Modern Logic: From Frege to Gödel: Whitehead and Russell

views updated

MODERN LOGIC: FROM FREGE TO GÖDEL: WHITEHEAD AND RUSSELL

In The Principles of Mathematics, published in 1903, Bertrand Russell (1872–1970) set out to establish the logicist view that "all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of logical principles" (2nd ed., p. xv) and also to explain "the fundamental concepts which mathematics accepts as indefinable" (ibid.). In the Principles this program is pursued with minimal recourse to symbolism, the systematic formal presentation being reserved for a proposed second volume. What in fact appeared as the sequel was the classic Principia Mathematica (3 vols., Cambridge, U.K., 1910–1913), written in collaboration with Alfred North Whitehead. The subject matter of Principia Mathematica considerably overlaps that covered by Frege in his Grundgesetze der Arithmetik, a work to which the authors acknowledge their chief debt on questions of logical analysis; in some respects, such as the demarcation between logical and metalogical theses, Principia Mathematica falls short of the standards of rigor observed in Frege's masterpiece. The symbolism adopted in Principia Mathematica derives largely from Peano, and the development of arithmetic and the theory of series is based on the work of Cantor.

We shall concentrate on the most important feature distinguishing Principia Mathematica from Frege's work, the attempt to avoid the contradictions which Russell found implicit in the fifth axiom of the Grundgesetze. This axiom licensed the transition from a concept to its extension and from an extension to the concept, a transition that appears to do no more than give formal expression to a platitude. For instance, the proposition "Stravinsky is a member of the class of composers" appears to be no more than a circumlocution for "Stravinsky is a composer." In general, it would seem reasonable to lay down as a law that x is a member of the class of ϕ 's if and only if x is ϕ —in Russellian notation, x εz̄ (ϕz ). ≡ .ϕx. But despite its platitudinous appearance, this principle turns out to harbor a contradiction, since corresponding to the concept "is not a member of itself" we have the class of all such things—that is, the class of all classes which are not members of themselves—and if we now ask whether this class is or is not a member of itself, we find that either way a contradiction arises: If it is a member of itself, then it satisfies the defining condition of such members, so it is not a member of itself, and if it is not a member of itself, it belongs to the class of such classes and so is a member of itself.

This contradiction was noted by Russell in 1901, and in subsequent years finding ways to avoid it formed one of his major concerns. His final analysis, incorporated into Principia Mathematica, attributed the contradiction, along with a number of analogous paradoxes, to a mode of reasoning involving a vicious circle, a circle that arises when we postulate a collection of objects containing members definable only by means of the collection as a whole. Russell regarded such collections as illegitimate totalities, to be avoided by observing his "vicious-circle" principle, "Whatever involves all of a collection must not be one of the collection." Appealing to this principle, Russell claimed that the values of a function cannot contain terms definable only by means of the function, and in place of an indiscriminate application of functions to arbitrary arguments he defined an ascending hierarchy of types, beginning with individuals and progressing through functions of individuals, functions of functions of individuals, and so forth, the only arguments which a function can significantly take being those of the immediately preceding type. In particular, a class cannot significantly be taken as an argument to its defining function, and the derivation of Russell's paradox is accordingly obstructed by ruling out both "x ε x " and its negation as ill-formed.

Apart from enabling us to block the derivation of paradoxes, Russell claimed, the theory of types based on the vicious-circle principle has a certain consonance with common sense. However, the principle itself (in the various nonequivalent forms given by Russell) can be challenged on the ground that it rules out circular procedures which are in no way vicious.

If the vicious-circle principle is rejected, it is natural to regard Russell's paradox as no more than a straightforward contradiction, the absurdities resulting from the abstraction schema (∃x )(y )(y ε x ≡ ϕ (y )) being no different in kind and requiring no different an explanation from those yielded by (∃x )(y )(Fyx ≡ ϕ (y )), where the membership relation is replaced by an arbitrary dyadic predicate. On this view the problem of finding consistent instances of the abstraction schema reduces to the analogous problem for the uninterpreted version, but although such an approach has its merits, it loses sight of an important feature of the system which the vicious-circle principle shapes via the theory of types. That is, the form of theory which the principle determines conforms to a natural conception of classes according to which they are, or at least could be, generated by a step-by-step procedure, the superstructure of classes of classes of classes, and so on, resting ultimately on the initial elements of lowest type. On the other hand, although it is natural to conceive of a domain of classes as initially secured by such a procedure, it would seem equally natural to relax this constructivist approach to the extent of allowing the specification of particular classes in the domain to proceed by characterizations in terms of the given totality, provided only that the consequent reflexivity does not embody a contradiction.