Homotopies in Classical and Paraconsistent Modal Logics

Bisimulations and van Benthem’s celebrated theorem provide a direct insight how truth preserving operations in modal logics work. Apart from bisimulations, there are several other operations in basic modal logic that preserve the truth (Blackburn et al., 2001). However, there is a problem. Given a modal model, and several bisimilar copies of it, there is no method to compare or measure the differences between bisimilar models apart from the basic model theoretical methods (i.e. they are submodels of each other, for instance). A rather negative slogan for this issue is the following: Modal language cannot distinguish bisimilar models. Since the modal language cannot count or measure, several extensions of the language has been proposed to tackle this issue such as hybrid logics and majority logics (Blackburn, 2000; Fine, 1972). However, such extensions are language based and introduce a non-natural, and sometimes counter-intuitive operators to the language, and often criticized as being ad-hoc. In this work, we will focus on the truth preserving operations rather than extending the modal language. In other words, we will ask the following question: Is there a truth preserving natural modal operation that can also distinguish the models it generates, and even compares them? We argue that such an operation exist, and the answer to that question is positive. However, to conceptualize our concerns and questions, we need to be careful at picking the correct modal logical framework. Kripkean models, in this respect, are criticized as they are overly simplistic and can overshadow some mathematical properties that can be apparent in some other modal models. Therefore, in this paper, we will concentrate on topological models for modal logics. On the other hand, note that topological models historically precede Kripke models, and are mathematically more complex allowing us to express variety of ideas within modal logic. Therefore, they can provide us with much stronger and richer structure of which we can take advantage. In this paper, we utilize a rather elementary concept from topology. We first introduce homemorphisms, and then homotopies to the modal logical framework, and show the immediate invariance results. On the other hand, from an application oriented point of view, we also have some applications to illustrate how our constructions