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 comprehensive 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 combination 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 consequence 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 translations. 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: (T1) Γ 1 ∆⇒ t(Γ) 2 t(∆) ∗The author wishes to acknowledge the partial support of FCT (Portugal) and FEDER (European Union), namely, via the Project FibLog POCTI / MAT / 37239 / 2001 of the Centro de Lógica e Computação (IST, Portugal) and the FCT grant SFRH / BD / 8825 / 2002. I am also grateful to Arnon Avron, Beata Konikowska, Carlos Caleiro, and Walter Carnielli for helpful related discussions.