  Hacking shows that, given certain background semantic assumptions (bivalence, valid inference preserves truth), any introduction and elimination rules meeting certain proof-theoretic conditions (subformula property, provability of elimination theorems for Cut, Identity, and Weakening) will uniquely determine a semantics for the constant they govern (Hacking 1979, 311–314). It is in this sense that these rules “fix the meaning” of the constant: “they are such that if strong semantic assumptions of a general kind are made, then the specific semantics of the individual logical constants is thereby determined” (313).