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Nino B. Cocchiarella, Logical Investigations of Predication Theory and The Problem o[ Universals, Humanities Press, Atlantic Highlands, New Jersey, 1986, 266 pp., $50.00.

Universals, in Nino Cocchiarella's sense, are predicables: whatever they are, they can be predicated of subjects. There are many philosophical accounts of the nature of these universals (Cocchiarella discusses various forms of nominalism, conceptualism and realism). His concern, in effect, is formalization. He associates each philosophical account with a formal theory of predication in a language with second-order predicate variables and, sometimes, devices to make subjects out of predicates and complex predicates out of simple ones. He places this work in the tradition of Carnap and Montague, his former teachers. The philosophical discussion is rather cursory; most of the book is devoted to the construction and investigation of formal systems and their semantics, all conducted with the painstaking rigour to be expected from Cocchiarella's previous work (some of which is indeed integrated into the monograph). Technical notions are defined with admirable explicitness; technical claims are proved in gruelling detail. Its readership is therefore restricted to those with some metalogical competence. It is best treated as a handbook of formal theories of predication with some notes on their possible ap- plications in philosophy. Unfortunately, its usefulness as such is diminished by the lack of any index or list of definitions, axioms or theories. It is nevertheless a valuable monograph.

Nominalism, for Cocchiarella, is the conjunction of three theses: ~ (1) the only predicables are predicate expressions; (2) co-extensive predi- cates are interchangeable salva veritate in predication theory; (3) there are only individuals. (1) is a reductive rather than eliminative view of universals: they exist, but are linguistic in nature. It is not assumed that all individuals are concrete individuals; (3) is consistent with the existence of sets (Cocchiarella uses set-theory freely in his treatment of both the syntax and semantics of nominalist theories). (2) will be discussed below. Cocchiarella argues that standard predicative second-order logic is appro- priate for nominalism when the predicate quantifiers are intel:preted substi- tutionally, such quantification being already implicit in the use of sche-

Linguistics and Philosophy 13: 265-271, 1990.

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matic predicate letters in first-order logic. The corresponding comprehension principle is:

(CP!) ( 3F n ) (Vxx) . . . (Vx , ) [F(x l . . . . . x , ) ~-->~]

where all occurrences of predicate variables in qt are free and the distinct individual variables x~ . . . . . x, , but not the n-place predicate variable F, are free in ~. Axioms and rules of inference are given, and a semantics with respect to which they are sound and complete. Since the predicate quantifiers are interpreted substitutionally, they are not presumed to range over all subsets of the domain of individuals; the semantics used in the standard incompleteness proof for second-order logic is not the one intended by nominalism. As Cocchiarella later notes, the nominalist could introduce a hierarchy of levels, with predicate variables at a given level taking as their substitution class the predicate constants definable at the previous level.

Cocchiareila's next section is called 'Nominalism and Modal Logic'. Since extensionalism - thesis (2) of nominalism as he defines it - leads to the complete collapse of modal distinctions by the usual sort of Frege/Church argument, this might appear to be something of a non- subject. However, the idea is to see whether (2) can be weakened. Exten- sionality entails but is not entailed by anti-essentialism, the denial of all non-trivial essentialist theses (i.e., those which discriminate between individuals). Let quasi-nominalism be the conjunction of (1), anti-essen- tialism and (3); it is thus a consequence of nominalism. In effect, Cocchia- rella discusses 'Quasi-Nominalism and Modal Logic'. Unfortunately, he does not make clear his reasons for associating (1) and (3) - the core theses of nominalism as traditionally understood - with either extensionalism or anti-essentialism. On the face of it, a philosopher could consistently com- bine (1) and (3) with some strong essentialist theses. The paradigm of Quine's nominalism may have distorted the presentation.

Cocchiarella goes on to argue in effect that the only notions of possibility and necessity of which quasi-nominalists might make sense are logical ones, on the grounds that non-logical modalities would have to be interpreted in terms of individuals such as propositions or possible states of affairs, which would satisfy non-trivial essentialist theses (pp. 48, 49). It is not explained why quasi-nominalists cannot treat these entities as linguistic or set-theoretic constructs to which they are in any case committed, nor why they cannot take non-logical modal operators as primitive.

What of the relation between quasi-nominalism and logical modalities? Cocchiarella interprets [] as logical truth, which naturally suggests that

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an assignment of values to variables should satisfy Z]~0 in a model just in case it satisfies q, in every model (complications arise when an individual which does not appear in the domain of every model is assigned to a variable). He notes that an essential incompleteness theorem follows, since for any first-order sentence ~b, -D~O is valid on this semantics if and only if ~O is not first-order valid, while [S]qt is valid otherwise; thus a recursively axiomatizable theory sound and complete for the semantics would, per impossibile, yield a decision procedure for first-order predicate logic. It is shown that this incompleteness extends to a Carnapian variant of the semantics on which an assignment satisfies [] 4J in a model just in case it satisfies qJ in every model with the same domain. One could restore completeness by following Kripke and using model structures with restric- ted sets of models, but Cocchiarella argues that this would not be faithful to the intended interpretation of []. Puzzlingly, he takes the incomplete- ness theorems to show the inappropriateness of the previous semantics for (quasi-)nominalism, since the incompleteness "is a consequence of realist assumptions regarding all of the logically possible extensions that might be assigned to predicate expressions (particularly as based upon a potentially infinite domain of discourse)" (p. 51). Why must these assump- tions be realist ones? As already noted, the nominalist may acknowledge sets, and in that case could regard every set as logically capable of being the extension of a predicate. Such a combination of views no longer results in a distinctively nominalist logic, but that is only to emphasize that not all philosophical differences boil down to logical ones.

