Powers and limitations of Urquhart-style semantics I: basic substructural logics

This paper addresses three kinds of binary operational semantics, called here Urquhart-style semantics, for basic substructural logics. First, we discuss the most basic substructural logic GL introduced by Galatos and Ono and its expansions with structural axioms and their algebraic semantics. Next, we provide one kind of Urquhart-style semantics, whose frames form the same structures as algebraic semantics, for those substructural logics and consider powers and limitations of this kind of semantics in substructural logic. We then introduce another kind of Urquhart-style semantics, whose canonical frames are based on prime theories, for DL, the GL with distributivity, and some of its non-associative expansions and extend it to the semantics with star operations for negations. Similarly, we consider powers and limitations of these two kinds of semantics in substructural logic.