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Pecularities of Some Three- and Four-Valued Second Order Logics

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Abstract

Logics that have many truth values—more than just True and False—have been argued to be useful in the analysis of very many philosophical and linguistic puzzles (as well, sometimes, in various computational-oriented tasks). In this paper, which is a followup to (Hazen and Pelletier in K3, Ł3, LP, RM3, A3, FDE, M: How to make many-valued logics work for you. Winning paper for the Canadian Schotch-Jennings Prize, one of the prizes of the Universal Logic competition in 2018; Notre Dame J Form Log 59, 2018), we will start with a particularly well-motivated four-valued logic that has been studied mainly in its propositional and first-order versions. And we will then investigate its second-order version. This four-valued logic has two natural three-valued extensions: what is called a “gap logic” (some formulas are neither True nor False), and what is called a “glut logic” (some formulas are both True and False). We mention various results about the second-order version of these logics as well. And we then follow our earlier papers, where we had added a specific conditional connective to the three valued logics, and now add that connective to the four-valued logic under consideration. We then show that, although this addition is “conservative” in the sense that no new theorems are generated in the four-valued logic unless they employ this new conditional in their statement, nevertheless the resulting second-order versions of these logics with and without the conditional are quite different in important ways. We close with a moral for logical investigations in this realm.

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Authors and Affiliations

  1. Department of Philosophy, University of Alberta, Edmonton, AB , T6G 2E1, Canada

    Allen P. Hazen & Francis Jeffry Pelletier

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  1. Allen P. Hazen
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  2. Francis Jeffry Pelletier
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Correspondence to Francis Jeffry Pelletier.

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This paper develops some issues that are suggested but not followed up in (Hazen and Pelletier [17]). That paper was the the winning paper of the Schotch-Jennings prize, the Canadian entry into the Universal Logic prize in logic for 2018. But that paper was committed to a volume honouring the founders of the FDE logic(s), Nuel D. Belnap, Jr., and J. Michael Dunn. We are grateful to Jean-Yves Béziau for his encouragement to follow up some loose ends of that paper, so that this paper could appear in the present issue of Logica Universalis.

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Hazen, A.P., Pelletier, F.J. Pecularities of Some Three- and Four-Valued Second Order Logics. Log. Univers. 12, 493–509 (2018). https://doi.org/10.1007/s11787-018-0214-7

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