Investigations into Ternary Relational Semantics

The introduction of the ternary relational semantics for relevance logic created a new approach to interpreting intensional logics. This project intends to contribute to the further development of relational semantics and to describe the history of these semantics. We also work on applying the approach to new logics and to logics that did not have this kind of semantics that gives a deeper understanding of a formal system. It is our expectation that using this approach we will be able to prove new results about certain intensional logics that are closely related to the most famous relevance logics such as the logic of ticket entailment and the logic of relevant implication.

       Relational semantics were introduced as interpretations of normal modal logics in the late 1950s by Saul A. Kripke.  The prototypical modal connectives such as ◊ (possibility) and □ (necessity) are unary; hence, they are interpreted by a two-place accessibility relation in the so-called possible-world semantics.  Intuitionistic implication can be interpreted as classical implication with respect to all future situations.  This means that the addition of a binary relation is sufficient to obtain a model for intuitionistic logic.  However, entailment is a richer connective than intuitionistic implication.  Attempts to give an interpretation of the implication connective of various relevance logics had some success, but was limited to fragments.  The idea to use a ternary relation to interpret a binary operation is quite straightforward from the point of view of representation theory.  A motivation for a variety of relevance logics is that → possesses different properties in them and this can be seen to reflect important philosophical distinctions in our understanding of various concepts of consequence relations.  Then, the ternary relation will not simply connect all situations; rather, the accessibility relation will have specific properties.  This introduces some technical complications — compared to the treatment of the basic normal modal logic K.

       The first applications of the relational semantic approach to relevance logics go back to the late 1960s–early 1970s.  An aim of this project is to provide an analytic history of the emergence of the relational semantics for relevance logics, which is greatly helped by fact that J. Michael Dunn was actively involved in many of the events.  We have already written two papers in which we scrutinized the work of several logicians who were working in distant parts of the globe without knowing about each other's efforts.  We are also working on a paper that analyzes a couple of unpublished manuscripts from the dawn of the ternary relational semantics for E and R. Relational semantics provide not only an alternative interpretation — compared to an algebraic interpretation — for a logic, but they come with a hefty layer of connotations.  The latter is a fertile ground for informal renditions of concepts to flourish.  It is a goal of the project to expand the range of informal interpretations in order to increase the applicability of relevance logics.  Our research also aims to define ternary relational semantics for logics that have not yet been fitted with such semantics.  We have designed such a semantics for R→ (proving soundness and completeness).  At the same time, we have highlighted the importance of the addition of the truth constant and fusion for T→, E→ and R→.  We expect that some further applications of the relational semantics approach to certain relevance logics will yield remarkable new results.

       This research project is funded by an Insight Grant (entitled “The third place is the charm: The emergence, the development and the future of the ternary relational semantics for relevance and some other non-classical logics”) awarded by the Social Sciences and Humanities Research Council of Canada.