Following earlier work, a modal logic for T-coalgebras is a functor L on a suitable variety. Syntax and proof system of the logic are given by presentations of the functor. This paper makes two contributions. First, a previous result characterizing those functors that have presentations is generalized from endofunctors on one-sorted varieties to functors between many-sorted varieties. This yields an equational logic for the presheaf semantics of higher-order abstract syntax. As another application, we show how the move to functors between many-sorted varieties allows to modularly combine syntax and proof systems of different logics. Second, we show how to associate to any set-functor T a complete (finitary) logic L consisting of modal operators and Boolean connectives.
A. Kurz and D. Petrişan, “Functorial Coalgebraic Logic: The Case of Many-sorted Varieties,” Electronic Notes in Theoretical Computer Science, vol. 203, no. 5, pp. 175–194, Jun. 2008. DOI: 10.1016/j.entcs.2008.05.025
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