Logical investigations of predication theory and the problem of universals 8870880702, 9788870880700

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NINO

B.

COCCHIARELLA

INDICES Monographs in Philosophical Logic and Formal Linguistics

LOGICAL INVESTIGATIONS

II

OF PREDICATION THEORY AND

Managing Editors

THE PROBLEM OF UNIVERSALS FRANZ GUENTHNER

UwE MoNNICH

Advisory Editors LENNART AQVIST

EDWARD KEENAN

NINO CoccHIARELLA

HAROLD LEVIN

joHN CoRCORAN

DONALD NUTE

Dov GABBAY

TERRENCE p ARSONS

jAAP HoEPELMAN

CHRISTIAN ROHRER

,,

HANS KAMP

KRISTER SEGERBERG

;I ()

'I

'

Editorial Assistant VIRGINIO SALA

d '. I

'I

BIBLIOPOLIS

Dedicated to my parents Carmela Saporito and Pasquale S. Cocchiarella

ISBN 88-7088-070-2 @ 1986 by « Bibliopolis, edizioni di filosofia e scienze » Napoli, via Arangio Ruiz 83

All rights reserved. No part of this book may be reproduced in any form or by any means without permission in writing- front the publisher Photoset by Milano

«

Studio in out »

Printed in Italy by « Grafitalia » ·Cereola, march 1986

CONTENTS

p.

9

»

11

NOTES TO THE INTRODUCTION

»

27

NOMINALISM

»

29

NOTES TO CHAPTER l

))

61

CONCEPTUALISM

))

65

»

99

Preface Introdnction 1. The Problem of the Predicable Nature of Universals, 12 - 2. Universality vs. Individuality, 13. - 3. Second Order Theories of Predication, 15. - 4. Predicativity vs. Impredicativity, 19. - 5. Internal vs. External Semantics, 24.

I.

1. First Order Theories, 32. - 2. The Referential Semantics of First Order Theories, 35. - 3. Standard Predicative Second Order Logic, 37. - 4. Nominalistic Semantics for Predicative Second Order Logic, 42. - 5. Soundness and Completeness with respect to Nominalistic Semantics, 44. - 6. Nominalism and Modal Logic, 47. - 7. Logical Truth as Validity in Every Domain of Discourse, 52. - 8. The Secondary Semantics of Universal Validity, 56.

/) /

II.

1. Conceptualism vs. Nominalism, 71. - 2. Constructive Conceptualism, 73. - 3. Substitutivity and Definibility in Constructive Conceptualism, 79. - 4. Identity in Nominalism vs. Identity in Constructive Conceptualism, 82. - 5. An External Semantics for Constructive Conceptualism, 85. - 6. Ramified Constructive Conceprualism, 88. 7. Holistic Conceprualism, 93. - 8. An External Semantics for Holistic Conceptualism, 95. - 9. Conceptualism and the Logical Modalities, 97. NOTES TO CHAPTER ll

8

III.

PREDICATION illEORY AND UNIVERSALS

REALISM

)) 105

PREFACE

» 161

Beginning with Aristotle's notion of a universal as that which can be predicated of things, I provide in this monograph separate logical analyses of what nominalism, conceptualism and realism take to be the predicable nature of universals. My position throughout is that such an analysis proceeds through the construction of a formal theory of predication on the one hand and a logical semantics on the other. I adopt and apply in this regard the formal and semantical techniques of my former teachers Rudolf Carnap and Richard Montague. One important way in which I differ from Carnap and Montague, however, is in our respective analyses of so-called "higher order" sentences - that is, sentences in which nominalized predicates, whether simple or complex, occur as the logico-grammatical subjects of other predicates. In this regard, whereas Carnap and Montague formulate and adopt one or another version of a theory of simple logical types as the framework within which to analyze such sentences, I formulate instead, relative to nominalism, conceptualism and realism, systems which do not require any grammatical type distinctions beyond those already found in standard second order predicate logic. All of the theories of predication formulated in this monograph, in other words, are second order theories, including those which contain a logic of nominalized predicates. Russell's paradox of predication, it turns out, can be resolved without resorting to a theory of types. The research reported in this monograph was supported in part by the National Endowment for the Humanities. An internal Grant-in-Aid of Research from Indiana University defrayed the cost for typing the manuscript.

