Understanding Negation Implicationally in the Relevant Logic R

Abstract

A star-free relational semantics for relevant logic is presented together with a sound and complete sequent proof theory (display calculus). It is an extension of the dualist approach to negation regarded as modality, according to which de Morgan negation in relevant logic is better understood as the confusion of two negative modalities. The present work shows a way to define them in terms of implication and a new connective, co-implication, which is modeled by respective ternary relations. The defined negations are confused by a special constraint on ternary relation, called the generalized star postulate, which implies definability of the Routley star in the frame. The resultant logic is shown to be equivalent to the well-known relevant logic R. Thus it can be seen as a reconstruction of R in the dualist framework.The defining feature of display calculus is the display property that every substructure of a sequent can be "displayed" through display rules. Cut elimination theorem can be proved in a generic way based on this property. Definition 4.3. (Antecedent/succedent parts) In X Y , we say X is an antecedent part (AP) and Y is a succedent part (SP). We define:• If W ; Z is an AP (SP) in X Y , so are W and Z, and • In any substructure W > Z of X Y , W (Z) is an AP (SP) in X Y .Definition 4.4. We say two sequents are display equivalent if they are derivable from each other using only display rules.

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