Analytic Tableaux for all of SIXTEEN 3
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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.
KeywordsTrilattice SIXTEEN3 Tableau calculi Functional completeness Truth entailment Falsity entailment Information entailment
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.
- 1.Baaz, M., Fermüller, C., Zach, R. (1994). Elimination of cuts in first-order finite-valued logics. Journal of Information Processing and Cybernetics, 29, 333–355.Google Scholar
- 3.Belnap, N.D. (1976). How a computer should think. In G. Ryle (Ed.), Contemporary aspects of philosophy (pp. 30–56). Stocksfield: Oriel Press.Google Scholar
- 4.Belnap, N.D. (1977). A useful four-valued logic. In J. Dunn, & G. Epstein (Eds.), Modern uses of multiple-valued logic (pp. 8–37). Dordrecht: Reidel.Google Scholar
- 6.Langholm, T. (1996). How different is partial logic? In P. Doherty (Ed.), Partiality, modality, and nonmonotonicity (pp. 3–43). Stanford: CSLI.Google Scholar
- 7.Muskens, R.A. (1995). Meaning and partiality. Stanford: CSLI.Google Scholar
- 10.Odintsov, S., & Wansing, H. (2014). The logic of generalized truth values and the logic of bilattices. In Studia Logica.Google Scholar
- 12.Rousseau, G. (1967). Sequents in many-valued logic I. Fundamenta Mathematicae, 60, 23–33.Google Scholar
- 15.Shramko, Y., & Wansing, H. (2011). Truth and falsehood: An inquiry into generalized logical values. In Trends in logic Vol. 36: Springer.Google Scholar
- 19.Wintein, S., & Muskens, R.A. (2012). A calculus for Belnap’s logic in which each proof consists of two trees. Logique & Analyse, 220, 643–656.Google Scholar
- 20.Wintein, S., & Muskens, R.A. (2014). From bi-facial truth to bi-facial proofs. Studia Logica. Online First.Google Scholar
- 21.Zach, R. (1993). Proof theory of finite-valued logics. Technical Report TUW-E185.2-Z.1-93. Institut für Computersprachen, Technische Universität Wien.Google Scholar