How to combine diagrammatic logics

We claim that combining “things”, whatever these things are, is made easier if these things can be seen as the objects of a category. Then things are related by morphisms, and the categorical machinery provides various tools for combining things, typically limits and colimits, as well as other constructions. What should be required from an object in a category to call it a logic? The analysis of various kinds of logic brings in light similar issues: which are the sentences of interest? what is the meaning of each sentence? how can a sentence be infered from another one? According to this analysis, each logic should have a syntax, a notion of model, a proof system, and the morphisms should, in some sense, preserve them. In section 2 we define the diagrammatic logics, which satisfy the required properties. Then in section 3 the category of diagrammatic logics follows, so that categorical constructions can be used for combining diagrammatic logics. As an example, a combination of logics using an opfibration is presented, in order to study computational side-effects due to the evolution of the state during the execution of an imperative program. This work stems from the study of the semantics of computational effects in programming languages. It has been influenced by the theory of sketches [Ehresmann 1968, Lellahi 1989] and the sketch entailments [Makkai 1997], by categorical logic [Lawvere 1963], the theory of institutions [Goguen and Burstall 1984], the use of monads in computer science [Moggi 1989], and many other papers and people. Diagrammatic specifications appeared in [Duval 2003].