Possible-Translations Semantics

This text aims at providing a bird’s eye view of possible-translations semantics ([10, 24]), defined, developed and illustrated as a very com-prehensive formalism for obtaining or for representing semantics for all sorts of logics. With that tool, a wide class of complex logics will very naturally turn out to be (de)composable by way of some suitable com-bination of simpler logics. Several examples will be mentioned, and some related special cases of possible-translations semantics, among which are society semantics and non-deterministic semantics, will also be surveyed. 1 Logics, translations, possible-translations Let a logic L be a structure of the form 〈S, 〉, where S denotes its language (its set of formulas) and ⊆ Pow(S)×Pow(S) represents its associated con-sequence relation (cr), somehow defined so as to embed some formal model of reasoning. Call any subset of S a theory. As usual, capital Greek letters will denote theories, and lowercase Greek will denote formulas; a sequence such as Γ, α,Γ ′ ∆′, β, ∆ should be read as asserting that Γ ∪ {α} ∪ Γ′ ∆ ′ ∪ {β} ∪∆. Morphisms between any two of the above structures will be called trans-lations. So, given any two logics, L1=〈S1, 1 〉 and L2=〈S2, 2〉, a mapping t: S1 → S2 will constitute a translation from L1 into L2 just in case the following holds