Guillermo Badia

A traditional aspect of model theory has been the interplay between formal languages and mathematical structures. This dissertation is concerned, in particular, with the relationship between the languages of relevant logic and Routley-Meyer models. One fundamental question is treated: what is the expressive power of relevant languages in the Routley-Meyer framework? In the case of finitary relevant propositional languages, two answers are provided. The first is that finitary propositional relevant languages are the fragments of first order logic preserved under relevant directed bisimulations. The second is that, when we restrict our attention to what can be labelled as De Morgan models, we can obtain an analogue of Lindström's theorem for finitary propositional relevant languages. Furthermore, it is shown that a preservation theorem characterizing the expressive power of infinitary relevant languages in classical infinitary languages follows as a consequence of an interpolation theorem for classical infinitary logic. In addition, algebraic characterizations of the classes of Routley-Meyer models axiomatizable in relevant propositional languages, incompactness of infinitary relevant propositional languages and the expressive power of quantificational relevant languages are discussed. A final chapter is devoted to the study of relevant languages as second order frame languages. In particular we devote our attention to the problem of which properties expressible by relevant languages are elementary and which are not. An algebraic characterization of such elementary properties together with some examples of non first order properties axiomatizable in relevant logic are given. Finally, a Sahlqvist-van Benthem algorithm showing that relevant formulas with a certain syntactic form express calculable first order properties at the level of frames is established.