In fact, truths of the form ’A&B implies B’ are not likely to depend on the above sentential scheme, but on the rules governing the use of the word ‘and’. These rules are likely to be semantic rules linking ‘and’ with ‘combine’,’join’,and ‘put together’,that is with the conjunctive meaning of ‘A’ combined with those of ‘B’.
Quine’s objection is therefore dismissed in that it does not apply to natural languages, and imposes an overly restrctive role on general conventions. On the other hand he is right to say that we will not find all the truths of mathematics and logic represented literally as linguistic rules and conventions.
Although Quine is critical of conventionalism in logic, he regards its potential as a philosophy of mathematics quite differently.
For set theory the linguistic doctrine has seemed less empty; in set theory, moreover, convention in quite the ordinary sense seems to be pretty much what goes on. Conventionalism has a serious claim to attention in the philosophy of mathematics, if only because of set theory.
(Quine,1966,page 108)
Conventionalism offers the beginnings of a descriptive account of the nature of mathematics, formulated in terms of its linguistic. It accommodates a fallibilist view of mathematics, and may account for both the objectivity of mathematical knowledge, through our necessary acceptance of linguistic rules, and for at least part of its genesis, via the acquisition of language. Since language connccts mathematics with other areas of knowledge, conventionalism has the potential to account for the applications of mathematics. Thus conventionalism is not refuted, and indeed may satisfy many of the adequacy criteria proposed earlier. Conventionalism is discussed