Dialgebraic Logics: Extended Abstract

Abstract

We extend the notion of a first–order signature in such a way that the type constructors used to define domain and codomain of the fundamental operations are taken to be a constituent part of the signature. Using the generative power of the type constructors and the fundamental types and operations we obtain a general construction of a category of typed terms which will be called syntactic category. Functors into the category of set from a syntactic category preserving the used type constructors represent models and terms with the constant type Bool as codomain represent properties.We demonstrate that in this way propositional logic and modal logic can be generated by a uniform constructions which differ only in the used type constructors of the corresponding syntactic categories. This observation leads to the conjecture that logics can be classified by the type constructors used on the syntactic level.