  The notion of determination of semantic value in a well-defined semantic framework is, at least, clear—unlike the general notion of determination of sense. However, as Gómez-Torrente points out, by concentrating on the fixation of reference (or semantic value) rather than sense, Hacking opens himself up to an objection not unlike the objection to permutation-invariance approaches we considered above (see also Sainsbury 2001, 369). Consider the quantifier \(\dq{W}\), which means “not for all not …, if all are not male widows, and for all not …, if not all are not male widows” (Gómez-Torrente 2002, 29). (It is important here that \(\dq{W}\) is a primitive sign of the language, not one introduced by a definition in terms of \(\dq{\forall}\), \(\dq{\neg}\), “male”, and “widow”.) Since there are no male widows, \(\dq{W}\) has the same semantic value as our ordinary quantifier \(\dq{\exists}\). (As above, we can think of the semantic value of a quantifier as a function from sets of assignments to sets of assignments.) Now let \(R\) be the standard introduction and elimination rules for \(\dq{\exists}\), and let \(R'\) be the result of substituting \(\dq{W}\) for \(\dq{\exists}\) in these rules. Clearly, \(R'\) is no less “purely inferential” than \(R\). And if \(R\) fixes a semantic value for \(\dq{\exists}\), then \(R'\) fixes a semantic value—the very same semantic value—for \(\dq{W}\). So if logical constants are expressions whose semantic values can be fixed by means of purely inferential introduction and elimination rules, \(\dq{W}\) counts as a logical constant if and only if \(\dq{\exists}\) does.