must each plenum be an indeterminate collectivity? Will every indeterminate collectivity be a plenum? The answer in each case is negative. Plena and indeterminate collectivities, although related in an important way, are not the same, and neither includes all cases of the other. Consider the collectivity C∈C of all self-membered collectivities. Although vast, that collectivity is not a membership plenum. Its subcollectivities include Ø, as does the power collectivity of every collectivity, for the same reasons that every set includes Ø as a subset. But Ø by definition has no members. It cannot then contain itself as a member, and so cannot be a member of the collectivity C∈C of all selfmembered collectivities. The collectivity C∈C is not a membership plenum. Whether it is a plenum in any other sense may depend on the other links possible between its members and its sub-collectivities, but at this point there seems to be no reason to think it is a plenum at all. Consider now the Russell collectivity, as invoked in former chapters: the collectivity CR of all non-self-membered collectivities. This clearly is an indeterminate collectivity. The assumption that it either does or does not contain itself as a member leads to contradiction. For CR, both CR ∈ CR and CR ∉ CR must fail. Is CR a plenum? Interestingly, it neither is nor is not a membership plenum. If CR were a membership plenum, it would contain all of its own sub-collectivities. But since every collectivity is a sub-collectivity of itself, were CR to contain all its sub-collectivities it would also contain itself as a member. That, we have seen, it does not. If CR were a membership non-plenum, on the other hand, there would be at least one sub-collectivity that was a non-member of CR. But consider any such collectivity Q. If Q is a (true) non-member of the collectivity of all non-self-membered collectivities, it must be selfmembered. The collectivity Q therefore has itself — Q — as a member. But Q is a sub-collectivity of CR. As such, every member of Q

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must be a member of CR. Since Q has Q as a member, Q must therefore be a member of CR, from which it follows that Q is non-selfmembered. Here again we have reached a contradiction. The Russell collectivity of all non-self-membered collectivities is therefore an indeterminate collectivity that is neither a membership plenum nor a membership non-plenum. Here as before, whether it qualifies as a plenum of any other sort depends on links other than membership possible between its elements and its sub-collectivities. The collectivity C∈C of all self-membered sets, we have seen, is not a membership plenum. If it is an indeterminate collectivity, it is at least not for the reasons that CR is. For CR, it cannot be maintained either that CR ∈ CR or that CR ∉ CR. For C∈C, on the other hand, it can be maintained without contradiction either that C∈C ∈ C∈C or that C∈C ∉ C∈C. Self-membership for C∈C, we might say, is underdetermined: either of two options is open, with nothing to direct us to one rather than the other. What consideration of these two collectivities indicates is that questions of indeterminateness and questions of plenary status are logically independent; answers to one do not necessarily give us an answer to the other. If so, there will be plena that are entirely determinate, as well as plena that are not. There will be indeterminate collectivities that are plenary, and indeterminate collectivities are not. There is, however, this important link between the two: • Every plenum must have an indeterminate collectivity as one of its sub-collectivities. Consider any collectivity C which pairs a unique element with each element of its power collectivity. This can be thought of in terms of a transform Tr: PC → C which matches elements of the power collectivity PC with unique elements of C. For members P of PC, let →Tr(p) represent the C-partner of element P of PC in such a matching, and let

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←Tr(c) similarly represent the power-set partner for members c of C. Now consider the collectivity C* of all c such that c ∉ ←Tr(c). C* is the collectivity of all elements of the original collectivity that are not members of the element of C's power set with which they are partnered. C* will be an element of the power collectivity PC, since each of its members is a member of C. But will →Tr(C*) ∈ C*, or not? Will that element of C that is the partner of C* be an element of C* or not? Both →Tr(C*) ∈ C* and →Tr(C*) ∉ C* yield a contradiction in precisely the manner of the Russell collectivity. C* must accordingly be an indeterminate collectivity. Given the generality of the proof, every plenum C must have an indeterminate sub-collectivity of such a form. And so, every plenum will contain an indeterminate subcollectivity. Does that entail that plena will themselves necessarily be indeterminate? There does not seem any reason to think so. Consider again the case of the Russell collectivity CR of all nonself-membered collectivities. Such a collectivity can be neither a member nor a non-member of itself. But the ground of indeterminacy regarding CR is its self-membership, not its status as a collectivity. The indeterminacy of CR need not, therefore, be contagious upward to a collectivity of all collectivities. There may be other grounds on which we have to conclude that the collectivity of all collectivities is itself indeterminate, but the indeterminacy of CR is not one of them. On these grounds, at least, it seems perfectly possible for there to be determinate plena, though all plena must contain indeterminate subcollectivities. 6. PLENA AND PARADOX We began by emphasizing that collectivities are not sets, and demand a logical basis beyond that afforded by classical logic associated with set theory. In exploring indeterminacy and indeterminate collectivities we find oscillatory phenomena, forms of dialectical argument, and visions of ascending hierarchies of semantic value that lie beyond

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the limits of familiar logical territory. Plena and plenary totalities, despite their intuitive obviousness, seem to violate the restraints of classical logic. If we are to do any or all of the following: • recognize familiar collectivities as real—the world’s foothills, for example, or our forgotten memories • accept forms of indeterminacy as genuine phenomena • conceptualize plenary totalities—all propositions, facts, or reality as a whole, for example we will need to forge new logical tools adequate to the task. Viewed through the lens of familiar logics, our explorations thro ughout have been in the realm of paradox. What might that realm look like if viewed through a different lens? NOTES 1

Patrick Grim, “There is No Set of All Truths.” Analysis 44 (1984), 206-208.

2

The function or order is going to be critical here—and also with some of the plena considered below (agenda and inventories, in specific). For joining the pair of sets {a,b},{c,d} and the pair {a,b,c}, {d} will yield one selfsame composite set, {a,b,c,d}. The uniqueness condition for plena is thus validated. However with ordered sets, the situation is resolved, seeing that {{a,b}, {c,d}} and {{a,b,c},{d}} are different.

3

Full actuality, as a totality, encompasses all the ways things are. The way the sun glints on the window, the way the train whistle sounds through the fog, and the way she loves me are all ways things are. To each sub-collectivity of the whole will correspond a way things are: things being that way, or the fact that they are that way. Much the same will hold for full possibilities; for total ways things might have been.

4

For more complete discussion see Grim, The Incomplete Universe (Cambridge, Mass., MIT Press, 1991).

5

George Boolos, “To Be is to Be a Value of a Variable (or to Be Some Values of Some Variables),” Journal of Philosophy 81 (1984) 430-439, and “Nominalistic Platonism,” Philosophical Review 94 (1985) 327-344, both reprinted in Logic, Logic and Logic (Cambridge, Harvard University Press, 1998); Agustín Rayo and Timothy Williamson, “A Completeness Theorem for Unrestricted First-Order Languages,” in J.C. Beall, ed., Liars and Heaps (Oxford: Clarendon Press, 2003), pp.

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NOTES

331-356. See also Vann McGee, “Universal Universal Quantification,” pp. 357364 in the same volume. Here we simply put these approaches aside, rather than pursuing the philosophical difficulties that they continue to raise. 6

See Georg Cantor “Beitraege zur Begruendung der transfiniten Mengenlehre, I” Mathematische Annalen, vol. 46 (1895) 481-512.

7

The alternative of accepting contradictions, invoked in Chapter 1, is further discussed in chapter 6.

8

See Patrick Grim, The Incomplete Universe (Cambridge, MA: MIT Press, 1991).

9

Nor, as discussed in chapter 2, does it amount to the specific degree of membership familiar in fuzzy logics.

Chapter Six LOGICAL OPTIONS 1. THE LOGICAL DEMANDS OF INDETERMINATE COLLECTIVITIES

M

uch of our conceptualization, deliberation, and reasoning is in terms of what we have termed collectivities: the world’s foothills, those now alive but for an atomic blast, the human species, tomorrow’s tasks, and future generations, for example. In order to take the elements of that conceptualization seriously—in order to take our own reasoning seriously—we have to take seriously the indeterminacy quite generally characteristic of such collectivities. Collectivities have members under all modalities: actual and potential members, past and future members, members identifiable or unknown. Most pointedly, familiar collectivities have members both determinate and indeterminate. Classical set theory is inadequate to deal with the fact that collectivities have members under all modalities, and classical logic is inadequate to deal with indeterminacy. Collectivities also include the self-amplificatory collectivities we have termed plenary totalities: the universe as a whole, the totality of things, all propositions, states of affairs, facts or truths. Plena such as these quite explicitly explode beyond the limits of classical logic and set theory. What is the broader logic required for these broader realms? At various points in the discussion we have dropped hints as to what such a logic would require, usually in terms of leaving the Law of Excluded Middle behind. For epistemically indeterminate collectivities there will be candidates for which neither membership nor non-membership is epistemically determinate. For semantically indeterminate collectivities, such as the world’s foothills, there will be candidates for which it is semantically determinate neither that they belong nor that they do not. For ontologically indeterminate collectivities, there will be candidates for which it is neither ontologically determinate that they belong

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nor that they do not. For logically indeterminate collectivities, there will be candidates for which membership remains forever in oscillation; candidate members which we are forced to conclude can logically neither belong nor not belong. In exploring explosive plenary totalities we took much the same tack. Plenary totalities need not themselves be indeterminate, but must in all cases have a proper subcollectivity that is logically indeterminate: a sub-collectivity for which membership and non-membership will be logically oscillatory and for which the Law of Excluded Middle will not apply. Our explorations have also led us to a hierarchical vision of compounded indeterminacies, in which not merely membership but determinate membership can be indeterminate, and determinate determinate membership can be indeterminate beyond that. Reflection on issues of indeterminacy leads quite naturally to a hierarchical picture of ascending orders. The present chapter pursues these issues further, with an emphasis on questions of logic. What is the logic to which indeterminate collectivities lead us? Our first outline below is perhaps the simplest, in accord with the denial of the Law of Excluded Middle appealed to throughout. But in fact, we want to suggest, that is not the only option. There are a number of different possibilities for appropriate logics, some of which themselves raise further questions regarding collectivities and plena. 2. AVERTING RUSSELL’S PARADOX At first glance, contemplations of plenary totalities seem immediately to evoke Russell’s paradox of ‘the set of all sets that are not members of themselves.’ But Russell’s paradox in original form need not be a threat here; the distinction between sets and collectivities enters to save the day. The problem arises for ‘the collectivity of all sets that are not members of themselves’ only on the false presupposition that such a collectivity qualifies as a set. It simply does not. Any similar inconsistency regarding ‘the collectivity of all collectivities that are not members of themselves’ can be avoided by insisting that while such a collectivity indeed exists, it is amorphous in that for this collectivity CR , both x ∈ CR and x ∉ CR can fail.

