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The main thesis of this paper will be best approached by raising a question of the philosophy of logic (or “meta-logic”) which most practicing logicians neglect to raise, presumably for the same reason that most mathematicians neglect to raise philosophical questions about mathematics: what is a ? The problem of defining what is meant by a “logical constant” (logical term, logical sign) is crucial for a satisfactory theory of logical truth, since it seems impossible to analyze the latter concept without using the concept of a logical constant. Definitions of logical truth which do not use this concept are easily shown to be unsatisfactory. If, for example, we define a logical truth as a statement which is true by the very meanings of its terms, we are either defining a concept of psychology, not of logic, or else the definition is implicitly circular. The former is the case if we interpret the definition to say that anybody who understands what the constituent terms of the statement mean (who understands, in other words, what proposition the sentence is used to expressed) will assent to it; and the definition is circular if it tells us that the statement will turn out to be derivable from logic alone once the definitions of its terms are supplied. Again, it is implicitly circular to define a logically true statement as one that cannot be denied without self-contradiction. For, surely, we want to say that is logically true if a contradiction is derivable from not- with the help of logic alone, without the use of factual premises. A definition which, , is free from the vice of circularity is the following one: a logical truth is a true statement which either contains only logical constants (besides variables) or is derivable from such a statement by substitution (this is essentially the definition preferred by Quine).1 It remains to be seen, however, whether this appearance will stand the test of analysis. The crucial question is obviously whether we could construct an independent definition of “logical constant.“

Ever since C. I. Lewis offered the concept of “strict implication,” defined explicitly in terms of logical possibility (−3 ≡ ~◊(.~)) and implicitly by the axioms of his system of strict implication, as corresponding to what is ordinarily meant by “deducibility” or “entailment,” there have been analytic philosophers who denied this correspondence. They denied it specifically because of the paradoxes of strict implication: that a necessary proposition is strictly implied by any proposition and an impossible proposition strictly implies any proposition. These theorems, it is maintained, do not hold for the logical relation ordinarily associated, both in science and in conversational language, with the word “entailment.” It is my aim in this paper to show that it is extremely difficult, if not downright hopeless, to maintain this distinction. I shall refer specifically to a subtle paper by C. Lewy (Lewy 1950), which deals with the intriguing problem of modal iteration, and which emphatically endorses the distinction here to be scrutinized.

A science can be in a highly advanced state, even though its logical foundations are far from being clarified. Mechanics, for example, reached a stage of astounding perfection by the end of the 18 century, mainly through the genius of Galileo and Newton, although much room was left for controversy about the meanings of its fundamental concepts: length, simultaneity, mass, and force. Even the meaning of the simple law of inertia remained controversial right up to Einstein’s “unification” of inertia and gravitation. Similarly, mathematics was not prevented from reaching breathtaking heights of perfection by the (dispute about foundations) which began in the 19 century and still continues.

In , an extensional system embodying the theory of types, Peano’s postulate qdistinct (natural) numbers have distinct (immediate) successors” is not formally derivable from purely logical axioms. The axiom of infinity (“the number of individuals is infinite”) is required for the proof that there is no finite cardinal such that equals +1. Russell argued as follows:1 if only n individuals existed, then the number + 1, being defined as the class of all classes that have + 1 members (or that would have exactly members if exactly one element were withdrawn from them), would be equal to the null class; for in that case no classes with + 1 members would exist. But by parity of reasoning, the successor of +1 would also be equal to the null class; therefore and +1, which on the hypothesis made are distinct numbers, would have the same successor. The usual reaction to this argument is that, without abandoning Russell’s conception of numbers as classes of similar classes, we fortunately do not need to postulate the axiom of infinity after all. For there are other ways of solving the logical paradoxes besides the theory of types, and once our constructive efforts are unimpeded by the latter, we can construct an infinite sequence of abstract entities without presupposing the existence of a single concrete individual: the null class, the unit class whose only member is the null class, the class whose members are the foregoing two classes, and so on. And once we have an infinite set of such abstract, though typically impure, entities, we can rest assured that no natural number will collapse into the null class. This is the approach of set theory, where such ghostly classes as the one just mentioned can be postulated to exist provided their definitions do not give rise to contradiction. However, I would like to re-examine Russell’s argument in order to see whether it isperhaps possible to get rid of the axiom of infinity without abandoning the (simple) theory of types.

