×

Truth and the liar in De Morgan-valued models. (English) Zbl 0989.03009

Summary: The aim of this paper is to give a certain algebraic account of truth: we want to define what we mean by De Morgan-valued truth models and show their existence even in the case of semantical closure: that is, languages may contain their own truth predicate if they are interpreted by De Morgan-valued models. Before we can prove this result, we have to repeat some basic facts concerning De Morgan-valued models in general, and we will introduce a notion of truth both on the object- and on the metalanguage level appropriate for such models. The definitions and the existence theorem are extensions of Kripke’s, Woodruff’s, and Visser’s concepts and results concerning three- and four-valued truth models.

MSC:

03A05 Philosophical and critical aspects of logic and foundations
03G25 Other algebras related to logic
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Barwise, J., and J. Etchemendy, The Liar , Oxford University Press, Oxford, 1987. · Zbl 0678.03001
[2] Belnap, N. D. Jr., and J. H. Spencer, “Intensionally complemented distributive lattices,” Portugaliae Mathematica , vol. 25 (1966), pp. 99–104. · Zbl 0158.27103
[3] Belnap, N. D. Jr., “A useful four-valued logic,” pp. 8–37 in Modern Uses of Multiple-Valued Logic , edited by J. M. Dunn and G. Epstein, D. Reidel, Dordrecht, 1977. · Zbl 0424.03012
[4] Bialynicki-Birula, A., and H. Rasiowa, “On the representation of quasi-boolean algebras,” Bulletin Academie Polen. Sci. Cl. , vol. 3 (1957), pp. 259–61. · Zbl 0082.01403
[5] Dunn, J. M., “Intensional algebras,” pp. 180–206 in Entailment , edited by A. R. Anderson and N. D. Belnap Jr., Princeton University Press, Princeton, 1975. · Zbl 0323.02030
[6] Dunn, J. M., “Relevance logic and entailment,” pp. 117–224 in Handbook of Philosophical Logic III , edited by D. Gabbay and F. Guenthner, D. Reidel, Dordrecht, 1986. · Zbl 0875.03051
[7] Gupta, A., and N. Belnap, The Revision Theory of Truth , The MIT Press, Cambridge, 1993. · Zbl 0858.03010
[8] Hermes, H., Einführung in die Verbandstheorie , Springer-Verlag, Berlin, 1967. · Zbl 0153.33203
[9] Kalman, J. A., “Lattices with involution,” Transactions of the American Mathematical Society , vol. 87 (1958), 485–91. JSTOR: · Zbl 0228.06003 · doi:10.2307/1993112
[10] Koppelberg, S., Handbook of Boolean Algebras , vol. 1, North-Holland, Amsterdam, 1989. · Zbl 0671.06001
[11] Kripke, S. A., “Outline of a theory of truth,” The Journal of Philosophy , vol. 72 (1975), pp. 690–716. · Zbl 0952.03513
[12] Leitgeb, H., Truth as Translation , Ph.D. thesis, University of Salzburg, Salzburg, 1998. · Zbl 0983.03003
[13] Martin, R. L., and P. W. Woodruff, “On representing ‘True-in-\(L\)’ in \(L\),” Philosophia , vol. 5 (1975), pp. 213–17. · Zbl 0386.03001
[14] McGee, V., Truth, Vagueness and Paradox , Hackett Publishing Company, Indianapolis, 1991. · Zbl 0734.03001
[15] Monteiro, A., “Matrices de Morgan caractèristiques pour le calcul propositionnel classique,” An. Acad. Brasil, Ci. , vol. 32 (1960), pp. 1–7. · Zbl 0094.00605
[16] Rescher, N., Many-valued Logic, McGraw-Hill, New York, 1969. · Zbl 0248.02023
[17] Rosser, J. B., Simplified Independence Proofs , Academic Press, New York, 1969. · Zbl 0209.30502
[18] Tarski, A., “Der Wahrheitsbegriff in den formalisierten Sprachen,” Studia Philosophica , vol. 1 (1935), pp. 261–405. · Zbl 0013.28903
[19] Tarski, A., “The semantic conception of truth and the foundations of semantics,” Philosophy and Phenomenological Research , vol. 4 (1944), pp. 341–76. · Zbl 0061.00807 · doi:10.2307/2102968
[20] Visser, A., “Four valued semantics and the Liar,” Journal of Philosophical Logic , vol. 13 (1984), pp. 695–708. · Zbl 0546.03007 · doi:10.1007/BF00453021
[21] Visser, A., “Semantics and the Liar Paradox,” pp. 617–706, in Handbook of Philosophical Logic IV , edited by D. Gabbay and F. Guenthner, D. Reidel, Dordrecht, 1989. · Zbl 0875.03030
[22] Woodruff, P. W., “Paradox, truth and logic. Part 1: paradox and truth,” Journal of Philosophical Logic , vol. 13 (1984), pp. 213–32. · Zbl 0546.03006 · doi:10.1007/BF00453022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.