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The collectivity of all self-excluding sets is clearly not a set; such a supposition entails a contradiction, the ultimate disqualification within the standard logic of sets. The situation of the collectivity CR of all collectivities that do not include themselves is something else again. It proves to be manageable because collectivity membership is not bivalent with x ∈ CR either definitely true or definitely false. Collectivity membership can be amorphous or indeterminate; CR can be neither a member of itself nor not without contradiction. Since collectivities are not sets they can put aside the burdens that sets must bear. But then what of the collectivity CS of all collectivities that either are not members of themselves or are neither members of themselves or not? Won’t CS have to be either a member of itself, a non-member, or neither a member or not? And won’t any of those three options again give us a contradiction? The most direct response here, and the response hinted at throughout, is a strong rejection of the Law of Excluded Middle. A strong rejection calls for complete truth-valuelessness—lack of semantic value entirely—rather than simply the prospect of a third or middle value. On such an approach, what is expressed by saying that a collectivity is neither self-membered or not is a rejection of each option, not the proposal of a third or middle form of membership. On such an approach, what is expressed by saying that a claim of membership is neither true nor false is a rejection of each option, not the proposal of an intermediate truth-value. For indeterminate collectivities, neither membership nor nonmembership for some items may be defined at all. For such collectivities and such items, both x ∈ C and x ∉ C will make no more sense than division by zero or the ratio 3/0. Logical compounds of these— such as ‘neither members of themselves nor neither members of themselves nor not’ in the specification of CS—will be equally undefined, making no more sense than ‘10 = 3/0 or 10 ≠ 3/0.’ 1 With a strong denial of the Law of Excluded Middle, the three options offered as exclusive for the strengthened CS will be rejected as inappropriate in the same way that the two options offered as exclusive in the case of CR are rejected as inappropriate. Here a parallel might be suggested to a propositionalist approach to the Liar.2 Such an approach, consistently pursued, maintains that the

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Liar sentence expresses no proposition, with a similar insistence regarding strengthened versions such as ‘This sentence is either false or expresses no proposition.’ If such a sentence expresses no proposition, it is quite natural to resist the crucial argumentative move that ‘it is therefore true after all’ because ‘that is what it says.’ If the sentence expresses no proposition, there is nothing that it says. Here, similarly, we deny that membership and non-membership are defined for all collectivities. Membership and non-membership may quite naturally remain undefined for ‘strengthened’ collectivities specified in terms of non-membership. A strong rejection of the Law of Excluded Middle is, we think, the most direct route to a logic appropriate for indeterminate collectivities. As indicated below, however, it is not the only route. 3. AVERTING CANTOR’S PARADOX Cantor’s power set theorem shows that the power set of any set is larger than the set itself, and therefore poses a crucial logical problem for any universal set of all sets. Won’t the same problem arise for plenary totalities—the totality of all things, for example, or all truths? It is helpful to encapsulate Cantor’s line of reasoning: A set S′ has greater cardinality than a set S if there is a one-to-one mapping of distinct elements of S onto distinct elements of S′ but is no mapping of distinct elements of S′ onto elements of S that does not leave out some element of S′. Consider then any set S and its power set PS. Distinct elements of S can be mapped onto distinct elements of PS simply by mapping them onto their singleton sets. But consider any mapping m proposed from PS onto S. For any such mapping, there will be the diagonal set S* of precisely those members of S which are not elements of that element of PS mapped onto them by m. To what element of S can m map S*? For any element s ∈ S proposed, s will have to either be a member of S* or not. Given the definition of our diagonal set S*, however, either alternative gives us a contradiction. There can then be no mapping of elements of PS into S; the power set of any set is larger than the set itself.

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Were this to apply to a plenum of all sets, we would have to say that the plenum was larger than itself; that it literally contained more members than it did, or that its members could not be put into one-toone correspondence with themselves by any relation, presumably including simple identity. Here again the strong rejection of the Law of Excluded Middle avoids contradiction. For a plenum of all sets, there is no guarantee that all its sub-collectivities are themselves sets. If they are not—if, in particular, that which plays the role of the diagonal set S* is not—then it need not hold that for any x either x ∈ S* or x ∉ S*. With an abandonment of the special situation of sets we lose such a presumption regarding membership. And without that presumption, we are no longer forced to say that the power set of a plenum is larger than the plenum itself. Indeed it cannot be, since plena are defined so as to contain distinct elements for each element of their power sets. Here again a strong rejection of the Law of Excluded Middle seems the most direct route. But it is not the only option. One might, for example, combine the following two theses. First, using the 1-to-1 relation of identity, • A plenum of all sets contains just as many members as it does. By the Cantorian diagonal reasoning above, however, • A plenum of all sets contains more members than it does.3 The result, of course, is contradiction. But here one might reject not the Law of Excluded Middle but the Law of Non-Contradiction instead—one of the options explored below. 4. GAPS AND GLUTS IN THE LOGIC OF COLLECTIVITIES One of the phenomena that has appeared repeatedly in our exploration of indeterminate collectivities is logical oscillation. Oscillation appears in the standard reasoning regarding Russell’s set SR of all and only the non-self-membered sets. Is SR itself self-membered? If it is, it must not be, since SR is specified as containing only non-self-

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membered sets. If it is not self-membered, it must be, since SR is a specified as containing all non-self-membered sets. Oscillation appears in similar reasoning regarding a collectivity CR of all and only non-self-membered collectivities. Given only a weak denial of the Law of Excluded Middle, in which ‘neither a member nor not a member’ is envisaged as a further membership status N, oscillation would appear again in asking whether the collectivity CZ of all and only collectivities that either are non-members or N-members of themselves is itself self-membered, non-self-membered, or N-self-membered. If CZ is either non-self-membered or N-self-membered, it must by specification be fully self-membered. If self-membered, on the other hand, it appears it must by specification be either non-self-membered or Nself-membered. Oscillation is also familiar, of course, in the reasoning of the Liar paradox: If ‘this sentence is false’ is true, it must be false. If false, it must be true. The direct response to such oscillations, we have proposed, is a strong denial of the Law of Excluded Middle. One avoids contradiction by envisaging gaps. In the case of logically indeterminate collectivities, membership or non-membership, having a property or lacking it, and truth or falsity can slip through the cracks of undefinedness. An alternative response, traditionally rejected as philosophically unpalatable but structurally quite close, is to embrace rather than attempt to avoid contradiction in such cases. In every case that we have proposed that a collectivity is neither a member of itself or not, one might propose instead that the collectivities at issue are both members of themselves and not. The cases of logical indeterminacy remain precisely as before, though in this alternative logic they are treated as cases of overdetermination rather than underdetermination. Selfmembership for CR, for example, oscillates not because it neither holds nor does not, but because it both holds and does not. On such an approach one abandons not the Law of Excluded Middle in all applications but the Law of Non-Contradiction in all applications. One embraces truth-value gluts instead of truth-value gaps. Both gaps and gluts raise immediate logical objection from the classical logician. But in both cases those immediate objections are fairly superficial. Against gaps, for example, it is argued that rejection

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of the Law of Excluded Middle leads quite directly to contradiction. To paraphrase Timothy Williamson, we have for example maintained: Not: either CR ∈ CR or not CR ∈ CR “By one of De Morgan’s laws, the negation of a disjunction entails the conjunction of the negations of its disjuncts.” Thus the sentence above yields: Not CR ∈ CR and not not CR ∈ CR. “The second conjunct … contradicts the first. There is no need to eliminate the double negation; [it] already contradicts itself.”4 The appropriate conclusion to draw, of course, is that a logic appropriate for gaps will not include this form of De Morgan. The fact that denial of excluded middle leads to contradiction in full classical logic, which comes with full De Morgan, is hardly surprising: a direct expression of the Law of Excluded Middle as (p v ~p) is classically equivalent to ~~(p v ~p), which by De Morgan is equivalent to ~(~p & ~~p), a clear form of the Law of Non-Contradiction. Not only does classical logic fail to countenance both gaps and gluts, but its prohibitions of each are equivalent. Williamson’s point is somewhat more subtle, however; his argument requires only the resources of standard Intuitionistic logic. Although denial of the Law of Excluded Middle is often taken as an invocation of Intuitionistic logic, however, that particular logic—tied to a peculiarly epistemological take on mathematics—is merely the historically most developed of a larger family. All members of that larger family of logics will reject the Law of Excluded Middle, but many will also obviate other aspects of Williamson’s argument. Rejection of the Law of Excluded Middle need not therefore force us into the arms of standard Intuitionism. Other options remain.5 Against gluts, on the other hand, the classical argument is that acceptance of contradictions would entail the trivial acceptance of any and all propositions. Suppose we maintain: CR ∈ CR and not CR ∈ CR

