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Modality, Semantics and Interpretations pp 167–178Cite as

An Explanation for the Ternary Relation R in the Relational Semantics of Relevance Logic

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Part of the book series: Logic in Asia: Studia Logica Library ((LIAA))

Abstract

Relational semantics is one of the most popular forms of semantics for relevance logic. However, this semantics, especially the ternary relation R lacks intuition, and that is why there are various interpretations for R. The original motivation for creating relevance logic is to give a better description of inference, thus if an interpretation for R could match this motivation better, then it will be more intuitive than the relational semantics and other interpretations. Therefore, this paper focuses on inference semantics, which could be viewed as a relational semantics with a ternary relation R, and R represents a relation among rules, premises and conclusions. These two forms of semantics are equivalent with respect to the relevance logic system \({\mathbf{{R}}_{+}}\). Actually, relational semantics could be regarded as a characterization of the formal structure of inference, and relevance logic may be defined as the logic of some kind of inference.

This research is supported by the National Social Science Fund of China (12&ZD119).

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Notes

  1. 1.

    For these clauses in Definition 10.2.1(3), their order has been changed from the order in Anderson and Belnap [2] for the convenience of the following propositions.

  2. 2.

    For any a, b, c, d in \(K, \mathrm{R}^{2}\mathrm{(ab)cd} = {}_{df}\exists x (\text {Rab}x\wedge \text {R}x\text {cd}),\text {R}^{2}\text {a(bc)d} = {}_{df}\exists x(\text {Ra}x\text {d}\wedge {Rbc}x)\).

  3. 3.

    In the period of this writing, we contacted N. Belnap about his idea of the semantics of relevance logic, especially about RS. He said “I spent many years thinking about the 3-termed relation for relevance logic. At the end, all I came away with as far as intuition goes was the three VERY short quotes in vol. II of Entailment that I found by Robert Meyer, Alistair Urquhart, and Kit Fine. The theory is formally very beautiful, but I have NO IDEA what it means. Since those three quotes are so short, I concluded that none of them was able to say anything further. “His evaluation of the issue that RS is short of intuition is that “But to my mind, it amounts to just ‘sand in the eyes”’.

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Acknowledgments

We are grateful to N. Belnap for his help and advice during the writing of this paper.

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Authors and Affiliations

  1. Department of Philosophy, Peking University, Beijing, China

    Beihai Zhou

  2. Institute of Philosophy, Chinese Academy of Social Sciences, Beijing, China

    Qing Jia

Authors
  1. Beihai Zhou
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  2. Qing Jia
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Correspondence to Beihai Zhou .

Editor information

Editors and Affiliations

  1. Institute of Logic and Cognition, Sun Yat-sen University, Guangzhou, China

    Shier Ju

  2. Institute of Logic and Cognition, Sun Yat-sen University, Guangzhou, China

    Hu Liu

  3. Japan Advanced Institute of Science and Technology, Ishikawa, Japan

    Hiroakira Ono

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Zhou, B., Jia, Q. (2015). An Explanation for the Ternary Relation R in the Relational Semantics of Relevance Logic. In: Ju, S., Liu, H., Ono, H. (eds) Modality, Semantics and Interpretations. Logic in Asia: Studia Logica Library. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47197-5_10

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