Saturday, June 28, 2008

Quantifiers and the Grammatical Definition of Logical Truth

The grammatical definition of logical truth, discussed here, is probably inadequate for all sorts of interesting languages, including English. The definition, from Quine's Philosophy of Logic, is this:
"a logical truth is a truth that cannot be turned false by substituting for lexicon. When for its lexical elements we substitute any other strings belonging to the same grammatical categories, the sentence is true." (2nd ed., 58)

For Quine - and I think he's right on this count - we treat a class of words as a grammatical category, as opposed to a class of particles yielding new grammatical constructions, just in case the category is big enough. For instance, in a language with lots of intransitive verbs, we treat those as comprising a grammatical category. If an L-structure has an infinite stock of variables, we treat variables (or argument-terms, more generally) as a grammatical category; if it has three variables, we might do well to treat each as a particle.

I take it that English has an infinite - or at least a very large - stock of quantifiers. This is because I think that, in English, the quantifiers translated by "(E_)" and "(A_)" in first-order logic belong to the same grammatical category as expressions such as "There are many", "There are ten", "There are one million", and "There are innumerable". If the literature on quantifiers in natural language says otherwise, please correct me. Also, there are usually an infinite number of quantifier-expressions in languages that support generalized quantification, right? Anyway, if quantifiers all belong to the same grammatical category, and we assume the grammatical definition of logical truth, then I can't think of a single logical truth containing a quantifier. For instance, "If Steve and Janice are cats, then there are some cats" would fail to be a logical truth, since "If Steve and Janice are cats, then there are innumerable cats" - gotten by "substituting for lexicon" - is false.

I imagine the Quinean response to all of this would be to say that, given a prior commitment to standard FOL, we should translate "There are n Fs" as "The class of all Fs has cardinality n." But it seems obvious to me that the average English speaker does not, as a matter of linguistic anthropology, commit herself to the existence of the class of all Fs in uttering "There are n Fs." The nominalist cannot properly respond, "No, there is no such thing as the set of all Fs." And besides, what if we substitute "self-member" for "F"?

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