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We can then introduce CR ∈ CR v P by addition, for any arbitrary proposition P. But we have not CR ∈ CR by conjunction elimination from our first step, which with our third step gives us P by disjunctive syllogism—for any arbitrary P. Here the appropriate conclusion to draw is simply that a logic appropriate for gluts will not include addition, conjunction elimination, and disjunctive syllogism in full form. Historically, it has been disjunctive syllogism that has been sacrificed, with quite complete logical development of a range of alternative paraconsistent logics.6 Gaps and gluts operate very much in parallel, and indeed in some respects seem to be stylistic variants of one another. Despite the fact that ‘neither P nor not P’ seems far more intuitively acceptable than ‘both P and not P,’ and thus gaps seem more intuitively acceptable than gluts, the formal consensus seems to be that the two alternatives are logically very much on a par.7 The deepest weaknesses in each case are philosophical rather than formal, and how much these qualify as weaknesses is open to argument. The difficulty for gaps is an apparent paucity of expressive power known as the ‘classification problem’. If CR is neither a member of itself nor a non-member, what is its appropriate membership classification? Shouldn’t there be some way we can categorize its membership status? If the Liar sentence is neither true nor false, what is it instead? Surely it must have some semantic value; what value is that? With regard to both CR and the Liar, an appeal to a strong denial of Excluded Middle must answer that there simply is no further value. Gaps are gaps, not additional values; Liar-like sentences simply lack

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semantic values, and there is no appropriate self-membership classification for CR. Just as that which expresses no proposition expresses no proposition of any semantic value, and just as ‘3/0’ is undefined rather than taking any particular numerical value, CR’s self-membership must have the indeterminacy of a full gap. Given the contrary—an expressible classification for CR’s self-membership, or any semantic classification for the Liar—oscillation would simply reappear in the form of CZ or the Strengthened Liar. The expressive limitations of gaps carry over to any reading of negation in a logic adequate for gaps, and to any notion of ‘false’ as ‘not true’ expressed in such a logic. What the Law of Excluded Middle insists is that p and its negation ~p are exhaustive. To deny the Law of Excluded Middle is therefore to employ a negation that is not exhaustive; given gaps, p and ~p need no longer cover all possibilities. If ‘false’ is to be defined as ‘not true’, ‘true’ and ‘false’ no longer cover all possibilities either. It is crucial within a genuinely gapped logic, moreover, that negation cannot be supplemented in any way that would restore exhaustiveness. One cannot introduce an additional negation for ‘something other than full p,’ for example, without reintroducing precisely the difficulties that gaps are intended to address. Critics of gaps can therefore charge that there is a concept of negation that logics adequate for gaps cannot express: the concept of an exhaustive ‘something other than p.’ Proponents of gaps, in response, can be expected simply to deny that there is any coherent concept here to be expressed. Gaps and gluts offer largely parallel approaches, and the parallels extend to expressibility. Within a logic adequate for gluts, there is no problem with negation being exhaustive: p and ~p can cover all the possible ground, as indeed can ‘true’ and ‘false’. The difficulty within a logic adequate for gluts is rather that negation cannot be exclusive: since the Law of Non-Contradiction does not hold, the fact that p does not exclude an additional possibility that ~p as well. The fact that something is true does not exclude the possibility that it is false as well. One way this expressive difficulty has been pressed against glut proponents is their apparent inability to express genuine disagreement with their critics. The critics may insist that p, the proponents of gluts that ~p, but as proponents of gluts they must recognize that the latter

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need not exclude the former. The critics may insist that the position of glut proponents is false, the proponents that it is true, but as glut proponents they must recognize that truth need not exclude falsity as well. For gluts as for gaps this expressibility is irremediable. Any attempt to introduce an additional exclusive negation—‘exclusively other than p’, for example, or ‘purely false’—would reintroduce parallels to precisely the difficulties that gluts were attended to address. Critics of gluts can therefore insist that there is a concept of negation that logics adequate for gluts cannot express: the concept of an exclusive contrary to p. Proponents of gluts must deny that there is any coherent concept here to be expressed. Although we will not fully pursue the issue here, the parallels between logics appropriate to gaps and gluts lead one to envisage a logic which includes both. In at least many applications, negation in English seems to carry with it implications of both exhaustiveness and exclusivity. Perhaps that is an ambiguity that is harmless in most contexts, but which requires logical disentanglement in some. We might envisage disentangling the two negations, using a negation ~ that is exclusive but not exhaustive and a negation ¬ that is exhaustive but not exclusive. For ¬, the Law of Excluded Middle will hold, but the Law of Non-Contradiction is not guaranteed. For ~, the Law of NonContradiction will hold, but Excluded Middle is not guaranteed. Full classical logic will hold for neither of our two negations alone: neither ~~p nor ¬¬p, for example, collapse to a simple p. For applicational conjunctions of our two negations, however, such as ‘~p & ¬p’, full classical logic will hold. 8 With regard to truth, we might still assume that ‘p is true’ is interchangeable with ‘p’, but would have to recognize two forms of falsity: that corresponding to ¬true and that corresponding to ~true. Implications can be illustrated in terms of a Liar sentence, but will carry over to issues of oscillatory membership as well. Because ~ is exclusive but not exhaustive, a gap-like treatment will be available in a dual negation logic for: This sentence is ~true.

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Because ¬ is exhaustive but not exclusive, a glut-like treatment will be available in a dual negation logic for: This sentence is ¬true. Familiar Liar-like oscillations would seem to demand something more; that the two negations be combined, for example in a Conjunctive Liar: This sentence is ~true and ¬true. Might such a sentence be true? If so, its status is both exclusive to ‘true’ and exhaustive with ‘true.’ But by exclusivity, it cannot be true: the assumption of truth leads to an exclusive opposition, or contradiction. In the traditional argument of the traditional Liar, we have only one alternative possibility: that the sentence is false, in a sense that blends both exhaustiveness and exclusivity. In a dual-negation logic we distinguish the two, and thus must ask ‘Is the sentence above ~true?’ and ‘Is it ¬true?’ Suppose this conjunctive Liar is ~true. Does it follow that it is true? No, since the sentence itself is a conjunction: we would need both conjuncts for truth, and this form of falsity alone does not give us that. Suppose it is ¬true. Does it follow that it is true? No, again because the sentence itself is a conjunction. Oscillation will reappear only if we insist that this sentence is false in both senses. At first glance it appears that in order to avoid oscillation we would simply have to say that the sentence at issue is not both ~ true and ¬ true, and can in that sense take our choice between a gap and a glut solution. That first glance, however, requires a second. Because the sentence is conjunction, declaring it false in only one sense—either ~true or ¬true—does not entail that the sentence is true after all. Only one of these options, however, turns out to be stable. It is consistent with the claim that this sentence—or any— is ¬true that it is also true: such is the nature of our glut-reflective negation ¬. Accepting this sentence as true, however, would commit us to accepting its first conjunct as true. But that first conjunct is the claim that the sentence is ~true, and ~ is

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an explicitly exclusive negation in our system; our principles for ~ disallow taking the sentence as both true and ~true. We therefore cannot take the Conjunctive Liar to be false in any sense of falsity consistent with truth. It must rather be false in a sense of falsity exclusive from truth: precisely the falsity of our gapped ~true. Within consistency at this level, at least, the stable analysis of our conjunctive Liar is that it is ~true.9 Our attempt here has been no more than a sketch of some aspects of a dual-negation logic. What even this sketch again illustrates, however, is the way gaps and gluts can complement one another in parallel situations. The Conjunctive Liar has a parallel in the Disjunctive Liar: This sentence is ~true or ¬true. If true, this sentence cannot be: if true it is ~true, which is exclusive of ‘true’. If it is ~true, it must be true, but must then have a value exclusive of ‘true’: contradiction. If it is ¬true, on the other hand, the same does not follow. ‘¬’ is exhaustive but not exclusive, leaving the possibility that the disjunction is true in virtue of its second disjunct: it is both true and ¬true, a distinctly ‘glut’ conclusion. Whereas the conjunctive Liar in a dual-negation logic calls for resolution by way of gap, the disjunctive Liar seems to demand a resolution by way of glut. The duality of our two negations extends to a duality of Liars for which each is required.

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5. ASCENDING HIERARCHIES OF VALUES Throughout, we have tended to characterize oscillations in terms of a strong denial of the Law of Excluded Middle. As indicated in the previous section, an alternative is to embrace gluts rather than gaps; a denial of the Law of Non-Contradiction that is in many ways parallel. There is a further alternative as well, parallel to the vision of a hierarchy of indeterminacy developed in chapter 4. Consider again the case of CR, the collectivity of all non-selfmembered collectivities. Is CR self-membered or not? One option is to take the route of gluts, taking oscillation to indicate that it is both selfmembered and not. Another option is to appeal to full gaps, taking oscillation to indicate that it is neither self-membered nor not, and treating the latter in terms of a strong denial of the Law of Excluded Middle. A third option is to adopt only a weakened denial of Excluded Middle. On such an approach CR has a third status: it is N-membered, neither self-membered nor not. What then of a collectivity CZ of all collectivities that are non-selfmembered or N-self-membered? CZ, we might insist, has a further status still. It is N2-membered: neither self-membered, non-selfmembered, nor N-self-membered. Beyond CZ, of course, will be a further troublesome collectivity, of all those collectivities that are neither self-membered, non-self-membered, N-self-membered nor N2 selfmembered. For it we will have to recognize a further self-membership status N3. The picture that results is of an ascending hierarchy of ever more complex membership values.10 Gap and glut approaches to membership form strong parallels to gap and glut approaches to the Liar. The same is true here: an alternative approach to Liar-like sentences is a similar ascending hierarchy of truth values. The Liar, we might insist, is neither true nor false: it has a status N. What then of the Strengthened, Liar, ‘This sentence is either false or N’? Its value will be N2. The semantic value for ‘This sentence is either false, N, or N2’ will be N3 . . . As indicated in chapter 4, this hierarchical ascension is also reminiscent of higher-order vagueness. There are tall men and short men, and men who qualify as neither tall nor short: there is no sharp line between tall and short. But neither is there a sharp line to be drawn be-

