Metaphysics and the New Logic by Warner Arms WickReview by: Carl G. HempelThe Journal of Symbolic Logic, Vol. 8, No. 1 (Mar., 1943), pp. 29-30Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2267981 .

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REVIEWS 29

particulier, la logistique voit ses advcrsaires se grouper autour du formalism de Peano et de Hilbert, et de l'intuitionnisme de Brouwer et de Weyl" is difficult to understand. It contradicts both the author's own previous mention of Peano and the prevalent estimate of Hilbert and his school as major contributors to mathematical logic. Even the Brouwerians are adversaries of logistic only in the sense that they regard its role as secondary to that of mathematical intuition; they have made their share of technical contributions to the subject.

The author contends that mathematical logic has some special connection with empiricism or nominalism in philosophy, and that formal modifications may be required in order to free it of its nominalistic implications. Apparently the system of Principia mathematics is meant, since, besides precursors, no other system is given substantial mention (a brief refer- ence to modal logics and to semantics suggests that these are considered to be departures from logistic proper). Whether the contention is meant in a purely historical sense or other- wise is not clear. It seems to the reviewer to be in any case hardly tenable.

ALONZO CHURCH

WARNER ARMS WICK. Metaphysics and the new logic. The University of Chicago Press, Chicago 1942, xiii + 202 pp.

The objective of this book is to examine the metaphysical foundations of logical empiri- cism. The author bases his inquiry on a distinction suggested by R. McKeon, of two funda- mental types of metaphysical systems: the holoscopic, illustrated by Platonic, and the meroscopic, illustrated by Aristotelian metaphysics. The former type is characterized in particular by the view that the first principles of knowledge and discourse can be found- though only in approximation-through a repeated reflexive application of scientific inquiry or discourse to itself, and that all knowledge is to some extent determined by the conceptual framework used in the inquiry that leads to it. The meroscopic view, on the other hand, is said to deny the possibility of a discursive justification of first principles of knowledge, leaving them to an intuition which is viewed as a necessary condition of all rational criticism.

The author argues that logical empiricism-he refers especially to the views of R. Carnap and C. W. Morris-is a holoscopic doctrine; to substantiate this contention he points to logical empiricism's interpretation of philosophy as logical analysis of language, and to the "conventionalistic" character of Carnap's "principle of tolerance" in logic; and he further attempts a detailed interpretation of some principles of Platonic metaphysics in terms of modern syntactical, semantical, and pragmatical theory. In this manner, he claims not only to be able to exhibit the holoscopic character of modern semiotic versions of philosophical analysis, but also to prove that at least some historically metaphysical doctrines are meaning- ful in the sense of the criteria of logical empiricism. In a similar fashion, the author tries to show that much of Aristotelian metaphysics can be meaningfully interpreted in terms of semiotic theory.

Dr. Wick's argument suffers, however, from at least two serious shortcomings. Fires, his presentation of the recent versions of syntax, semantics, and pragmatics contains various inaccuracies and even definite falsehoods. Thus, e.g., he fails clearly and consistently to distinguish between the concepts of sentence expressible in a language, true sentence of a language, and confirmed or scientifically accepted sentence of a language. This leads him to incorrect assertions such as that "roughly, a sentence is semantically true in SI if it desig- nates what is desig; -ted by an accepted sentence of S2" (p. 80), and "if my metalanguage contains a sentence to the effect that squares are round, then, as measured by it, 'Ein Viereck ist rund' is semantically true in German" (p. 81) .-And secondly, the author's way of inter- preting metaphysical views as semiotical principles often requires an unbearable stretching of the meaning and connotation of the original formulation. Thuns, the holoscopic idea of an Absolute, which is unattainable by a finite series of reflexive elucidations of knowledge by itself, but which is, as it were, the ideal limit of this process, is said to have its logical re- incarnation in the theory of transfinite classes, and in the idea of a language containing vari- ables of transfinite levels (pp. 70, 71). Similarly, 1lato's Theatetus would pem to have anticipated some of G6clel's results (p. 154).-With this degree of liberality, any meta- physical formulation whatsoever appears to be amenable to an intelligible semiotic "inter- pretation," but it seems more than doubtful whether this approach can offer any help either

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30 REVIEWS

in doing justice to the historical metaphysical system or in clarifying the foundations of logical analysis. CARL G. HEMPEL

FREDERIC B. FrrCH. A basic logic. The journal of symbolic logic, vol. 7 (1942), pp. 105-114. See erratum, ibid., p. iv.

Identifying as a system any recursively specifiable class of propositions, Professor Fitch construes 'the system K is definable in the system L' as meaning that every true proposition (and no false proposition) of the form

----- is a member of K

is itself a member of L. The system of logic which this paper is concerned with formulating is presumably basic in the sense that every system of logic is definable in it. Since in the author's judgment all systems (as distinct from the calculi which express them) are com- binatory, consideration is restricted to combinatory systems from the outset.

The system which Professor Fitch offers as basic may be formulated in a calculus K, de- scribable as follows. The sole primitive operation of K is juxtaposition within parentheses, interpreted as application of function to argument. The undefined signs of K consist of an unspecified and infinite class of non-overlapping expressions plus ten specified combinators. Five of the latter express identity, class-membership, propositional conjunction, proposi- tional alternation, and the proper ancestral of a relation respectively. Two others are anal- ogous in meaning to Curry's 'B', and the three remaining to Curry's 'W' and 'V', and Rosser's 'T' respectively. As axioms K contains all identities of the form '(=ab)' where 'a' and 'b'

are one and the same expression. Each of the nine other specified primitive combinators is governed by an appropriate transformation rule, or rule of inference. K contains, via con- textual definitions, class-abstracts and relational abstracts, and avoids Russell's paradox by omitting negation.

The system formulated in K is not proved basic, but the author proffers strong arguments in support of his belief lhatit is basic. These turn upon his concept of representation. Where U is the class of all expressions in K's vocabulary, where F and G are arbitrary sub- classes of Up and where 'g' and 'a' are members of U, 'g' is said to represent G in F if every 'a' satisfies this condition:

'a' e G= '(ga)' e F.

(Representation of other kinds of syntactical classes and relations is also defined by the author.) In virtue of one of the nine transformation rules mentioned above, '(ga)' is equiva- lent to 'a e g'. Accordingly, where G and F are calculi formulating systems and 'g' represents G in F, '' may be construed as a name of G, and the system formulated in G will be definable in the system formulated in F. If every system has a combinatory formulation represented in F, F itself therefore formulates a basic system.

The author's reasons for believing that every calculus whose vocabulary is included in U is represented in K may be roughly abridged as follows. Combinatory versions of the func-

tional calculus of first order, the functional calculus of order w (simple theory of types), and

the ramified Principia, are all represented in K. Such calculi, with slight extensions, can

serve as meta-languages for almost any conceivable combinatory calculus. But it can be

proved that any combinatory calculus is represented in K if its metalanguage is repre- sented in K. Therefore, there is good reason to believe that any U-calculus is represented in K.

If in addition we assume that U is sufficiently rich in means of expression to contain a

formulation of every logical system and that K really does formulate a system, then the

following four propositions are forthcoming in view of certain previously proved theorems. (1) The decision problem is insoluble for the system formulated in K. (2) Every system is definable in this formuland of K, i.e., K formulates a basic logic. (3) Every class defin-

able in any system of logic is definable in the formuland of K. (4) The class of propositions which are not members of the formuland of K is not definable in any system of logic.

GEORGE D. W. BERRY

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