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Understanding Negation Implicationally in the Relevant Logic R

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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.

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Authors and Affiliations

  1. Department of Philosophy, Kyoto University, Yoshida Honmachi, Sakyo-ku, Kyoto, Japan

    Takuro Onishi

  2. Department of Philosophy, National University of Singapore, 3 Arts Link, Block AS3, Singapore, Singapore

    Takuro Onishi

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  1. Takuro Onishi
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Correspondence to Takuro Onishi.

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Onishi, T. Understanding Negation Implicationally in the Relevant Logic R. Stud Logica 104, 1267–1285 (2016). https://doi.org/10.1007/s11225-016-9676-x

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  • DOI: https://doi.org/10.1007/s11225-016-9676-x

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