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tween those who are tall and those who fall into this third category. If lack of a sharp line is taken to indicate an intervening interval or category, we are clearly committed to a similar ascending hierarchy of categories of tallness, with the number of categories increasing by powers of two at each step. As also noted in chapter 4, the width of those categories against an objective scale may also grow increasingly smaller at each stage, approaching a limit of 0. That limit, however, is merely an abstract limit within an infinite series; the hierarchy of higher-order values will be an infinite hierarchy nonetheless. A hierarchy of values is one way of attempting a logic adequate for indeterminate collectivities. It raises further questions, however: How many values are at issue, and what is their structure? In interesting ways, this turns our examination onto itself. The question of values appropriate to a logic adequate for collectivities and plena turns us to questions of collectivities and plena themselves. 6. THE COLLECTIVITY OF VALUES If we envisage an ascending hierarchy of membership values for collectivities, what is the totality of values we envisage? If indeterminacy forces a hierarchy of indeterminacies, what is the totality of indeterminacy values? If self-referential paradoxes are to be dealt with in terms of a hierarchy of semantic values, how many semantic values are there? Consider again the hierarchy of membership values. Selfmembership for the collectivity CR of all non-self-membered collectivities is assigned status N. Self-membership for the further collectivity CZ of all collectivities that are non-self-membered or N-selfmembered is assigned status N2. In terms of N2 we can characterize a collectivity the membership of which can only be N3. To this point the pattern of generation follows that of the finite ordinals. But it is also clear that we can go beyond that. For consider the collectivity Cω of all those collectivities that are either non-selfmembered or whose membership is one of the series just outline: N1, N2, N3… Self-membership in this collectivity must have a further value, which we will designate Nω. Using that additional value we will be able to characterize a further collectivity with a further value

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Nω+1, then Nω+2, leading eventually to values N2ω, N2ω+1, Nω+2…, and on to ω2, ω3, ω4, . . . ωω and beyond. The values required for a hierarchical treatment of collectivity membership map onto the ordinals. Much the same, for much the same reasons, will be true for hierarchical truth-values. Starting with just two truth-values—true and false—the standard Liar seems to force us to a third value N, or ‘neither true nor false’. But there will then be a strengthened Liar ‘this is either false or N’ which forces us to a further value N2, and so on. That will generate a series in terms of which we can envisage an Omega Liar ‘This sentence is either false or has one of the values N, N2, N3 . . . ’ and so forth. The hierarchy of truth-values, like that of membership values, will track the ascending ordinals. Higher-order vagueness might initially be thought to follow a more restricted pattern. At each progressive division between ‘tall’ and ‘short’, or between ‘tall’ and ‘neither tall nor short’ we are forced to envisage a further category, but that alone might seem to generate only a countable series of indeterminacy values, mapping onto the countable ordinals but not into the transfinite. Hartry Field has argued that hierarchy even here is forced into the transfinite. Consider a figure at one end of the progressively divided line from ‘tall to short’ who always comes out positive: not merely determinately tall, not merely determinately determinately tall, but …determinately determinately tall; a candidate who comes out determinately-positive throughout all the infinite iterations of ‘determinately’. The step from that category to anything less, Field argues, demands a further indeterminacy value just as much as did any previous step—but indeterminacy at that point can be labeled only by means of a corresponding to a transfinite ordinal.11 On this picture, the hierarchy of vagueness indeterminacies will parallel that of membership and semantic values in tracking the full ordinals. Each of these hierarchies is self-expansive: Each step in the hierarchy forces a further step. In none of the cases can we envisage a complete list of all hierarchical values. The Burali-Forti paradox makes it clear why we cannot envisage a complete list in terms of sets. The ordinals are order-types for sets. Because each ordinal can be uniquely characterized in terms of the ordered set of ordinals preceding it,

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there can be no set of all ordinals. If there were, it would have an order-type corresponding to an ordinal beyond it, contradicting our assumption that there was no ordinal beyond it. In at least the case of membership and semantic values, it is also clear from internal content that there can be no complete list of values. Were there a finished totality of all membership-values, we could envisage a collectivity of all collectivities that have some value within the totality other than simple self-membership. That collectivity would force a membership value beyond our supposed totality. Were there a finished totality of all semantic values SV, we could envisage a Liar sentence This has some value of SV other than ‘true’ which would force a semantic value beyond our supposed totality of semantic values. The semantic values, like the membership values, expand with every step in the hierarchy and expand beyond any attempt at finished collection. Just as appeals to gaps and gluts each seem to face problems of expressibility, so does appeal to hierarchy: expressibility of a concept of ‘all the values’ would offer again the tools for a collectivity characterization or a strengthened version of the Liar that would demand a value beyond ‘all the values’. Mapping onto the ordinals, the membership and semantic values of the envisaged hierarchies cannot form a set and cannot form a finished collection. But collectivities beyond sets, often intrinsically unfinished or expansive in form, have been precisely the subject of our explorations here. Can the ordinals be envisaged as a collectivity rather than as a set? Can the membership categories of the hierarchy appropriate to collectivities, and the semantic values appropriate to a closely related logic, themselves be conceived as collectivities? The ordinals, as order-types of sets, cannot themselves form a set. But of course set-limitations offer no prohibition against conceiving of the ordinals as a collectivity. We might go on to conceive of collectivity-ordinals: of order-types of collectivities. Does the Burali-Forti paradox appear here in a new guise? This is unexplored territory, and we can perhaps only sketch some initial possibilities. One possibility is that collectivities may not al-

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ways have order-types. Order-types for collectivities will fail to characterize all collectivities, and the collectivity even of all order-types might be a collectivity that lacks an order-type. In that case a collectivity of all collectivity order-types would not force us to a further order type. The following may be equivalent. Crucial to the Burali-Forti paradox is the fact that the ordinals, as order-types for sets, are themselves well-orderable. Would the collectivity-ordinals be similarly well-orderable? If the answer is ‘no’, we would escape the implication of a collectivity-ordinal beyond the collectivity-ordinals. Note also that in the realm of indeterminacy the answer to such a question might be ‘neither yes or no’, once again exempting us from one of the assumptions of the set-level paradox. The suggestion is that the membership values of a hierarchy intended to deal with collectivities, and of semantic values of a companion logic, must themselves be conceived—like the ordinals they map onto—in terms of collectivities rather than sets. Whether those collectivities must be envisaged as indeterminate collectivities is a distinct possibility, but at present remains an open question. Might these hierarchies of value form plena? On the characterization given so far, there seems to be no reason to think so. A membership plenum is a collectivity that contains each of its subcollectivities as a distinct element. This is not true of the ordinals, and thus the fact that our hierarchies of values map onto the ordinals does not indicate that they constitute full plena. The ordinals do have a related property: each of an ordinal’s distinct elements is a subset of that ordinal. What is required for membership plena is the converse—that each subset appear also as an element. That requirement is not satisfied by the ordinals. There is a further conception of membership and semantic values, however, on which each of these forms a more extensive group than that generated in ordinal fashion above. Phrased in terms of semantic values, this conception turns on the answer to a fairly simple question: What distinguishes semantic values? On what criterion are two semantic values to be thought of as distinct? One very plausible answer is this:

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• Two semantic values are distinct if they carry distinct implications for some candidate propositions. Were two values V and V’ the same, after all, we would expect the claim that p is V and the claim that p is V’ to carry the same implications for all p. Given that sufficient condition for a distinction between semantic values, we will have to recognize semantic values beyond those generated by the ordinal stream envisaged to this point. For consider the values ‘false’ and ‘N’ in the form generated above, and also the further value ‘false or N’. Consider also three candidate propositions: This sentence is false. This sentence is N. This sentence is false or N. The value ‘false’ applied to the first and third sentences will carry the implication that they are true. Applied to the second, however, it need not carry that implication. ‘N’ applied to the second and third will carry an implication of truth, but not if applied to the first. ‘False or N’ applied to the three sentences, on the other hand, will imply truth in all three cases. On the sufficient criterion outlined, then, ‘False or N’ is a semantic value distinct from either of its components. On such an argument, the ordinal hierarchy of semantic values outlined fails to exhaust them all. Every combination by disjunction of elements in that hierarchy will constitute a further distinct semantic value. Will this be enough to show that the semantic values form a plenum? Apparently not. For to constitute a plenum, each sub-collectivity must correspond to a distinct member. What the argument above establishes is that the sub-collectivity composed of ‘false’ and ‘N’ will correspond to a member distinct from each: the semantic value ‘falseor-N’. But there are clear counter-examples to the idea that this will hold for arbitrary sub-collectivities. Consider, for example, the subcollectivity of ‘false’ and ‘false or N’. Here a disjunction ‘false or

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false or N’ simply reduces to ‘false or N’, failing to give us a distinct semantic value and thus failing to establish the collectivity of semantic values as a plenum. What is the logic required for indeterminate collectivities and plena? We have sketched a range of possibilities: a logic of gaps, a logic of gluts, a logic combining the two, and a logic incorporating an ascending hierarchy of values. It is intriguing to note that the values required in this last suggestion must themselves form a collectivity— precisely the subject matter of our study as a whole. NOTES 1

As indicated below, we consider this strong denial of the law of excluded middle an appropriate response to what has been called the “classification problem” (see J.C. Beall, “True, false and paranormal,” Analysis 66 (2006), 80-83). If a sentence is neither true or false, what is its classification? If neither p nor ~p, what is the status of p? Although we will not develop the point here, that question itself assumes a weak denial of the law of excluded middle, which does allow a status for those things that fall through the cracks. This presupposition is precisely what a strong denial refuses commitment to.

2

See J. Howard Sobel, “Lies, Lies, and More Lies: A Plea for Propositions,” Philosophical Studies 67 (1992) 51-69; James Cargile’s review of Barwise and Etchemendy’s The Liar, Noûs 24 (1990) 757-73; Yehoshua Bar-Hillel, “Do Natural Languages Contain Paradoxes,” Studium Generale 19 (1966) 391-397, reprinted in Bar-Hillel, Aspects of Language (Jerusalem: Magnes Press, 1970), and Richard Cartwright, “Propositions,” in R. J. Butler, ed., Analytical Philosophy (Oxford: Oxford University Press, 1962), reprinted with addenda in Cartwright’s, Philosophical Essays (Cambridge, MA: MIT Press, 1987).

3

The idea of a plenary collectivity that has more members than it does continues and completes a conceptual trajectory in a particularly appealing way. Finite sets are those every proper subset of which is smaller than they are. Infinite sets are those some proper subsets of which are of the same size as the set itself. Plena extend the trajectory: some proper sub-collectivities of plena, we might say, are larger than the plena themselves.

