Identicals by Logic

Andrea Salvador (Universitá della Svizzera Italiana)

By uttering a declarative sentence, we say things are a certain way. Also, we can say the same thing by uttering different sentences. Following a recent custom, I call “generalised identities” (GIs) the statements which say that two statements say the same thing in the previous sense. The present paper tackles the question of what GIs are true by logic, focusing on propositional logic.  Only a few recent proposals by Dorr 2016, Correia 2016, Brast-McKie 2021 and Elgin 2023 target the propositional logic of GI, with the last three using the framework of truthmaker semantics (TMS). My proposal differs from theirs in two ways. The first is that I accept the idempotence of conjunction and disjunction, contra Elgin and Dorr, and reject both distribution axioms for conjunction and disjunction, contra Correia. In fact, I argue that one should not accept the distributivity of conjunction over disjunction but reject that of disjunction over conjunction, as Correia does. Second, all the last three proposals admit in their language only GIs which do not contain GIs as proper sub-formulas. Thus, their logics do not validate certain principles for GI that should be valid just because they cannot be stated in their language. Moreover, their semantics lack exact verifiers and falsifiers for GIs. Since, to my knowledge, there is no available TMS with exact verification and falsification clauses for GIs, my paper fills this lacuna by providing such clauses. Here is my proposal. Let us include in the state spaces some states which identify some propositions with other propositions and also states that distinguish certain propositions from other propositions. If s identifies P and Q, it is an identity-state for them, while if it distinguishes them, it is a difference-state for them. Models are based on state spaces with a domain of propositions closed under fusion and include a set of possible states and possible worlds, which are just special possible states, such that every possible state is part of a possible world. Furthermore, I put some constraints on the models to ensure that they provide adequate identitystates and difference-states and enough of them. By letting “A ≡ B” mean that one would say the same thing by uttering “A” or uttering “B” and by adding two functions giving us the verifiers and falsifiers of atoms and thereby of complex formulas, we can finally give exact verification and falsification clauses for GIs. Say that a state s exactly verifies A ≡ B iff s identifies the verifiers of A with those of B and the falsifiers of A with those of B, while s exactly falsifies A ≡ B iff either s distinguishes the verifiers of A from those of B, or it distinguishes the falsifiers of A from those of B.  Provided a definition of classical logical consequence, I show that the resulting logic captures the correct principles for GI. For example, it avoids my previous objections to Dorr, Elgin, Brast-McKie and Correia. Most notably, I show that GIs entail necessarily true biconditionals, without being logically equivalent to them, and that the statements identical by logic are those having the same exact verifiers and falsifiers in every model—as one would expect.