Quinean indeterminacy and forcing

Abstract

Levin uncontroversially takes Quine's thesis of the indeterminacy of trans? lation to be that 'there can be two translations 5" and S" of a sentence S, both faithful to all data about the occasions of use of S, such that S' is obviously inequivalent to S"" (p. 25). He goes on to outline the relevant features of Cohen's notion of forcing, an implication-like relation holding between finite consistent sets of sentences ('conditions') and sentences of ZF. (His special interest in an example of indeterminacy not involving inscrutability of reference requires him to say a little more than this about forcing, but further specification here would be irrelevant to the simple points I want to make.) The forcing relation can be defined (via G?del numbering) by a two-place predicate F(c, p) of ZF itself, but it suits Levin's purpose to deal in terms of ZFF, the system which results from adding the predicate F as a new primitive to the vocabulary of ZF, and the definition of forcing as a class of axioms. Since the only primitive predicate of ZF is 6, the only primitive predicates of ZFF are g and F, so that any trans? lation of e and F determines a translation of every sentence of ZFF via the recursive formation rules for wffs of ZF.