4

Timothy Williamson, Vagueness (New York: Routledge, 1994), p. 189. Williamson’s full argument is in terms of denial of bivalence, but this is the crucial step. See also Paul Horwich, Truth (Oxford: Blackwell, 1990), p. 76, and David Barnett, “Indeterminacy and Incomplete Definitions,” The Journal of Philosophy, 105 (2008) 167-191.

5

The work of Hartry Field is particularly promising in this regard, both for links to problems of vagueness and for an outline of a weak and non-truth-functional conditional P → Q that flowers into classical material implication with the addition of excluded middle assumptions regarding P and Q. See Hartry Field, “A RevengeImmune Solution to the Semantic Paradoxes,” Journal of Philosophical Logic 32

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NOTES

(2003) 139-177, “The Semantic Paradoxes and Paradoxes of Vagueness,” in JC Beall, ed., Liars and Heaps (Oxford: Clarendon Press, 2003), pp. 262-311, and Stephen Yablo, “New Grounds for Naïve Truth Theory,” pp. 312-330 in the same volume. As both Field and Stephen Yablo emphasize, there may be alternatives that are equally or even more satisfactory. 6

See Graham Priest, An Introduction to Non-Classical Logic, 2nd ed. (New York: Cambridge University Press, 2008), In Contradiction, (New York: Oxford University Press, 2006), and Doubt Truth to be a Liar (New York: Oxford University Press, 2006). A different approach, through development of a semantics of combinatory worlds, appears in Nicholas Rescher and Robert Brandom, The Logic of Inconsistency (Oxford: Blackwell, 1979).

7

See for example Terence Parsons, “True Contradictions,” Canadian Journal of Philosophy, 20 (1990) 335-354 and Hartry Field, Saving Truth from Paradox.

8

Given gaps, the fact that something is exclusive of something exclusive of p is no guarantee that it is precisely p. The fact that something is exhaustive with something exhaustive with p is similarly no guarantee that it is p itself. Something both exhaustive with and exclusive of something both exhaustive with and exclusive of p, on the other hand, is guaranteed to be p itself.

9

One might of course take gluts even higher, maintaining that our conjunctive Liar is not true, recognizing the contradiction that it is also therefore true and therefore exclusive of truth by its first conjunct, but embracing that contradiction as well: the sentence is both true and exclusive of truth. One might take that course, but the point here is that one need not; the option of a gapped ‘false’ for this particular Liar is available as an alternative. How far up acceptance of contradiction needs to go is in fact an issue on which Graham Priest has held various views at various points in the development of his dialetheism. In early work higher-order and meta-linguistic contradictions are accepted as the price of accepting contradictions; in later work his views are significantly more subtle. See Timothy Smiley (1993). “Can Contradictions Be True? I,” Proceedings of the Aristotelian Society, suppl. Vol. 58 (1993) 17-33 and Patrick Grim, “What is a Contradiction?” in Graham Priest, J.C. Beall, and B. Armour-Garb, The Law of Non-Contradiction: New Philosophical Essays (Oxford: Oxford University Press 2005) 49-72.

10

Although we will not pursue it here, a similar hierarchy of gluts rather than gaps is possible as well.

11

Hartry Field, Saving Truth from Paradox (Oxford: Oxford University Press, 2008) 103-104.

Chapter Seven NUMBER BEYOND NUMBER 1. NEW OPTIONS, NEW NUMBERS

N

umbers seem sharp, definite and discrete: indeed the realm of number seems a paradigmatic realm of the sharp, definite and discrete. Number theory, of which classical arithmetic is the simplest instantiation, is a matter of clear operations on the realm of the sharp, definite, and discrete. Just as sharp set theory can be seen as a foundation for sharp number theory, however, the theory of indeterminate collectivities offers a route into numbers of a different sort: inherently vague and indistinct numbers, with a corresponding arithmetic appropriate to indeterminate quantities. Our goal in this chapter is to introduce forms of number theory both beneath number as standardly conceived and beyond. 2. NUMBER BENEATH NUMBER One route to indeterminate numbers has long been before us, embedded in common language. We can deal conceptually with the concept of 2 items, of 20, of 1099. But we also deal with inherently indefinite quantities: • A small handful of gifts, • A bunch of letters, • A small number of incidents, • A minor percentage • A tiny proportion

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• A lot of sailboats, • A large number of voters, • On many occasions, • In the vast majority of cases, • A zillion stars We reason with these indefinite quantities on a daily basis. And the fact that standard arithmetic is not built for them indicates only that there are quantitative considerations that escape its reach, namely the quantitative considerations appropriate not to sharp but to indefinite quantities. We know, for example, that: • If the kids get a bunch of candy for Halloween, and will get even more the next week, they will have a significant amount of candy in a relatively short period. • If there are only a few ships in the harbor, and only a few more are expected later, there certainly won’t be a large number of ships in the harbor. • If some phenomenon holds in the overwhelming majority of cases, and another in only a few isolated cases, the first is far more frequent and a much safer bet. • If Mike will be here a long time, and Bill for only a moment, Mike will be here much longer than Bill will. Both individually and as a culture, we may have been able to reason in terms of indefinite numerical quantities—many, a few, a mere handful, or lots—even before we could reason in terms of sharper number concepts. ‘A few’ may not be learned as a disjunctive ambiguity across 2, 3, or 4; the latter may rather be precisifications of ‘a few,’

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with those precisifications learned later and derivatively. Indefinite quantities may be the older numbers beneath number. What is the indefinite arithmetic appropriate to indefinite quantities? Despite the fact that ‘a few’ and ‘many’ may be older than ‘precisely 7’ and ‘95’, a first grasp of the arithmetic appropriate to the former may be easiest with terminology that clearly is derivative of the latter: ‘roughly 7’ and ‘approximately 95’. What is the arithmetic appropriate to ‘roughly’ and ‘approximately’? It is clear that the formal relation of reflexivity holds for approximate equality. Any quantity is approximately equal to itself. Symmetry holds as well: if A is approximately equal to B, B will be approximately equal to A. The formal relation that fails for approximate equality is transitivity: the fact that A is approximately equal to B, and B approximately equal to C, does not guarantee that A will be approximately equal to C. The further the chain to which we attempt to apply transitivity, the less likely ‘approximate equality’ will carry down the chain: given any significant difference between each element of a chain of 100 increasing items, the first and last will undoubtedly not be approximately equal even if each two adjacent pairs are approximately equal. The familiar facts of Sorites series can be thought of in precisely these terms: any two men in the ‘forced march’ may be approximately equal in terms of baldness, though the men at the ends of the series are clearly not. Strict numerical equality is an equivalence relation. Approximate equality is not. The breakdown in transitivity for approximate equality will carry over to familiar arithmetical operations. Though x is approximately 7 and y is approximately 9, (x + y) is not guaranteed to be approximately 16 and (x · y) is not guaranteed to be approximately 63. One way of modeling this fact—though it is merely a way of modeling it—is to note that what counts as ‘approximately 7’ is standardly keyed to a particular context and that the context carries something like a penumbra of variance within which ‘approximately’ is appropriate. If one models that penumbra in sharp terms—as a single integer on either side, so that 6 and 8 are ‘approximately 7’ in a given context but 5 and 9 are not, it is clear that sums or products of two numbers that are approximately x and y may nonetheless exceed the penumbral limitations through compounding. 6 is within an integer of 7, 8 is within an

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integer of 9, but (6 + 8) is not within an integer of (7 + 9) and (6 X 7) is not within an integer of (7 X 9). The general lesson outlined in terms of ‘roughly’ and ‘approximately’ also holds for the indeterminate quantifiers listed above. The lesson holds with particular clarity for arithmetical operations that combine indeterminate and sharp quantities. If you have only a few items in the basket and add precisely one, you will still have only a few items in the basket. Repeat the operation, however, and the failure of transitivity carries you beyond the indefinite limits of ‘a few’. If you have a great many items and lose only a few, you will still have a great many; repeat the operation, however, and the failure of transitivity carries you below the indeterminate bounds of ‘many’. 3. INDEFINITE NUMBERS FROM INDEFINITE COLLECTIVITIES The indefinite membership characteristic of many collectivities offers an additional route into indeterminate numbers. Here collectivities indeterminate in the various ways catalogued above offer numbers indeterminate in each of those ways. Consider for example: • The number of people who shook Lincoln’s hand. • The percentage of voters who will vote for the Republican Presidential candidate in the next election. • The number of visible stars. • The number of atoms in this block of Californium that will lose their electrons over the course of the next thirty days. • The number of residents of Hiroshima in 1945 who would be alive today if the United States had not dropped the atomic bomb • The number of tall men in Pittsburgh • The number of foothills in Dawson County

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Each of these is an indefinite or indeterminate number. The first three are epistemically indeterminate numbers. The fourth is ontologically indeterminate in a strong sense, while the fifth carries the indeterminacy of radical contingency. The final two are semantically indeterminate numbers.1 Indeterminate numbers from collectivities pose a problem for standard conceptions of numeration and counting. Numbers are introduced in set theory via the idea of a one-to-one correspondence between members of distinct sets. That distinctness breaks down with collectivities. To be sure, quantitative comparisons may be possible in some cases even though numerical measures are not. Thus if C has both determinate and indeterminate members, and if every item that is not determinately a member of C’s complement is determinately a member of C′, then C will indeed be smaller than C′. But such comparisons will not generally possible, and only rarely will they give us anything numerically quantifiable that is transportable from one context to another. From the mathematician’s point of view it will be awkward that indeterminate collectivities may lack a definite size. Consider for example the collectivity consisting of all integers n for which it is true that • A pile of n grains of sand does not suffice to make a heap. Clearly 1, 2 and 3 will belong to this collectivity and no integer greater than 1,000,000 will do so. So the size of the collectivity will have to be somewhere between three and one million. But no definite size can be ascribed to it. While this collectivity indeed has magnitude, no specific measure can possibly be assigned to it. The numerical indefiniteness of collectivities is vividly illustrated by the classical Sorites Paradox, discussed further in the next section. It is clear that one grain of sand does not make a heap. If n sandgrains are insufficient to do so, adding one single grain to achieve n + 1 will not suddenly give us a heap either. Here the totality of heaps constitutes an indefinite collectivity. Piles of less than 20 sand grains are clearly out, piles of 100 or more are clearly in, and those inbetween are indefinite. Observe however, that it is not merely the boundary region -‹ ›- between IN and OUT that is indefinite, but the boundary re-

