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Analytic Tableaux for all of SIXTEEN 3

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Abstract

In this paper we give an analytic tableau calculus P L 1 6 for a functionally complete extension of Shramko and Wansing’s logic. The calculus is based on signed formulas and a single set of tableau rules is involved in axiomatising each of the four entailment relations ⊧ t , ⊧ f , ⊧ i , and ⊧ under consideration—the differences only residing in initial assignments of signs to formulas. Proving that two sets of formulas are in one of the first three entailment relations will in general require developing four tableaux, while proving that they are in the ⊧ relation may require six.

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Notes

  1. Many-sided calculi for many-valued logics also result from the methods used in, for example, [1, 12, 13, 21], but [8] explains why these methods are not applicable in the case of a functionally complete variant of Belnap’s logic. Wansing [16] also makes use of many-sided sequents.

  2. We use the terms ‘entailment relation’ and ‘consequence relation’ purely for convenience in the case of the information entailment relation ⊧ i . Strictly more correct perhaps would be ‘necessary approximation’, a term that was used in [19] for a ‘consequence’ relation based on Belnap’s A4 lattice.

  3. This simplicity has a pay off because it is now very easy to implement the logic. Mainly for their own amusement and instruction the authors have written a simple theorem prover 16T A P, loosely based on (the propositional part of) Beckert & Posegga’s [2] lean T A P marvel. It similarly exploits Prolog’s left-to-right depth-first evaluation mechanism, but the code unfortunately is far less concise than Beckert & Posegga’s.

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Acknowledgments

We would like to thank the referee for encouraging words and helpful comments. Stefan Wintein wants to thank the Netherlands Organisation for Scientific Research (NWO) for funding the project The Structure of Reality and the Reality of Structure (project leader: F. A. Muller), in which he is employed. Reinhard Muskens gratefully acknowledges NWO’s funding of his project 360-80-050, Towards Logics that Model Natural Reasoning.

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Authors and Affiliations

  1. Tilburg Center for Logic, Ethics, and Philosophy of Science (TiLPS), Tilburg University, Tilburg, Netherlands

    Reinhard Muskens

  2. Faculty of Philosophy, Erasmus University Rotterdam, Rotterdam, Netherlands

    Stefan Wintein

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  1. Reinhard Muskens
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  2. Stefan Wintein
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Correspondence to Reinhard Muskens.

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Muskens, R., Wintein, S. Analytic Tableaux for all of SIXTEEN 3 . J Philos Logic 44, 473–487 (2015). https://doi.org/10.1007/s10992-014-9337-3

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  • DOI: https://doi.org/10.1007/s10992-014-9337-3

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