Mates’ Puzzle 1 A Note on the Relationship Between Mates’ Puzzle and Frege’s Puzzle Marc A. Moffett University of Colorado, Boulder Abstract In this note I argue that, relative to certain largely uncontroversial background conditions, any instance of Mates’ Puzzle is equivalent to some instance of Frege’s Puzzle. If correct, this result is surprising. For, barring the radical move of rejecting the possibility of synonymous expressions in a language tout court, it shows that there is no strictly lexical solution to at least some instances of Frege’s Puzzle. This forces the hand of theorists who wish to provide a semantic (rather than pragmatic) solution to Frege’s Puzzle. The only option open will be modify in one way or another the standard formulation of semantic compositionality. 1. Introduction. In this note I will argue that, relative to certain largely uncontroversial background assumptions, some instance of Frege’s Puzzle concerning the substitution of co- referential singular terms can be derived from any instance of Mates’ Puzzle concerning the unrestricted substitution of synonymous expressions. 1 Conversely, it will be clear that the argument can equally well be run in reverse so that any instance of Mates’ Puzzle may be derived from some instance of Frege’s Puzzle. Thus, relative to the indicated background conditions, any instance of Mates’ Puzzle is equivalent to some instance of Frege’s Puzzle. If correct, this result is nontrivial. For, while a great deal of philosophical work has gone into finding a solution to Frege’s Puzzle, many of the resulting theories fail to generalize 1 Frege observed that redundant identity statements involving singular terms (i.e., statements of the form ‘a = a’) have different doxastic properties than true non-redundant identity statements (i.e., statements of the form ‘a = b’, where ‘a’ and ‘b’ are co-referential singular terms). In particular, while the former are trivial truths of logic (and, hence, almost universally believed), the latter are typically nontrivial, perhaps even expressing empirically significant discoveries (and, hence, frequently disbelieved). Frege’s insights may be extended. For it follows from these considerations that co-referential terms will not generally be substitutable salve veritate into contexts reporting beliefs. That is, even though the sentence ‘x believes that a = a’ is true, the sentence ‘x believes that a = b’ may not be true. As is well known, Frege proposed to solve this puzzle by positing a distinct kind of semantic value for singular terms. According to Frege, even though ‘a’ and ‘b’ both refer to the same object, they nevertheless differ in their sense (Sinn). The sense of a term may be thought of as a descriptive concept or mode-of-presentation of the referent. Thus, according to Frege, the failure of co-referential terms to substitute salve veritate is accounted for by