Bisimulations for Boolean relevant logics

SERGIO ARTURO CELANI

Proceedings of the 6th conference of mathematics Dr. Antonio A. R. Monteiro

INMABB

Lugar: Bahia Blanca; Año: 2001 vol. 6 p. 105 - 117

0327-9170

The notion of bisimulation is an important tool for the study of the model theory of classical modal logic and also plays a crucial role in theoretical computer science. This notion was introduced by J. van Benthem under the name of p-relation. Some very important results, as for example preservation and denability results, are proved using bisimulation as the fundamental tool. R. Routley and R. K. Meyer [Stud. Log. 32, 51-66 (1973; Zbl 0316.02029)] provided a Kripke-style semantics for some Boolean relevant logics using a ternary accessibility relation. This semantics is an adaptation of the relational semantics for relevant logics. With a ternary relation dened on a set X we can dene two binary operations on the power set algebra P(X): an operator  called fusion, and a (relevant) implication ). The operation  can be considered as a genuine modal operator of necessity. So, an extension of classical propositional calculus PC by means of an operator of fusion  can be seen as a particular polymodal logic. So, the model theory of this type of logics is a particular case of the model theory for classical modal logic ML. But the implication ) is not a modal operator, and consequently the results of model theory developed for ML are not directly applicable to logics with this kind of implication. Thus, it is natural to ask if the questions typical to model theory of ML remain valid for some Boolean relevant logics. We address some of these questions in the present work. If we like to prove results on the model theory of BR following the lines of the results for ML, we need an adequate notion of bisimulation. G. Restall [An introduction to substructural logics. Routledge (1999)] introduced a notion of bisimulation for some substructural logics, including some relevant logics. In this paper we will apply this notion of bisimulation to the Boolean relevant logic BR