Abstract logics, logic maps, and logic homomorphisms.

*(English)*Zbl 1131.03006Summary: What is a logic? Which properties are preserved by maps between logics? What is the right notion for equivalence of logics? In order to give satisfactory answers we generalize and further develop our topological approach [in: J.-Y. Beziau (ed.), Logica universalis. Towards a general theory of logic. Basel: Birkhäuser. 35–63 (2005; Zbl 1082.03041)] and present the foundations of a general theory of abstract logics which is based on the abstract concept of a theory. Each abstract logic determines a topology on the set of theories. We develop a theory of logic maps and show in what way they induce (continuous, open) functions on the corresponding topological spaces. We also establish connections to well-known notions, such as translations of logics and the satisfaction axiom of institutions [T. Mossakowski, J. Goguen, R. Diaconescu and A. Tarlecki, ibid. 113–133 (2005; Zbl 1080.03028)]. Logic homomorphisms are maps that behave in some sense like continuous functions and preserve more topological structure than logic maps in general. We introduce the notion of a logic isomorphism as a (not necessarily bijective) function on the sets of formulas that induces a homeomorphism between the respective topological spaces and gives rise to an equivalence relation on abstract logics. Therefore, we propose logic isomorphisms as an adequate and precise notion for equivalence of logics. Finally, we compare this concept with another recent proposal presented in [C. Caleiro and R. Gonçalves, ibid. 99–111 (2005; Zbl 1081.03012)].