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The logic determined by Smiley’s matrix for Anderson and Belnap’s first-degree entailment logic

The logic determined by Smiley’s matrix for Anderson and Belnap’s first-degree entailment logic The aim of this paper is to define the logical system Sm4 characterised by the degree of truth-preserving consequence relation defined on the ordered set of values of Smiley’s four-element matrix MSm4. The matrix MSm4 has been of considerable importance in the development of relevant logics and it is at the origin of bilattice logics. It will be shown that Sm4 is a most interesting paraconsistent logic which encloses a sound theory of logical necessity similar to that of Anderson and Belnap’s logic of entailment E. Intuitively, Sm4 can be described as a four-valued expansion of the positive fragment of Lewis’ S5 or, alternatively, as a four-valued version of S5. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Applied Non-Classical Logics Taylor & Francis

The logic determined by Smiley’s matrix for Anderson and Belnap’s first-degree entailment logic

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The logic determined by Smiley’s matrix for Anderson and Belnap’s first-degree entailment logic

Abstract

The aim of this paper is to define the logical system Sm4 characterised by the degree of truth-preserving consequence relation defined on the ordered set of values of Smiley’s four-element matrix MSm4. The matrix MSm4 has been of considerable importance in the development of relevant logics and it is at the origin of bilattice logics. It will be shown that Sm4 is a most interesting paraconsistent logic which encloses a sound theory of logical necessity similar to that of Anderson and...
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Publisher
Taylor & Francis
Copyright
© 2016 Informa UK Limited, trading as Taylor & Francis Group
ISSN
1958-5780
eISSN
1166-3081
DOI
10.1080/11663081.2016.1153930
Publisher site
See Article on Publisher Site

Abstract

The aim of this paper is to define the logical system Sm4 characterised by the degree of truth-preserving consequence relation defined on the ordered set of values of Smiley’s four-element matrix MSm4. The matrix MSm4 has been of considerable importance in the development of relevant logics and it is at the origin of bilattice logics. It will be shown that Sm4 is a most interesting paraconsistent logic which encloses a sound theory of logical necessity similar to that of Anderson and Belnap’s logic of entailment E. Intuitively, Sm4 can be described as a four-valued expansion of the positive fragment of Lewis’ S5 or, alternatively, as a four-valued version of S5.

Journal

Journal of Applied Non-Classical Logics – Taylor & Francis

Published: Jan 2, 2016

Keywords: many-valued logics; four-valued logics; Smiley’s four-element matrix; relevant logics; modal logics

References

  • Reasoning with logical bilattices
  • The value of the four values
  • Entailment and relevance
  • Completeness proofs for the systems RM3 and BN4
  • Intuitive semantics for first-degree entailments and ‘coupled trees’
  • Partiality and its dual
  • What is strict implication?
  • An interpretation of Łukasiewicz’s 4-valued modal logic
  • The non-relevant De Morgan minimal logic in Routley-Meyer semantics with no designated points
  • A lattice of implicative extensions of regular Kleene’s logics