Truth As a Logical Constant, With an Application to the Principle of Excluded Middle

Abstract

What is it that is susceptible of truth or falsity? The most frequently proffered candidate is the proposition. Others not infrequent are statement, assertion, belief. The question, as well as the proposed answers, shrouds an ambiguity which is, from the point of view of the theory of truth, fatal. That the ambiguity (hereunder to be unravelled) should have persisted in the philosophy of logic into recent times is rather surprising since, as I shall try to show, all the resources for instituting the required distinctions with the maximum of formal clarity are available in the symbolism of modern truth-functional logic. I begin with a criticism of the customary definition of truth function. It is usual to find the following kind of definition in logical texts: A truth function is a statement whose truth or falsity depends solely on the truth value of its components. The objection to this is simply that a truth function is not a statement and cannot, in itself, be spoken of as true or as false. Indeed, the functionality of truth functions, as of any type of function whatever, consists specifically in the fact that they are devoid of value-truth value, in this case. Of course, if a truth value is assigned to the components of a truth function, a value for the whole function is automatically determined. But the fact is, and this will emerge more clearly in due course, that there is a difference of considerable theoretical significance between assigning a truth value to a component of a truth function and remarking of a statement that it is true or that it is false. It is of interest to recall that in Frege's symbolism, a special sign was employed to mark the functionality of the atomic variables of truth functional logic. That symbol is the horizontal line, which, in the Begriffsschrift period, he called the content stroke and later simply the horizontal. The sign 'h-' which is often called the assertion sign of Frege ought rather to be called the assertion sign of Russell and Whitehead (see Principia Mathematica (Cambridge, 1964), Introduction, p. 8). In the context of Frege's symbolism, the sign 'h-' stands for a combination of the assertion sign proper, represented by the vertical line, and the content stroke, represented by the horizontal line. What corresponds to the function 'p' is, in Frege, not a simple letter such as 'A', but '-A' which, as it were, carries its functionality on its face. A philosophically urgent question now arises: What is the logical, or if