Conceptualists hold the primary predicables to be concepts; they also deny that there are only individuals, on the grounds that concepts are not individuals but unsaturated cognitive structures realizable in thought (p. 71; little is said about the relevant sense of 'unsaturated'). Two logically independent divergences from nominalism thus seem to be at issue. Schematically, the nominalist and the conceptualist identify universals with saturated features of language and unsaturated features of thought respectively; someone might instead identify them with unsaturated features of language or with saturated features of thought.

Cocchiarella distinguishes between two kinds of conceptualism at an increasing distance from nominalism. Constructive conceptualism, asso- ciated with Konrad Lorenz, treats concept-formation as a process of phylogenetically programmed change in the functioning of the central nervous system, in which each stage is somehow reducible to and predictable from its predecessor. Holistic conceptualism, "associated with Piaget and Vygotsky, denies each of these theses of phylogenetic program- ming, reducibility and predictability. Cocchiarella relates these genetic

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issues to logical ones with the suggestion that the holistic but not the constructive conceptualist leaves room for impredicative concept-forma- tion. The holistic conceptualist holds that "once the human child has reached the stage of adolescence or what Piaget calls formal operational thought, the powers of the intellect are such as to allow for idealized transitions to a limit through the (idealized) closure of a pattern of reflec- tive abstraction that can accommodate impredicative concept-formation" (p. 69), although it is reported that "even the post-adolescents of primitive societies do not appear to have any impredicative concepts as part of their repertoire" (p. 99). In contrast, impredicative concept-formation is said to be inexplicable in terms of phylogenetic programming. These connec- tions are hard to follow. Presumably the holistic conceptualist allows that the stage of formal operational thought is realized by some brain state or other: how could there be an a priori argument that such a state is incapable of being phylogenetically programmed? Moreover, we have so little idea of what an explanation of predicative concept-formation in terms of phylogenetic programming would be like that to rule out the possibility of a similar explanation of impredicative concept-formation seems prema- ture.

Cocchiarella gives the following comprehension principle for construc- tive conceptualism:

(CCP!) (VG1) . . . ( V G , ) ( 3 F n ) ( V x ~ ) . . . (Vxn)[F(x~ . . . . . x,) ~--~O]

where, in addition to the constraints on (CP!), tp is to be free of predicate constants and G1 . . . . . Gi are all the predicate variables in $. (CCP!) is a weakening of (CP!). In effect, it states only the closure of the set of predicable concepts under first-order logical operations. As Cocchiarella remarks, it therefore does not require the existence of any concepts at all; he points out that the constructive conceptualist allows open for- mulae which do not stand for concepts to be meaningful (formulae with predicate quantifiers are an obvious example).

(CCP!) does not seem to fit well with Cocchiarella's further claims about constructive conceptualism. Two problems may be mentioned.

First, Cocchiarella says that an interpreted formula stands for a concept according to constructive conceptualism only if it is biologically possible to form a capacity to identify those individuals which satisfy it (p. 76). It is unclear by what right such a whiff of verificationism has entered the conceptualist's position, but the point to be made here is that (CCP!) seems to allow the construction of concepts which do not obey the verificationist constraint out of those which do. Consider the follow- ing instance:

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(V GE)(3F')(Vx)[ F(x) ,~, (Vy) G(x, y)].

If the domain of individuals is unsurveyable, it might be biologically possible to tell of any given pair of individuals whether they stand in a certain relation without being biologically possible to tell of any given individual whether it bears that relation to at least one member of the domain, rather as in cer ta insys tems 'y is a proof of x' stands for a decidable relation although 'x has a proof' does not stand for a decidable property. Perhaps individual as well as predicate quantifiers should be banned from @ in (CCP!); alternatively, the verificationist constraint should be modified.

Second, Cocchiarella says that constructive conceptualism requires the domain of the predicate quantifiers to be at most countably infinite, whatever the cardinality of the domain of individuals, since the intellect can construct at best a potential infinity of concepts (p. 102). However, consider the following theorem, the universal generalization of an in- stance of (CCP!):

(Vy)(VG2)(3F1)(Vx)[ F(x) ~ G(x, y)].

