ON SUBSTRUCTURAL LOGICS PRESERVING DEGREES OF TRUTH
Josep Maria Font-Llagunes · 2007
The purpose of this paper is to discuss how some ideas coming from the many-valued logic world can be introduced in a sensible way into the world of substructural logic; namely, the ideas around what does it mean for a logic to say that it preserves degrees of truth. The two mentioned subject areas are by their origin rather far apart; I would like to exemplify how the recent evolution of research in the field of substructural logics, and the application of central techniques from abstract algebraic logic, has revealed such borderline issues and has facilitated their investigation. Degrees of truth are of course ubiquitous in the literature on manyvalued and fuzzy logic, as one of the interpretations of non-classical truthvalues. However, even logics admitting semantics with more than two degrees of truth are in general cast to preserve just one of them, “absolute” truth. Less discussed is the idea of a logic “preserving degrees of truth”. It was considered, in algebraic terms, in [14] and in [19]. Both assume an ordering relation between degrees of truth, so that the idea appears as related only to ordered algebras. Here I would like to give it a broader spectrum of application. The idea of a logic preserving degrees of truth is often presented as opposed to that of a truth-preserving logic; I will try to demonstrate that