Charles Sayward

Peter Geach proposed a substitutional construal of quantification over thirty years ago. It is not standardly substitutional since it is not tied to those substitution instances currently available to us; rather, it is pegged to possible substitution instances. We argue that (i) quantification over the real numbers can be construed substitutionally following Geach's idea; (ii) a price to be paid, if it is that, is intuitionism; (iii) quantification, thus conceived, does not in itself relieve us of ontological commitment to real numbers.

This paper is a discussion of Frege's maxim that it is only in the context of a sentence that a word has a meaning. Quine reads the maxim as saying that the sentence is the fundamental unit of significance. Dummett rejects this as a truism. But it is not a truism since it stands in opposition to a conception of meaning held by John Locke and others. The maxim denies that a word has a sense independently of any sentence in which it occurs. Dummett says this denial is inconsistent with the fact that people understand sentences they have never heard before. The maxim is defended against this attack.

Hilary Putnam suggests that the essence of the realist conception of mathematics is that the statements of mathematics are objective so that the true ones are objectively true. An argument for mathematical realism, thus conceived, is implicit in Putnam's writing. The first premise is that within currently accepted science there are objective truths. Next is the premise that some of these statements logically imply statements of pure mathematics. The conclusion drawn is that some statements of pure mathematics are objectively true. A key principle assumed is that if one statement logically implies a second, then if the first is objectively true so is the second. A question about this principle is raised and answered. The problem with the argument is with the second premise.