1. Logical Realism vs. Holistic Conceptualism, 108. - 2. The Essential Incompleteness of Logical Realism, 110. - 3. Natural Realism and Conceptualism, 113. - 4. On the Logic of Natural Realism, 115. 5. Natural Realism and Modal Logic, 118. - 6. An External Semantics for Modal Natural Realism, 120. - 7. A Completeness Theorem for Modal Natural Realism, 124. - 8. Modal Logical Realism, 134. 9. Possibilism and Actualism in Modal Logical Realism, 137. - 10. Logical Realism and Essentialism, 143. - 11. Possibilism and Actualism in Holistic Conceptualism, 150. NOTES TO CHAPTER III

IV. ON THE LOGIC OF NOMINALIZED PREDICATES AND ITS PHILOSOPHICAL INTERPRETATIONS

)) _ f65

1. Some Philosophical Views of Nominalized Predicates, 166. - 2. Terminology, 168. - 3. A Minimal Logic for Nominalized Predicates, 170. - 4. Russel's Paradox of Predication, 173. - 5. Identity as Syncategorematic, 178. - 6. The Consistency of T'", 182. - 7, The Theories of Homogeneous, Heterogeneous and Cumulative Simple Types as Second Order Logics, 188. - 8. The Consistency of HST'" Relative to Monadic HST•, 193. - 9. On the Relative Consistency of Monadic HST•, 199. - 10. On The Consistency of the Unrestricted Comprehension Principle (CP'"), 205.

V.

NOTES TO CHAPTER IV

)) 212

COMPLEX PREDICATES AND THE A - OPERATOR

)) 215

1. A Generalized Logical Syntax, 215. - 2. The Problem of Complex Predicates, 218. - 3. The Minimal Logic for Nominalized Complex Predicates, 220.. - 4. Conceptual Platonic Realism and Other Extensions of AM'"('°), 225. - 5. The Theses of Extensionality and lntensionality with Nominalized Complex Predicates, 232. - 6. Modal ·Logic and Nominalized Complex Predicates, 234. - 7. The Principle of Rigidity Revisited, 239. NOTES TO CHAPTER V

VI. TWO FREGEAN SEMANTICS FOR NOMINALIZED COMPLEX PREDICATES 1. Fregean Frames and Intensional Models, 244. - 2. A Generalized Completeness Theorem for Extensions of DAM'" + (Dl;:xtt), 248. 3. The Relative Consistency of DA.HST'" + (OExt'") and DAT'" + (DExt'"), 254. - 4. The Relative Consistency of HST:o + (OExt'") and T:o + (DExt'"), 257. - 5. A:n Alternative Fregean Semantics, 259.

)) 241

)) 243

Nino B. Cocchiarelta Indiana University

INTRODUCTION

PREDICATION THEORY AND THE PROBLEM OF UNIVERSALS

Predication theory has been a subject of philosophical concern since at least the writings of Plato and Aristotle. It is in its way the locus of a number of philosophical issues both in metaphysics and epistemology, not the least of which is the problem of universals. The latter problem, sometimes all too simply put as the question of whether there are universals or not, is especially germane to the notion of predication since a theory of universals is at least in part a semantic theory of predication; and it is just to such a theory that we must turn in any philosophical investigation of the notion of predication. In doing so, however, we need not assume the truth or superiority of any one theory of universals over another. Indeed, an appropriate preliminary to any such assumption might well consist of a comparative analysis of some of the different formal theories of predication that can be semantically associated with these different theories of universals: for just as the latter provide a semantics for the former, it is only through the logical syntax of a formal theory of predication that the logical structure of a theory of universals can be rendered perspicuous. That, in any case, is the principal methodological assumption for the approach· to the problem of universals we shall undertake in the present monograph where we will be more concerned with the construction and comparison of the abstract logical systems that may be associated with different theories of universals than with the metaphysical or epistemological issues for which they were originally designed. It is our hope and expectation, however, that these comparative formal analyses will be instrumental toward any philosophical decision as to whether to adopt a given theory of universals or not.