It allows one to form monadic concepts from dyadic ones by treating some terms of the latter as parameters. Now suppose that the domain of individuals is uncountable and that G is a dyadic, reflexive, anti- symmetric concept, for example, the concept: x is a (not necessarily proper) part of y. It is easy to show that the above theorem will yield distinct concepts F when distinct individuals y are used as parameters, and thus that the domain of predicates in this case is uncountable. The mistake may have been to take countability as the set-theoretic representation of potential infinity. A kind K is potentially infinite (in the sense applicable to the kind concepts we possess) just in case K cannot have infinitely many instances, but for any natural number n K can have at least n instances. The 'cannot ' here means biological impossibility; Cocchiarella says that the concepts we possess are "at best" potentially infinite, presumably because it may not be biologically possible for us to construct a trillion trillion concepts. If all possible members of K can be constructed from a finite number of initial elements by the recursive application of a finite number of operations (as with sequences of marks), the set of all possible instances of K, if one may speak of such a thing, is indeed at most countable. However, in the present case any member of a completed, uncountable domain of individuals can be used as a parameter; since the number of operations to be recursively applied to the finite initial stock of concepts is thus itself uncountable, it is not surprising that the set of all possible instances of the kind so constructed

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is uncountable too. Perhaps in the statement of (CCP!) x~ . . . . . xn should be stipulated to be all the free variables in ~, so that individuals could not be used as parameters (although Cocchiarella himself uses them as such in an application of (CCP!) on p. 102); alternatively, the coun- tability constraint should be dropped. As Cocchiareila points out, the constraint does not affect validity in his semantics, for a version of the Lowenheim-Skolem theorem is provable.

Constructive conceptualism can also be ramified, by iteration of its characteristic process of reflection on concepts. The result is a predi- cative half-way house to holistic conceptualism with its impredicative comprehension principle (CP), which is like (CP!) except that bound predicate variables are allowed in ~.

Cocchiarella provides set-theoretic semantics for both constructive and holistic conceptualism, with corresponding completeness proofs. The semantics is external, in the sense that it is unfaithful to the concep- tualists' intended meaning, since it represents unsaturated entities (concepts) by saturated ones (sets). It is also argued that both theories must reject the logical modalities, since incompleteness theorems would otherwise be forthcoming on the intended semantics - and in such a way as absurdly to commit the conceptualist to uncountably many concepts. However, this commitment arises from the treatment of logical necessity as a kind of quantifier over logically possible worlds (p. 98). It might be argued that this is not the treatment of the logical modalities that the conceptualist intends.

Cocchiarella divides realism into natural and logical varieties. The natural realist holds that universals - properties and relations - are mind-independent parts of the causal order, the logical realist that they belong to an automonous, abstract realm; the latter view may be either Platonist or Fregean, depending on whether the universals are supposed to be saturated or unsaturated. The fundamental thesis of logical realism is said to be that "every class of individuals is the extension of some logically real universal" (p. 109); it does not seem to follow from the assumption that universals transcend concrete existence. Logical realism is thus committed to the strong comprehension principle (CP). In contrast, natural realism has no comprehension principle, since it holds that the existence of a universal can be uncovered only by a posteriori investi- gation.

Some versions of natural realism assert the non-existence of unin- stantiated universals - the thesis of moderate realism:

(MR) ( V F n ) ( 3 X l ) . . . ( 3 x n ) F ( x l . . . . . X n ) .

Others rule out only the existence of uninstantiable universals:

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(~MR) (VF n) O ( 3 x l ) . . . ( 3 x . ) F ( x l , . . . , x . ) .

(There is a parallel difference between co-extensiveness and necessary co-extensiveness as criteria of identity for universals.) The work of a natural realist such as D. M. Armstrong suggests other constraints. Perhaps, for example, one should require xl . . . . . x, in (MR) and (~MR) to be pairwise distinct; how could a natural relation be incapable of holding between distinct terms? It could also be argued that a natural universal should have, or be capable of having, counter-instances as well as instances. In the other direction, Armstrong's Principle of Conjunctive Universals might be used as a limited comprehension principle for natural realism: "If U is a universal, and U' is a distinct universal, and there are particulars which have U and U', then U & U' is a universal" ( A Theory of Universals H: Universals and Scientific Realism, Cambridge University Press, Cambridge, (1978) at p. 170; cp. pp. 30-36).

Natural realism is described as providing the "ontological underpin- nings" of both forms of conceptualism (p. 114). This seems to reflect the socio/biological twist which Cocchiarella has given to conceptualism, witnessed by his choice of Lorenz, Piaget and Vygotsky k as its pro- tagonists, rather than any commitment of conceptualist doctrine as traditionally understood.

Much of Cocchiarella's work on realism is taken up with issues about its interaction with modality: actualism, possibilism and essentialism. Various set-theoretic semantics are developed for the different systems; since they reflect logical realism's insistence that all classes are the extensions of universals and natural realism's almost complete neutrality in this respect, it is not surprising that the two theories turn out to be incomplete and complete respectively for these semantics.

For reasons of space, the concluding chapters - 'On the Logic of Nominalized Predicates and its Philosophical Interpretations', 'Complex Predicates and the h-operator' and 'Two Fregean Semantics for Nominalized Complex Predicates' - will not be discussed. Like the rest of the book, they contain a not easily digested variety of systems and results. In view of this richness, it may be perverse to complain that more attention to recent philosophical work in the area would have widened the range of options that could have been considered. A thin layer of philosophy has been used to gloss a solid work of logic.

University College Oxford OX1 4BH U.K.

TIMOTHY WILLIAMSON