Semantics for a basic relevant logic with intensional conjunction and disjunction (and some of its extensions) (original) (raw)

2008, Mathematical Structures in Computer Science

This paper proposes a new relevant logic B + , which is obtained by adding two binary connectives, intensional conjunction and intensional disjunction , to Meyer-Routley minimal positive relevant logic B + , where and are weaker than fusion • and fission +, respectively. We give Kripke-style semantics for B + , with →, and modelled by ternary relations. We prove the soundness and completeness of the proposed semantics. A number of axiomatic extensions of B + , including negation-extensions, are also considered, together with the corresponding semantic conditions required for soundness and completeness to be maintained. † Dunn's general approach is algebraic, where each logical connective is characterised as an operation on distributive lattices, which 'distributes' in each of its places over at least one of ∧ and ∨, leaving ∧ or Y. Gao and J. Cheng 146 • † , and shares with + ‡. Then, additional axioms or rules can be added to make coincide with •, and with +. This qualifies and as weaker versions of intensional conjunction and disjunction, respectively. To give a semantics for B + , we apply Dunn's strategy (Dunn 1990), that is, we use n + 1-placed accessibility relations to model n-placed connectives. The semantics is defined by adapting and extending the traditional relational semantics for relevant logics. There are four ternary relations: R 1 and R 2 for →; S 1 for ; and S 2 for. To construct canonical models, as well as theories, we define dualtheories and antidualtheories such that R 1 , R 2 , S 1 , S 2 are canonically defined as derivatives of operations on theories and anti-dualtheories. The crucial tools for completeness are extensions or reductions of a given theory or anti-dualtheory to a prime theory. Then, by well-known standard techniques, together with our extra definitions, we can establish the soundness and completeness of the proposed semantics for B +. Furthermore, we consider a number of axiomatic extensions of B + (including negation-extensions with negation modelled by the Routley ' * '-operation), together with the corresponding semantic conditions to ensure that soundness and completeness are maintained. 2. The basic system B + 2.1. An axiom system for B + B + is expressed in a language L, which has the two-place connectives →, ∧, ∨, and , parentheses (and), and a stock of propositional variables p, q, r, ... Formulas are defined recursively in the usual manner. We use the following scope conventions: the connectives are ranked , , ∧, ∨, → in order of increasing scope (that is, binds more strongly than , binds more strongly than ∧, and so on), otherwise, association is to the left. A, B, C, ... will be used to range over arbitrary formulas. We begin by giving an axiom system for B + , which is defined in the same way as that of Priest and Sylvan (1992) and Restall (1993) § : Axioms