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gion between IN and -‹ ›- as well. Indefinite boundaries are themselves indefinite in extent. Are collectivities really something different from sets? The answer lies in noting that the concept of membership indefiniteness simply does not figure in standard set theory but plays a crucial role with collectivities. The conceptual machinery of set theory leaves out of sight modalities that are crucial within collectivities; set theory is no more adequate to handle collectivities than eclipses. Despite the indeterminacy or indefiniteness of numbers conceived in terms of collectivities, there are many things we can know about them with considerable precision. Consider for example the indeterminate number of males m who shook Lincoln’s hand, and the indeterminate number of females f who did so. Our knowledge that all people are male or female allows us to conclude with precision that m + f = the number of people who shook Lincoln’s hand, despite the fact that each of these is an epistemically indeterminate quantity. The logical relations of properties definitional of indeterminate numbers can give us precision in arithmetical operations, in other words, even where the numbers themselves do not. We know that the number of atoms a in this block of Californium that will lose their electrons over the next 30 days, plus the number of atoms b that will lose their electrons over the following 50 days, will together equal precisely the number of atoms c that will lose their electrons over the next 80 days—this because of the additive character of our definitional specifications and despite the indeterminacy of the numbers to which those specifications apply. These examples of precision despite indeterminacy depend on logical relations between the descriptive characterizations of those numbers: it is the fact that all people are male or female that guarantees that the indeterminate number of females who shook Lincoln’s hand plus the indeterminate number of males who shook Lincoln’s hand equals precisely the indeterminate number of people who shook Lincoln’s hand. It is arguable that a wide range of precise numerical manipulations remain, despite the indeterminacy of numbers at issue and independently of logical relations between descriptive characterizations. If the indeterminate number of women that shook Lincoln’s hand is w and the indeterminate number of boys in the audience for the Gettysburg

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address is b, there is no logical relation between the descriptors for the two numbers. It nonetheless seems safe to assume that w + b equals b + w and that 2w · 3b = 6bw. Algebraic relations between variables, because they hold whatever the numbers at issue, seem a prime candidate to hold for indeterminate numbers. One interpretation of variables is as unknown quantities—an interpretation particularly appropriate for epistemically indeterminate quantities. Another interpretation, proposed by Russell, is of variables as ambiguous names, an interpretation perhaps appropriate for other forms of indeterminacy. It might be proposed, therefore, that algebraic relations hold for indeterminate numbers even if those numbers cannot be identified with the entities to which a full arithmetic is appropriate. An intriguing observation here is that axiomatic real algebras have logical properties importantly different from those of arithmetic. Thus it is a familiar fact that no axiomatic arithmetic can be both consistent and complete.2 It is a less familiar fact that such a limitation does not hold for axiomatic real algebras without the notion of ‘a natural number’, which can be shown to be both consistent, negation-complete, and decidable.3 Formalization for operations on indeterminate quantities—the number beneath number—can be formally complete in ways that no formalization for the same operations on natural numbers can. These considerations hold for the first four items on the list above. Whether they also hold for the final two examples—those turning on semantic indeterminacy—is perhaps more open to debate. There is an indeterminate number of men in Pittsburgh who are both tall and rich. There is an indeterminate number who are rich but not tall. But does the number of men who are rich and tall, plus the number of men who are rich and not tall, equal precisely the number of men who are rich? Where the source of indeterminacy is vagueness, it may be less tempting to think that there is a solid but undetermined number beneath the characterization at issue. ‘Tall’ and ‘not tall’ offer descriptive characterizations that are not as discretely exclusive as ‘males’ and ‘females’ or ‘next 30 days’ and ‘following 50 days’. We often speak of a man who is ‘neither tall nor not tall’, for example, or—perhaps equivalently—of a man ‘both tall and not tall’. If ‘tall’ and ‘not tall’ are not guaranteed to be both exhaustive and exclusive, the number of tall men plus those who are not tall may not equal precisely the number of

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men in total. It may be, therefore, that semantic indeterminacy resists formal treatment—even the formal treatment of an algebra—where epistemic and ontological indeterminacies do not. 4. NUMERICALLY INDETERMINATE QUANTIFIERS AND THE SORITES

To this point indeterminate collectivities have been deployed to introduce indeterminate quantities. It is also possible to offer an analysis of the former, however, in terms of the latter. In particular, it is possible to offer an analysis of semantic vagueness in terms of a buried quantifier that is numerically indeterminate. On such a view, and contrary to the great bulk of philosophical work on vagueness, the source of vagueness is not to be found in vague monadic predicates such as ‘bald’, ‘tall,’ or ‘old’. The true source of vagueness, on such a view, lies beneath these, in a mechanism using a buried quantifier operative over the comparatives ‘balder’, ‘taller’, and ‘older’.4 Philosophers have been obsessed with just two quantifiers—‘all’ and ‘some.’ Linguists recognize far more, including ‘a few’, ‘several’, ‘many’, ‘lots of’ and ‘almost all.’ The important characteristic of these quantifiers is that they are members of the list with which began: numerical terms that are essentially indeterminate. None of these vague quantifiers are reducible to ∀x or ∃x, and none are specifiable in terms of a precise number or percentage. Each is an essentially vague or indeterminate quantifier. It is another vague quantifier of the same family, we might propose, that lies buried under vague terms in general. We might use ‘Zx’ to represent a quantifier that might be expressed as: • For the great bulk of xs… • For the overwhelming majority of xs… • For a large percentage of xs… It might be better for our purposes were we able to introduce Zx ostensively, as it might be introduced in teaching a language. With piles of

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beans, grapefruits, or ball bearings we are confident we would be able to introduce you to the quantifier Zx in a short period of time. For at least paradigm cases, familiar vague monadic predicates can be defined derivatively in terms of such a quantifier. Someone is ‘bald’ just in case they are balder than the great bulk of a given comparison class. They are ‘tall’ if they are taller than the overwhelming majority of the population at issue. Someone is ‘old’ if it is they are older than a large percentage of the people at issue. Using ‘Bx’ for ‘x is bald’ and ‘xBy’ for ‘x is balder than y’, we can outline vague monadic predicates as applications of a buried and numerically vague quantification over comparatives: Bx ↔ Zy(xBy) The quantifier Zx is indefinite in two very intuitive ways, as will be expressions built from it. There is, first of all, no answer to the question ‘Precisely how many (or what proportion) constitute ‘the great bulk of’? In this regard Zx is like the quantifiers ‘many,’ ‘lots of’, ‘almost all’, ‘a few,’ and ‘a significant number of,’ none of which correlate with any sharp number or point proportion of the population. Like these, Zx is essentially numerically indeterminate. Zx is also indefinite as to the comparison class. Balder than the great bulk of whom? Taller than the overwhelming majority of what group? These issues are specified by context, and can only be so specified. Because ‘bald’ is to be understood in terms of Zx applied over a comparison class, it followed immediately that ‘x is bald’ will be true only with reference to a comparison class. This is not a weakness of such a theory of vagueness, but a strength; it is precisely how we use ‘bald’, ‘tall,’ and ‘nice.’ The hirsute man at the bald men’s convention may be the bald man at the hirsute men’s convention. The old kid in kindergarten may be all of 6 1/2. The young guy in the nursing home is 59. The key to understanding vagueness, on this account, lies in understanding the buried quantifier Zx. The familiarity of ∀x and ∃x are often obstacles to understanding Zx, however, and it is clear that the logic appropriate to ∀x and ∃x will fail for Zx.

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Consider a comparison class of men numbering in the millions, the relation ‘x is balder than y’ mapped onto that class, and a notion of ‘x is bald’ defined as ‘balder than the overwhelming bulk of’ those men. Consider a ‘forced march’, in which we line the men up in terms of relative baldness. As we step down the line of candidates from the most bald to the least, will there be a particular point at which we pass from an individual who is ‘balder than the overwhelming bulk of’ the members of the group to a next individual who is not ‘balder than the overwhelming bulk’? Will there be a transition step such that ∃x∃z(Zy(xBy) & z is next in line behind x & ~Zy(zBy)) ? Clearly not. The explanation for the lack of a transition step lies precisely in the nature of ‘the overwhelming bulk of’ quantifier. If our men are lined up in terms of numbers of hairs, we will correspondingly deny a transition step phrased in terms of our derivative ‘is bald’: ~∃x∃z(Bx & z has one more hair than x & ~Bz). There is no step at which we go from ‘bald’ to ‘not bald’ precisely because there is no step at which we go from ‘balder than the great bulk of’ to ‘not balder than the great bulk of’. If we maintain that there is no transition step, are we not forced to the classically equivalent ∀x∀z(Bx & z has one more hair than x → Bz) ? No. This too we can deny. Quantifier Negation holds only on the assumption of Excluded Middle, which here comes down to the assumption that ∀x(Bx v ~Bx). But if this abbreviates ∀x(Zy(xBy) v ~Zy(xBy)), then it is clear from the nature of our quantifier Zy that Excluded Middle simply will not hold. ‘Balder than the great bulk of’ gradually loses applicability as we walk down the line, and thus there will be cases in which we will not want to maintain either that x is balder than the great bulk of the comparison class or that x is not

