Abstract
My purpose in this paper is to argue that the classical notion of entailment is not suitable for non-bivalent logics, to propose an appropriate alternative and to suggest a generalized entailment notion suitable to bivalent and non-bivalent logics alike. In classical two valued logic, one can not infer a false statement from one that is not false, any more than one can infer from a true statement a statement that is not true. In classical logic in fact preserving truth and preserving non-falsity are one and the same thing. They are not the same in non-bivalent logics however and I will argue that the classical notion of entailment that preserves only truth is not strong enough for such a logic. I will show that if we retain the classical notion of entailment in a logic that has three values, true, false and a third value in between, an inconsistency can be derived that can be resolved only by measures that seriously disable the logic. I will show this for a logic designed to allow for semantic presuppositions, then I will show that we get the same result in any three valued logic with the same value ordering. I will finally suggest how the notion of entailment should be generalized so that this problem may be avoided. The strengthened notion of entailment I am proposing is a conservative extension of the classical notion that preserves not only truth but the order of all values in a logic, so that the value of an entailed statement must alway be at least as great as the value of the sequence of statements entailing it. A notion of entailment this strong or stronger will, I believe, be found to be applicable to non-classical logics generally. In the opinion of Dana Scott, no really workable three valued logic has yet been developed. It is hard to disagree with this. A workable three valued logic however could perhaps be developed however, if we had a notion of entailment suitable to non-bivalent logics.
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REFERENCES
Anderson, A. R. and Belnap, N. D.: 1975, Entailment, Vol. 1, Princeton University Press, Princeton.
Gazdar, G.: 1979, Pragmatics, Academic Press, London.
Kempson, R. M.: 1975, Presupposition and the Delimitation of Semantics, Cambridge University Press, Cambridge.
Priest, G.: 1987, In Contradiction, Martinus Nijhoff, Dordrecht.
Routley, R., Meyer, R. K., Plumwood, V. and Brady, R. T.: 1982, Relevant Logic and its Rivals, Ridgeview.
Scott, D.: 1970, Advice on modal logic, in K. Lambert (ed.), Philosophical Problems in Logic, Reidel, Dordrecht, pp. 143-173.
Stalnaker, R.: 1974, Pragmatic presuppositions, in Munitz and Unger (eds.), Semantics and Philosophy, NYU Press, New York.
Strawson, P. F.: 1952, Introduction to Logical Theory, Methuen, London.
Van Fraassen, B. C.: 1968, Presupposition, implication and self-reference, J. Philos. 65.
Van Fraassen, B. C.: 1969, Presuppositions, supervaluations and free logic, in K. Lambert (ed.), The Logical Way of Doing Things, Yale University Press, New Haven.
Van Fraassen, B. C.: 1971, Macmillan, New York.
Van Fraassen, B. C.: 1975, Incomplete assertion and belnap connectives, in Hockney, Harper and Freed (eds.), Contemporary Research in Philosophical Logic and Linguistic Semantics, Reidel, Dordrecht, pp. 43-70.
Wilson, D.: 1975, Presuppositions and Non-Truth Conditional Semantics, Academic Press, London.
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Michael, F.S. Entailment and Bivalence. Journal of Philosophical Logic 31, 289–300 (2002). https://doi.org/10.1023/A:1019959811200
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DOI: https://doi.org/10.1023/A:1019959811200