12

1.

PREDICATION TIIEORY AND UNIVERSALS

TuE PROBLEM OF THE PREDICABLE NATURE OF UNIVERSALS

The original use of the term "universal" goes back to Aristotle according to whom a universal is that which can be predicated of things (De Interpretatione, 17 a 39). We shall retain the core of this notion throughout this essay and assume that whatever else it may be a universal has a predicable nature and that it is this predicable nature which is what constitutes its universality. Nothing follows from that assumption, however, regarding whether a universal is (1) merely a predicate expression (nominalism) of some language or other; (2) a concept (conceptualism) in the· sense of a sociobiologically based cognitive ability or capacity to identify, collect or classify, and characterize or relate things in various ways; or (3) a real property or relation existing independently of both language and the natural capacity humans have for thought and representation (realism). We propose to take each of these interpretations or theories of universals seriously in what follows at least to the extent that we are able to asso~ date each with a formal theory of predication. Our particular concern in this regard, moreover, will be with the explanation each provides of the predicable nature of universals, i.e., of that in which the universality of universals consists. Our discussion and comparison of nominalism, conceptualism and realism, accordingly, will not deal with the variety of arguments that have been given for or against each of them, but with how each as a theory of universals may be semantically associated with a formal theory of predication. Our assumption here, as indicated abo¥e, is that insofar as such an associated formal theory of predication provides a logically perspicuous medium for the articulation of the predicable nature of universals as understood by the theory of universals in question, then to that extent the formal theory may itself be identified with the explanation which that theory of universals provides of the predicable nature of universals. It is in the sense of this assumption, moreover, that we understand a philosophical theory of predication to be a formal theory of predication together. with its semantically associated theory of universals.

INTRODUCTION

2.

13

UNIVERSALITY VS. INDIVIDUALITY

One important part of the problem of universals which we will be concerned with in this essay is the problem whether universals are real individuals (in the logical sense) or whether their predicable nature is such as to exclude their also having an individual nature. According to some theories, for example, universals cannot be individuals because having a predicable nature amounts to being what in some sense is only an unsaturated structure: though whether such a structure exists independently of causal contexts, or is realizable, i.e., can be saturated, only in cognitive contexts, etc., will depend on the particular theory of universals in question. There are other theories, in any case, which take universals to have an individual as well as a predicable nature, {e., to have a nature under which other universals (including perhaps even themselves) may be predicated of them in essentially the same sense in which they are themselves predicated of individuals. Semantically, this part of the problem of universals has to do with how we are to interpret the role of nominalized predicates in ordinary and scientific discourse. Thus, e.g., while it is generally agreed that the surface grammar of a verb such as "runs'', "walks", "loves", etc., is different from that of its corresponding verbal noun, "running", "walking", "loving", etc., and that similarly the grammar of a predicate adjective, such as "wise", "pious'', "triangular", etc., is different from that of its corresponding nominalization, "wisdom", "piety", "triangularity", etc., it is not generally agreed whether there really are individuals that are the referents of verbal nouns and nominalized predicates, or, even if it be assumed that there are such individuals, whether they are (or can be) the universals that are semantically associated with the corresponding verbs and predicate expressions. It should be noted here, however, that the distinction between individuals and universals which is in question is not the Aristotelian distinction between primary substances and forms (universals), where primary substances, by definition, are entities which cannot be predicated at all. That universals are not substances in this sense is of course obvious and unproblematic: but that in no way settles the question whether universals are individuals, or, equivalently, whether all individuals are substances in the above sense. In general, in other words, when we are comparing different philosophical theories of predication, i.e., when it is the different