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balder than the great bulk of the comparison class. Excluded Middle fails for Zy(xBy) and thus for Bx, and Quantifier Negation fails with it. The crucial test for any theory of vagueness is the Sorites. In imagination we line up men with increasing numbers of hairs, each perhaps with one more hair than his predecessor. A standard formulation for the Sorites uses the induction principle ∀x∀z(Bx & z has one more hair than x → Bz). If baldness is definable in terms of a vague quantifier over the comparison adjective ‘balder’, this amounts to: ∀x∀z(Zy(xBy) & z has one more hair than x → Zy(zBy)). Such a principle will clearly be false. As we walk down our line of men, it will eventually be clear that we no longer have a ‘great bulk of’ the population for a candidate to be balder than. The induction principle fails. If we deny the induction principle, however, are we not forced to admit that there is a crucial transition step?: ∃x∃z(Bx & z has one more hair than x & ~Bz)? No, for this amounts to ∃x∃z(Zy(xBy) & z has one more hair than x & ~Zy(xBy)), which we also deny. The character of Zx makes it clear why predicates incorporating such a quantifier will not obey Excluded Middle. Only with a principle of Excluded Middle would an inference from the negation of the induction principle to an existential claim of a crucial transition step hold. An understanding of ‘bald’ as incorporating the buried Zx quantifier allows us to see how both the existential and universal quantifications will fail. There is one further desideratum that any account of vagueness targeted for the sorites should satisfy. The notion of a crucial transition step—the notion that

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∃x∃z(Bx & z has one more hair than x & ~Bz) is inherently implausible. The fact that the Zx account denies such a claim is therefore a point in its favour. The Zx account also denies the induction principle, however: ∀x∀z(Bx & z has one more hair than x → Bz). Unlike the transition step, this generalization is taken to be eminently plausible. If it is false, why are we so tempted to think it is true? Although the induction principle above is false, and must be so, it has a close relative that may well be true: Zx∀z(Bx & z has one more hair than x → Bz). All that has changed is the first quantifier, from ∀x to Zx. Unlike the induction principle, this is not a universal quantification over x at all. It is instead a vague quantification: For the great bulk of cases, the overwhelming majority of cases, if x is bald and z has one more hair then z is bald as well. Because it holds ‘for the great bulk of cases’, it may be quite generally usable as a rule of thumb for individual applications. Because it falls short of a full universal quantification, however, it does not force us to the paradoxical reasoning of the Sorites. 5. NUMBER BEYOND NUMBER Explorations of indeterminate collectivities open up a realm of indeterminate number beneath the determinate. But considerations of plena also open up a realm of number beyond number as standardly conceived. How many facts are there? The idea of a complete listing of all facts with some numerical limit is manifestly impossible. For consider a list F of all and only factual statements, and the particular statement: • The list F of stated facts fails to have this statement on it.

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If this statement is on the list, it clearly does not state a fact, so F is not after all a list purely of factual statements. The statement must therefore be left off the list. But in that case the statement will be true and list F cannot be complete. Facts can never be listed or numbered in toto because there will always be further facts—facts about the entire list or its number—that a supposedly complete list could not manage to register.5 Plena like the totality of facts take us beyond sets and correspondingly beyond number as set-theoretically conceived. We know, via Cantor’s Power Set Theorem, that there will be an uncountably infinite number of subsets of any countably infinite set of items. If conceived of in terms of a set of all facts, each of the subsets of an infinitely complex world’s objects gives rise to a fact uniquely characteristic of that particular subset. If the number of such subsets is more than countably infinite, so too must the number of facts be uncountably infinite. As already noted, there can be no merely finite plena. Any plenum must contain all its subsets as elements, which gives us sets of its subsets as further elements, sets of its sets of its subsets, and so on. Nor can there be a plenum that is merely countably infinite. For suppose any plenum with countably infinite elements. Among its subsets will be all finite and infinite collectivities of those elements. But that gives us a structure reflecting the reals rather than the natural numbers, and thus the plenum must be at least nondenumerably infinite. The important point is that such an argument can be repeated for any number proposed for the elements of a plenum, of any order of nondenumerable infinity. The elements of any set of that order will be outnumbered by its subsets, and for each of those subsets there will be a distinct member of the plenum. Plena extend beyond any set-size survey of them. For any notion of number defined in terms of sets, plena are literally beyond number. In this sense plena continue and complete a certain conceptual trajectory. Finite sets are those every proper subset of which is smaller than the whole. Infinite sets have some proper subsets equal in size to the whole. Plena have proper subcollectivities larger than the whole. Plena are thus oversize collectivities in that from the angle of standard set theory they are immeasurably large—so massive as to resist

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the meaningful assignment of a cardinal number. By Cantor’s Theorem, sets inevitably have more subsets than members. Plena, in contrast, are large enough to contain as many elements as they have subsets. Can plena be compared in terms of size? Despite the fact that they are beyond set-size number, the answer appears to be ‘yes’. As standard set theory has it, one set is of lesser or equal cardinality than another whenever the membership of the former can be mapped into that of the latter in such a way that different members are always mapped to different members: • card (S1) ≤ card (S2) iff there is a mapping m from S1 into S2 such that whenever x ∈ S1 & y ∈ S1 & x ≠ y then m(x) ≠ m(y)) And so S1 and S2 will have the same cardinality whenever both card (S1) ≤ card (S2) and also conversely card (S2) ≤ card (S1). An analogous relationship can be implemented for plena by way of logical-conceptual correspondences. Consider, for example, a correspondence between items and sets: • To any set we associate as its paired item that set itself, and moreover— • To any item we associate as its paired set the set containing as its only member that item itself. We might similarly implement a coordination relationship between sets and formulable truths via the following correspondence: • To any formulable truth t we associate the set {t} consisting of that truth alone, and moreover— • To any set S we associate the truth S = S to the effect that it is self-identical.

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With respect to things/items, sets, and formulable truths we obtain the result that these three plena are equinumerous. For every possibility, there is a distinct truth: that what is at issue is possible. For every truth, there is the distinct possibility of its truth. Symbolically formulable truths can therefore be coordinated with formulable possibilities via the pairing x ≈ ◊x and formulable falsehoods can be coordinated with formulable truths via the pairing x ≈ not-x Moreover, for every truth (syntactically understood) there is a unique corresponding item/thing/object, namely its characterizing formulation. And conversely for every thing x there is a unique corresponding truth, viz. the thesis x = x. Whether this equinumerosity can be extended across the board for plena—so that one could simply speak of “the cardinality of a plenum”—is an open question. Weak inductive evidence would suggest, however, that an affirmative answer is not implausible. In addition to numerosity, there is also the matter of structure. Structural order is a key feature of plena. Thus let P1, P2, P3 be subcollectivities of some plenum P. Then {P1, P2} correlates with a unique P-member, as does {{P1, P2}, P3}. Moreover {P1, P3} correlates with a unique P-member, as does {P1, {P2, P3}}. The uniqueness of the correspondence correlation provides the basis for an organizing structure. The branching at issue here precludes plena from taking on a linear order. The explosive nature of plena blocks the road to the standard Cantorean process for well-ordering sets. As Cantor himself insists in the Grundlagen, sets are well-orderable and increasable manifolds, and he insisted that manifolds which fail to conform to these conditions are not sets. But that is exactly what plena do. Here a note is in order concerning the function under which subsets can be seen to be members of a collectivity or plenum. Consider a collectivity of all facts F and subcollectivities F1, F2, and F3. Consider

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{F1, F2} and {{F1, F2}, F3} on the one hand and {F1, F3} and {F1, {F2, F3}}. For each of these we can generate a distinct fact—for example, a fact of the following form: The collectivity {{F1, F2}, F3} contains precisely the collectivities {F1, F2} and F3. The collectivity {F1, {F2, F3}} contains precisely the collectivities F1 and {F2, F3}. Not every function on subcollectivities generated in the structural order noted will preserve that uniqueness, however. Consider for example a function of adjunctive compilation, which gives us merely the total semantic content of a collectivity of facts, including the total semantic content of any subcollectivities all the way down. Under adjunctive compilation, {{F1, F2}, F3} will give us the same fact as {F1, {F2, F3}}. The collectivity of all facts does constitute a plenum, but it is facts that reflect the structure of collectivities or other truths, rather than facts that adjunctively compile semantic content, that makes its status as a plenum clear. The Cantorean tradition sees cardinal numbers in terms of sets, a cardinal number being defined by the set of all sets whose members can be coordinated one-to-one with those of given set. On this basis plena are non-set collectivities that extend beyond set-theoretical number; there will be no such thing as the number of elements in a plenum just as there will be no such thing as the number of all numbers. But the situation becomes altered when we move from sets to collectivities. With number seen in the light of collectivities there is no basis for denying plena access to numerosity. Our argument throughout has been that the conceptual universe contains both indeterminate collectivities and plena. The logic demanded for a clear recognition of these—a recognition required both to make sense of our own conceptualization and to make sense of the universe accessed through that conceptualization—will be a new logic, or a new set of logics, but by no means an impossible logic. And that logic, so we submit, there will come new concepts of quantity and

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magnitude: number both beneath number as standardly conceived, and beyond it. NOTES 1

David H. Sanford makes an eloquent appeal for indeterminate numbers appropriate to a many-valued treatment of vagueness in “Vague Numbers,” Acta Analytica 17 (2002), 63-73. In his treatment, he admits, “Vague Numbers” stands for an awareness of the problem rather than a serious attempt at theoretical solution. This chapter attempts some steps toward the latter.

2

Kurt Gödel, 1931: “On Formally Undecidable Propositions of Principia Mathematica and Related Systems,” in Jean van Heijenoort, ed., From Frege to Gödel: A Source Book in Mathematical Logic 1879-1931 (Cambridge, MA: Harvard Univ. Press, 1967), pp. 596-616.

3

Alfred Tarski, 1930, published 1948: A Decision Theory for Elementary Algebra and Geometry prepared for publication by J. C. C. McKinsey, (Santa Monica, Ca: The Rand Corporation, 1948. Revised edition Berkeley: Univ. of California Press, 1951); G.B.B. Hunter, Metalogic. (Berkeley: Univ. of California Press, 1971).

4

The discussion of the buried quantifier parallels Grim, “The Buried Quantifier: An Account of Vagueness and the Sorites,” Analysis, 65 (2005) 95-104.

5

Granted, these deliberations assume a great deal about the nature of facts. But this is not the place for unraveling the relevant complications.

