Relevance Logic - Semantics

Semantics

Relevance logic is, in syntactical terms, a substructural logic because it is obtained from classical logic by removing some of its structural rules (e.g. explicitly of some sequent calculus or implicitly by "tagging" inferences of a natural deduction system). It is sometimes referred to as a modal logic because it can be characterized as a class of formulas valid over a class of Kripke (relational) frames. In Kripke semantics for relevant logic, the implication operator is a binary modal operator, and negation is usually taken to be a unary modal operator. As such, the accessibility relation governing the operator is ternary rather than the usual binary ones that govern unary modal operators often read as "necessarily".

A Kripke frame F for a propositional relevance language is a triple (W,R,*) where W is a set of indices (or points or worlds), R is a ternary accessibility relation between indices, and * is a unary function taking indices to indices. A model M for the language is an ordered pair (F,V) where F is a frame and V is a valuation function mapping sets of worlds (propositions) to propositional letters. Let M be a model and a,b,c indices from M. An implication is defined

• .

Negation is defined

• .

One obtains various relevance logics by placing appropriate restrictions on R and on *. Details need to be filled in.