14

PREDICATION THEORY AND UNIVERSALS

analyses of the nature of predication that are in question, it is the distinction between universals and individuals, and not the distinction between universals and substances, which is at issue. From the logical point of view, what the problem at issue has to do with is the logical or formal nature of individuals vis-a-vis the logical or formal nature of universals, i.e., we are concerned here with the complementary notions of universality and individuality in the logical sense; and, of course, it is in this sense that these notions must be relativized to a philosophical theory of predication, i.e., a formal theory of predication which is semantically associated with a theory of universals. It is only through the logico-grammatical roles of subject and predicate expressions in the logical forms of a philosophical theory of predication, in other words, that the complementary notions of individuality and universality can be rendered logically perspicuous. · Accordingly, when we speak hereafter of the individuals of a given philosophical theory of predication we shall mean no more and no less by · their individuality (in that theory) than the purported fact (as advocated by an ideal proponent of the theory) that these are the entities which can. be referred to by means of the subject expressions of that theory, and in particular that these are the entities which an ideal proponent of the theory would take to be the values of its bound individual variables. This, essentially, is all that individuality in the logical sense amounts to - where of course it is always to be understood that the logic in question is that which is articulated in the logical forms of the theory in question. It is in this sense also that we shall occasionally speak of the individuals of a given philosophical theory as the logical subjects of that theory. Similarly, in speaking of the (n-ary) universals of a philosophical theory of predication, we shall mean the entities that are represented primarily, if not solely, by the (n-place) predicate expressions of that theory; and, in particular, since quantification with respect to (n-place) predicate variables will be allowed in all of the theories to be considered here, these are the entities that are understood to be indicated by the bound (n-place) predicate variables 1• This notion of indication, however, is not to be confused with the referential relation between the individuals of a given theory and its bound individual variables. Indeed, if universals are not individuals in that theory, then this notion of indication will not be a relation in that theory in anything at all like the sense in which the relata of a relation must be individuals.

15

INTRODUCTION

3.

SECOND ORDER THEORIES OF PREDICATION

All of the philosophical theories of predication which will be considered in this monograph, it should be noted, are second order logics based on the conditions under which each has been semantically associated with a theory of universals. Essentially, what this means is that exactly two types of quantifiers, one with respect to subject positions and the other with respect to predicate positions, are allowed in the formal component of each of these theories. In particular, they will all contain the well/armed formulas (wffs) of standard second order logic (though they will in general differ in the logistic which is assigned to these wffs). These, in fact, will be the only wffs of the theories to be considered in chapters I-III of this essay, while the wffs in chapters IV-VI will go beyond them by including a well-formedness condition which allows for the occurrence of nominalized predicates. It is of course assumed that the wffs of any given theory are to be understood as representing the logical forms of the assertions that are possible in ·that theory; and for this reason we shall also refer to the wffs of a philosophical theory as the propositional forms of that theory.2 In regard to logical constants, 3 we shall use the following with the indicated reading:

&

v

v 3:

A.

the the the the the the the the the

negation sign (material) conditional sign (material) biconditional sign conjunction sign disjunction sign universal quantifier existential quantifier identity sign A.-o~rator.

As is well-known, we do not need to take all of these signs as primitive logical constants; and for convenience, especially when giving inductive definitions or proofs based on the well-formedness conditions of wffs, we shall assume that ~, -, V, and = are the only logical constants actually occurring in wffs. The remaining constants will then be understood as serving the purposes of abbreviatory devices of the metalanguage. (We will not use the A-operator until chapters V and VI.)