Name Index Anderson, C. Antony 11n2 Armour-Garb, Bradley, 92n9 Bar-Hillel, Yehoshua, 91n2 Barnett, David, 42n4, 91n4 Barwise, Jon, 91n2, 11n4 Bealer, George, 11n3 Beall, J. C., 58n6, 91n1, 92n9 Berkeley, George, 28-30, 37-38, 41n2, 42n3 Boolos, George, 60, 71n5 Boyer, David, 42n7 Brandom, Robert, 92n6 Burali-Forti, Cesare 87-89 Cantor, Georg, 43, 59, 62, 72n6, 76-77, 105-107 Cargile, James, 91n2 Cartwright, Richard, 91n2 De Morgan, Augustus, 79 Diaconis, Persi, 25n6 Downing, Lisa, 42n3 Etchemendy, John 91n2 Field, Hartry, 87, 91-92n5, 92n7, 92n11 Fine, Kit, 25n3 Gallois, Evariste, 42n2 Gödel, Kurt, 38. 109n2 Grim, Patrick, 11n5, 55n2, 58n4, 71n1, 72n4, 72n8, 92n9, 109n4 Holmes, Susan, 25n6 Horwich, Paul, 91n4 Hunter, G. B. B., 109n3 Jürgens, Hartmut, 58n5

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Kamp, J. A., 25n3 Kant, Immauel, 64-65 Mar, Gary, 58n2 McGee, Vann, 72n5 McKinsey, J. C. S., 109n3 Montgomery, Richard, 25n60 Moss, L., 11n4 Myhill, John, 11n4 Parsons, Terence, 92n7 Peitgen, Heinz-Otto, 58n5 Pitcher, George, 42n3 Priest, Graham, 92n6, 92n9 Rayo, Agustín, 60, 71n5 Rescher, Nicholas, 25n1, 245n5, 25n7, 92n6 Russell, Bertrand, 5, 6, 43, 47, 53-55, 68, 70, 74-77, 99 Sanford, David H., 109n1 Saupe, Dietmar, 58n5 Smiley, Timothy, 92n9 Sobel, J. Howard, 91n2 Sorensen, Roy A., 25n2, 41n1 St. Denis, Paul, 58n2 Tarski, Alfred, 56, 109n3 Van Fraassen, Bas, 25n3 Williamson, Timothy, 25n2, 42n639, 41n1, 60, 65n6, 71n5, 79, 91n4 Yablo, Stephen, 92n5 Zalta, Edward N., 11n2

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Ontos

NicholasRescher

Nicholas Rescher

Collected Paper. 14 Volumes Nicholas Rescher is University Professor of Philosophy at the University of Pittsburgh where he also served for many years as Director of the Center for Philosophy of Science. He is a former president of the Eastern Division of the American Philosophical Association, and has also served as President of the American Catholic Philosophical Association, the American Metaphysical Society, the American G. W. Leibniz Society, and the C. S. Peirce Society. An honorary member of Corpus Christi College, Oxford, he has been elected to membership in the European Academy of Arts and Sciences (Academia Europaea), the Institut International de Philosophie, and several other learned academies. Having held visiting lectureships at Oxford, Constance, Salamanca, Munich, and Marburg, Professor Rescher has received seven honorary degrees from universities on three continents (2006 at the University of Helsinki). Author of some hundred books ranging over many areas of philosophy, over a dozen of them translated into other languages, he was awarded the Alexander von Humboldt Prize for Humanistic Scholarship in 1984. ontos verlag has published a series of collected papers of Nicholas Rescher in three parts with altogether fourteen volumes, each of which will contain roughly ten chapters/essays (some new and some previously published in scholarly journals). The fourteen volumes would cover the following range of topics: Volumes I - XIV STUDIES IN 20TH CENTURY PHILOSOPHY ISBN 3-937202-78-1 · 215 pp. Hardcover, EUR 75,00

STUDIES IN VALUE THEORY ISBN 3-938793-03-1 . 176 pp. Hardcover, EUR 79,00

STUDIES IN PRAGMATISM ISBN 3-937202-79-X · 178 pp. Hardcover, EUR 69,00

STUDIES IN METAPHILOSOPHY ISBN 3-938793-04-X . 221 pp. Hardcover, EUR 79,00

STUDIES IN IDEALISM ISBN 3-937202-80-3 · 191 pp. Hardcover, EUR 69,00

STUDIES IN THE HISTORY OF LOGIC ISBN 3-938793-19-8 . 178 pp. Hardcover, EUR 69,00

STUDIES IN PHILOSOPHICAL INQUIRY ISBN 3-937202-81-1 · 206 pp. Hardcover, EUR 79,00

STUDIES IN THE PHILOSOPHY OF SCIENCE ISBN 3-938793-20-1 . 273 pp. Hardcover, EUR 79,00

STUDIES IN COGNITIVE FINITUDE ISBN 3-938793-00-7 . 118 pp. Hardcover, EUR 69,00

STUDIES IN METAPHYSICAL OPTIMALISM ISBN 3-938793-21-X . 96 pp. Hardcover, EUR 49,00

STUDIES IN SOCIAL PHILOSOPHY ISBN 3-938793-01-5 . 195 pp. Hardcover, EUR 79,00

STUDIES IN LEIBNIZ'S COSMOLOGY ISBN 3-938793-22-8 . 229 pp. Hardcover, EUR 69,00

STUDIES IN PHILOSOPHICAL ANTHROPOLOGY ISBN 3-938793-02-3 . 165 pp. Hardcover, EUR 79,00

STUDIES IN EPISTEMOLOGY ISBN 3-938793-23-6 . 180 pp. Hardcover, EUR 69,00

ontos verlag Frankfurt • Paris • Lancaster • New Brunswick 2006. 14 Volumes, Approx. 2630 pages. Format 14,8 x 21 cm Hardcover EUR 798,00 ISBN 10: 3-938793-25-2 Due October 2006 Please order free review copy from the publisher Order form on the next page

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NicholasRescher

Nicholas Rescher

Autobiography Second Edition

This revised edition of his Autobiography brings up-to-date Rescher’s account of his life and work. The passage of years since the publication of an autobiographical work makes for its growing incompleteness. Moreover, the passage of time is bound to bring some new perspectives to view. This new edition comes to terms with these circumstances. Since the publication of the previous version Rescher’s philosophical work has made substantial progress, betokened by the publication of over a score of new books that mark an ongoing expansion of his philosophical range. Then too, the internet has brought to light interesting new information about Rescher’s family background and antecedence. Overall the book affords a detailed, vivid, and highly personalized picture of the life and work of someone who counts as one of the most prolific and many-sided contemporary thinkers.

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Frankfurt • Paris • Lancaster • New Brunswick 2010. 419 Seiten Format 14,8 x 21 cm Paperback EUR 49,00 ISBN 978-3-86838-084-2

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NicholasRescher

Nicholas Rescher

Free Will An Extensive Bibliography With the Cooperation of Estelle Burris

Few philosophical issues have had as long and elaborate a history as the problem of free will, which has been contested at every stage of the history of the subject. The present work practices an extensive bibliography of this elaborate literature, listing some five thousand items ranging from classical antiquity to the present.

About the author Nicholas Rescher is University Professor of Philosophy at the University of Pittsburgh where he also served for many years as Director of the Center for Philosophy of Science. He is a former president of the Eastern Division of the American Philosophical Association, and has also served as President of the American Catholic Philosophical Association, the Americna Metaphysical Society, the American G. W. Leibniz Society, and the C. S. Peirce Society. An honorary member of Corpus Christi College, Oxford, he has been elected to membership in the European Academy of Arts and Sciences (Academia Europaea), the Institut International de Philosophie, and several other learned academies. Having held visiting lectureships at Oxford, Constance, Salamanca, Munich, and Marburg, Professor Rescher has received six honorary degrees from universities on three continents. Author of some hundred books ranging over many areas of philosophy, over a dozen of them translated into other languages, he was awarded the Alexander von Humboldt Prize for Humanistic Scholarship in 1984. In November 2007 Nicholas Rescher was awarded by the American Catholic Philosophical Association with the „Aquinas Medal“

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Frankfurt • Paris • Lancaster • New Brunswick 2009. 309pp. Format 14,8 x 21 cm Hardcover EUR 119,00 ISBN 13: 978-3-86838-058-3 Due December 2009

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NicholasRescher

Nicholas Rescher

On Rules and Principles A Philosophical Study of their Nature and Function The present book is a natural outgrowth of Rescher’s longstanding preoccupation with the rational systematization of our knowledge as manifested in such earlier works as Cognitive Systematization (Oxford: Blackwell, 1979), and Complexity (New Brunswick: Transaction Publishers, 1998). Accordingly, the role of principles in human affairs is crucial and ubiquitous. Principology, the theory of principles—underdeveloped through it may be—is accordingly bound to find a significant place in the sphere of philosophical inquiry regarding matters of thought and action. About the Author Nicholas Rescher is University Professor of Philosophy at the University of Pittsburgh where he also served for many years as Director of the Center for Philosophy of Science. He is a former president of the Eastern Division of the American Philosophical Association, and has also served as President of the American Catholic Philosophical Association, the American Metaphysical Society, the American G. W. Leibniz Society, and the C. S. Peirce Society. An honorary member of Corpus Christi College, Oxford, he has been elected to membership in the European Academy of Arts and Sciences (Academia Europaea), the Institut International de Philosophie, and several other learned academies. Having held visiting lectureships at Oxford, Constance, Salamanca, Munich, and Marburg, Professor Rescher has received six honorary degrees from universities on three continents. Author of some hundred books ranging over many areas of philosophy, over a dozen of them translated into other languages, he was awarded the Alexander von Humboldt Prize for Humanistic Scholarship in 1984. In November 2007 Nicholas Rescher was awarded by the American Catholic Philosophical Association with the „Aquinas Medal“.

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Frankfurt • Paris • Lancaster • New Brunswick 2010. 220 Seiten Format 14,8 x 21 cm Hardcover EUR 89,00 ISBN 978-3-86838-089-7

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