LOGIC AND THE CONCEPT OF ENTAILMENT(1950)

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 logical constant? 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 p is logically true if a contradiction is derivable from not-p with the help of logic alone, without the use of factual premises. A definition which, prima facie, 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.“