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Algebraic Kripke-Style Semantics for Relevance Logics

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Abstract

This paper deals with one kind of Kripke-style semantics, which we shall call algebraic Kripke-style semantics, for relevance logics. We first recall the logic R of relevant implication and some closely related systems, their corresponding algebraic structures, and algebraic completeness results. We provide simpler algebraic completeness proofs. We then introduce various types of algebraic Kripke-style semantics for these systems and connect them with algebraic semantics.

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Notes

  1. These semantics are not necessarily non-operational, e.g., semantics for relevance logics generally have a star (∗) operation for negation (see [21]).

  2. As Yang pointed out, there is “another trend of four-valued semantics for relevance logic …, combining Routley–Meyer semantics for relevance logic and Dunn’s four-valued semantics for the logic of first-degree entailments.” ([61], p. 257).

  3. Here the term ‘algebraic semantics’ is used in the broad sense that it means not merely algebraic semantics in the strict sense, the semantics distinguished from matrix semantics, but also matrix semantics, cf. see [14].

  4. We use the notation & in place of ∘ to express the fusion in order to avoid unnecessary confusion because the latter notation is also used as a restricted operational symbol in Section 3.

  5. For the more exact definition, see Section 3.1.

  6. In [28–30], these systems were introduced without much consideration of relevance itself.

  7. For unexplained notions, notations, and terminology of universal algebra and algebraic semantics used in this paper, see [10, 11, 13, 14, 28].

  8. A weakly implicative logic \(\mathcal {L}\) is a logic satisfying A1, (mp), transitivity (\(\varphi \rightarrow \psi , \psi \rightarrow \chi \vdash \varphi \rightarrow \chi \)), and symmetrized congruence, i.e., congruence w.r.t. connectives, see [11, 13] for more details.

  9. A unital quantale is a complete FL-algebra introduced in [44], and every unital quantale is isomorphic to a phase structure (see [45]).

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  1. Department of Philosophy, Chonbuk National University, Rm 307, College of Humanities Blvd. (14-1), Jeonju, 561-756, Korea

    Eunsuk Yang

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I must thank the referees for their helpful comments and suggestions for improvements to this paper.

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Yang, E. Algebraic Kripke-Style Semantics for Relevance Logics. J Philos Logic 43, 803–826 (2014). https://doi.org/10.1007/s10992-013-9290-6

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