Relational semantics for distributive substructural logics
by
Tomoyuki Suzuki
Japan Advanced Institute of Science and Technology

In the present talk, we discuss relational semantics for distributive substructural logics, i.e. substructural logics over FL satisfying the distributive law. Many attempts have been done of introducing relational semantics for substructural logics, but in most cases these semantics are unsatisfactory in their tractability and generality if we compare them with relational semantics for modal logics.

Apparently, the success of relational semantics for modal logics comes mainly from Stone and Jónnson-Tarski duality. So, it would be natural to restict our attention to logics with the distributive law so that we can expect to have a powerful relational semantics for them. In fact, Routley and Meyer have developed to some extent relational semantics with ternary relations for relevance logics that have the distributivity. Moreover, the class of distributive substructural logics includes many of important non-classical logics like superintuitionistic logics, many-valued logics and fuzzy logics.

Starting from the definition of DFL frames and general DFL frames, we will show how far we can develop a theory of relational semantics for distributive substructural logics in parallel with that for modal logics. This will include correspondence theory and completeness with respect to descriptive frames. Also, comparisons of our relational semantics with existing relational semantics for subclasses of distributive substructural logics, including intuitionistic and relevance logics, will be discussed.

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Date received: April 30, 2007