Negation in Relevant Logics (How I Stopped Worrying and Learned to Love the Routley Star)

Abstract

Negation raises three thorny problems for anyone seeking to interpret relevant logics. the frame semantics for negation in relevant logics involves a ‘point shift’ operator *. Problem number one is the interpretation of this operator. Relevant logics commonly interpreted take the inference from A and ~ A ⋁ B to B to be invalid, because the corresponding relevant conditional A ⋀ (~A ⋁ B) → B is not a theorem. Yet we often make the inference from A and ~ A ⋁ B to B, and we seem to be reasoning validly when we do so. Problem number two is explaining what is really going on here. Finally, we can add an operation which Meyer has called Boolean negation to our logic, which is evaluated in the traditional way: x ⊨ −A if and only if x ⊭ A. Problem number three involves deciding which is the ‘real’ negation. How can we decide between orthodox negation and the new, ‘Boolean’ negation? In this paper, I present a new interpretation of the frame semantics for relevant logics which will allow us to give principled answers to each of these questions.