In reference to Mr.E.J. Nelson’s recent article on “Contradiction and the Presupposition of Existence” (Nelson 1946), I want to comment upon a fundamental assumption involved in the author’s arguments which I find highly questionable. The following is the paradox for the solution of which Mr. Nelson elaborates a distinction between “the necessary conditions of the existence of a proposition” and “the necessary conditions of its truth (exclusive of its existence)

Most contemporary logicians seem to be in agreement that Russell’s hierarchy of orders of functions and of orders of propositions, dictated by the vicious-circle principle “whatever is defined in terms of a totality cannot be significantly said to belong to that totality,” is unnecessary. For, so the argument goes, the simple theory of types suffices for the solution of the paradoxes; the rest of the paradoxes, however, which Russell proposed to solve (or avoid, whichever term be more suitable) at one stroke by his comprehensive vicious-circle principle are paradoxes which can be overcome by due observance of distinctions of levels of language. It is, of course, understandable that logicians would eagerly embrace the Tarski-Ramsey-Carnap method of solving the nonlogical paradoxes (liar paradox, Grelling paradox, Berry’s paradox, etc.), since the vicious-circle principle led to the embarrassing dilemma “either accept the axiom of reducibility or reject large portions of classical mathematics.”

The principle of verifiability is that famous criterion of propositional significance whose discussion, I venture to guess, has occupied more space in the publications of contemporary analytic philosophers than discussion of any other topic. It is not the purpose of this discussion to add a little more water to the ocean of literature on the precise meaning or formulation and the general implications of this principle, which is often referred to as the very soul of the movement called variously “logical positivism” and “logical empiricism.” A comprehensive review of this kind has, indeed, been made unnecessary by C. G. Hempel’s excellent contribution to a recent issue of dedicatedto the theme “logical empiricism.”

In his highly significant and provocative book, , Gilbert Ryle undertakes to show that the Cartesian theory of two worlds, the physical world characterized by publicity and the mental world characterized by privacy, inaccessibility to all but one, is a “myth” created by misunderstandings of language. His method is an excellent example of theWittgensteinian method of diagnosing the origin of puzzling philosophical theories as pointless, confusing departures from ordinary language. If, for example, a man puzzles how on earth it is possible ever to verify a proposition about the future since, after all, one cannot observe an event that has not yet occurred, his puzzle is of the kind which can be effectively dissolved by the Wittgensteinian treatment.

I understand the business of the philosophy of science to be painstakingly careful analysis of concepts, principles, and methods used in science. This broad statement fails, of course, to differentiate such analysis of concepts as inevitably occurs in a developed science itself (e.g. analysis of the concepts “simultaneity,” “absolute motion,” “energy,” etc. in physics) from such analysis as is more likely to be the professional concern of the philosopher of science . The most natural object of distinctively philosophical analysis concerned with science would seem to be the very activity, or class of activities, defining science in general, rather than some one specific science. As it goes without saying that one such activity characteristic of science is , the analysis of the concept of predictability is a vital task of the philosophy of science. Ability to predict is a virtue marking a good scientist; ability toanalyze clearly the concept of predictability is a virtue marking a good philosopher of science.

Since the time when Carnap published his classical contribution to the analysis of scientific language, “Testability and Meaning” (Carnap 1937), it has come to be universally recognized by competent philosophers of science that the concept of is inadequate for the analysis of specifications of meaning going on in science. Instead the term has been incorporated into the essential terminological furniture of the meta-language in which scientific procedures are described. In Carnap 1937, Carnap initially explained the need for reduction sentences in connection with the problem ofdefining dispositional predicates.