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## Towards a Proof-Theoretic Semantics of Equalities
## Frege's QuestionGottlob Frege opened his seminal paper distinguish between Modern As appealing as this view might be, there are (at least) two problems with it. First, it comes with a concept of rigid designators. Second, it is not applicable to mathematics, because mathematical equations hold equally in every possible world. While many approaches try to attack the problem from a semantic perspective, here we would like to provide a syntactic account, which takes up Frege's original question of the (epistemic2) difference of ## Equality Versus IdentityTo set the stage for the further discussion we would like to assume that we always use a first-order language for which we may have some non-logical axioms, and a fixed structure with universe A in which this language is interpreted3 by an interpretation function (Gothic) ones for elements of the structure. Equality is understood as the relation In this setting we can recast Frege's first observation that “if we were to regard equality as a relation between that which the names ' would thereby be expressed of a thing to itself, and indeed one in which each thing stands to itself but to no other thing.” [5, p. 157]. In our terminology we may say that we are not concerned with the semantical identity relation a ≡ a, but with the syntactical equality relation It is standard to axiomatize equality in first-order logic as a universal congruence relation, i.e., an equivalence relation compatible with all operations (functions and relations). As such, it mimics on the syntactic level just the properties which identity exhibits on the semantic level. But the domain of identity is simply A × A, and the elements of A are unique in the sense that a ≡ a, but a /≡ b for two elements a, b ∈ A. On the syntactic side, however, the equality relation is defined for terms, and, clearly, two terms, though being interpreted by the same object a, may well be different. Frege introduced the |

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