PREDICATION THEORY AND UNIVERSALS

16

Because we shall also consider how each of these formal theories of predication are affected when supplemented with a modal logic regarding necessity and possibility, we shall take D

0

the necessity operator the possibility operator

to be logical constants as well (of the supplemented theories). Here again redundancy allows us to assume that D is taken as primitive with O presumed defined as ~o~. Though it is unnecessary for logical syntax, we shall also use parentheses, brackets and commas as auxiliary signs. These aid us in our visual identification of displayed wffs; but it may be assumed that the displayed wffs which appear with parentheses, brackets and commas do so only in the metalanguage, e. g., that they are all really being generated only in some form of Polish notation (where auxiliary signs are unnecessary and do not occur). We also assume that juxtaposed signs are concatenated signs, i.e., that concatenation is to be represented by juxtaposition. In regard to variables, we assume the existence of a potential infinity of individual variables and, for each natural number n, a potential infinity of n-place predicate variables. (We identify propositional variables with 0-place predicate variables.) All of these potential infinities can be generated from the use of one sign together with numerical subscripts and superscripts (or a certain super- and subscripted number of primes); and therefore the assumption of the availability of such variables even in the most restrictive systems to be considered here seems fully justified. The atomic wffs of the minimal set of pure second order formulas will be expressions either of the form (x = y), where x, y are individual variables, or of the form Fn(XI, ... , Xn), where Fn is an n-place predicate variable and XI, ... ,Xn are all individual variables (not necessarily distinct). (For n = 0 we take Fn(XI, ... ,Xn) to be just Fn itself.) We shall use "x", "y'', "z'', with or without numerical subscripts, to refer (in the metalanguage) to individual variables, and "Fn'', "Gn'', "Hn" and "Rn'', with or without numerical subscripts, to refer (in the metalanguage) to n-place predicate variables. ("Rn" will be used only when n > 1.) We shall usually delete the superscript when the context makes clear the degree or number of subject positions that go with the predicate variable in question. We identify the mbject positions of an atomic wff by means of the individual variables occurring in that wff; and we identify the single predi-

INTRODUCTION

17

cate position of an atomic wff of the form Fn(x1, ... ,Xn) as the position in which Fn occurs in that wff. Thus, while an atomic wff may contain many subject positions, it will contain at most one predicate position. (As a logical constant, the position for the identity sign in an atomic wff is not to be taken as a predicate position, nor in general is the identity sign to be taken as a predicate constant. This does not mean that the identity sign cannot stand for a binary relation in any of the theories of universals to be considered here, however, but only that it need not do so unless certain second order existence conditions are stipulated, as well perhaps as that it cannot do so under the denial of these conditions. Both alternatives will in fact be separately realized in some of the theories to be considered here.) The extended set of atomic wffs for the theories in chapter 4 of this essay will contain not only all members of the minimal set but also all expressions obtained from members of the minimal set by replacing one or more occurrences of individual variables by predicate variables, i.e., it will consist of all expression either of the form (a = b) or of the form Fn(aI, ... an), where a, b, aI, ... ,an are predicate or individual variables. We shall use "a" and "b", with or without numerical subscripts, to refer (in the metalanguage) to both predicate and individual variables. We speak of the subject position occurrences of predicate variables as nominalized ocrnrrences of these variables or as predicate nominalizations, and we note that a nominalized occurrence of a predicate variable is allowed to occur in one or more of the subject positions that go with that predicate variable. (Adding suffixes such as "-ity'', "-ness" or "-hood" to mark nominalized occurrences of predicate variables would be superfluous here, incidentally, since such occurrences are already identified as subject position occurrences. 4 ) We extend this grammar in chapter V and VI by allowing for the nominalization of complex predicates as well, but we shall avoid discussing this extension in these introductory remarks. Pure second-order wffs, accordingly, can be defined inductively as the members of the smallest set K containing all of the pure atomic wffs of the extended set and such that ~cp, (cp ~ rfl), (Vx)cp, (VFn)cp are in K whenever cp, rfl are in K, xis an individual variable, n is a natural number, and Fn is an n-place predicate variable. (We also require that D cp E K if cpE K, when considering modal extensions of the theories to be investigated here. In general, however, unless otherwise indicated by the context in question, we shall assume wffs to be modal free.) We shall use "cp'', "rfl", "x'', to refer (in the metalanguage) to wffs (of what ever sort), and we

18

PREDICATION 1HEORY AND UNIVERSALS

identify the subject and predicate positions of a wff with the subject and predicate positions of the atomic wffs occurring in that wff. By an applied instance of a pure second-order wff