Quantification over the real numbers

✍ Scribed by Arthur I. Fine

• Publisher: Springer Netherlands
• Year: 1968
• Tongue: English
• Weight: 390 KB
• Volume: 19
• Category: Article
• ISSN: 0031-8116
• DOI: 10.1007/bf00463056

✦ Synopsis

IN RECENT publications Ruth Marcus has gone back to Russell to revive a way of reading quantified formulas? Where F is a formula with the single free variable 'x,' the suggestion is to read '(x)F' as 'every substitution instance of F is true' and to read '(3x)F' as 'at least one substitution instance of F is true.' While unpleasant to the ears, perhaps, Marcus points out that this way of rendering quantified formulas obviates some of the difficulties which have been raised by Strawson and, in the case of quantified modal logics, it provides an easy way of seeing that Ouine's charge of essentialism is not correct. In subsequent discussion s Quine has not accepted Marcus' suggestion as just another way of reading quantified formulas but has urged, rather, that this is a "new interpretation." According to Quine, it is indeed an interpretation which won't do and, among the reasons for that is the suggestion that it won't do for "the example of real numbers."

Let us see what is involved. We can suppose that F is a one-place predicate, that N is the class of names which may meaningfully be substituted in F. Call '(N:F)' the class of statements 'Fn' for n in N and let O be the class of objects to which the property (I, expressed by F can be applied. (We assume that each predicate "expresses" a property, but not necessarily the converse.) Consider the following two assertions:

( 1 ) All the statements in (N:F) are true.

(2) All the objects in O have property ~.