## Notre Dame Journal of Formal Logic

Achille C. Varzi

#### Abstract

Lewis has argued that impossible worlds are nonsense: if there were such worlds, one would have to distinguish between the truths about their contradictory goings-on and contradictory falsehoods about them; and this--Lewis argues--is preposterous. In this paper I examine a way of resisting this argument by giving up the assumption that in so-and-so world' is a restricting modifier which passes through the truth-functional connectives. The outcome is a sort of subvaluational semantics which makes a contradiction 'A and not-A' false even when both 'A' and 'not-A' are true, just as supervaluational semantics makes a tautology 'A and not-A' true even when neither 'A' and 'not-A' are.

#### Article information

Source
Notre Dame J. Formal Logic Volume 38, Number 4 (1997), 621-639.

Dates
First available in Project Euclid: 10 December 2002

http://projecteuclid.org/euclid.ndjfl/1039540773

Digital Object Identifier
doi:10.1305/ndjfl/1039540773

Mathematical Reviews number (MathSciNet)
MR1648856

Zentralblatt MATH identifier
0916.03016

#### Citation

Varzi, Achille C. Inconsistency without Contradiction. Notre Dame J. Formal Logic 38 (1997), no. 4, 621--639. doi:10.1305/ndjfl/1039540773. http://projecteuclid.org/euclid.ndjfl/1039540773.

#### References

• [1]Anderson, A. R., N. D. Belnap, Jr., and J. M. Dunn, Entailment: The Logic of Relevance and Necessity, vol. 2, Princeton University Press, Princeton, 1992.
• [2]Belnap, N. D. Jr., A useful four-valued logic,'' pp. 8--37 in Modern Uses of Multiple-Valued Logics, edited by J. M. Dunn and G. Epstein, Reidel, Dordrecht, 1977.
• [3]Bencivenga, E., Free semantics for indefinite descriptions,'' Journal of Philosophical Logic, vol. 7 (1978), pp. 389--405.
• [4]Bencivenga, E., Free semantics,'' Boston Studies in the Philosophy of Science, vol. 47 (1981), pp. 31--48.
• [5] Collins, J., and A. C. Varzi, Unsharpenable vagueness,'' Proceedings of the Conference on Methods in Philosophy and the Sciences, New York, 1997.
• [6] Fine, K., Vagueness, truth, and logic,'' Synthese, vol. 30 (1975), pp. 265--300.
• [7] Fitting, M., Kleene's logic, generalized,'' Journal of Logic and Computation, vol. 18 (1992), pp. 225--56.
• [8] Fodor, J. A., and E. Lepore, What cannot be evaluated cannot be evaluated, and it cannot be supervalued either,'' The Journal of Philosophy, vol. 93 (1996), pp. 516--35.
• [9]Gärdenfors, P., Knowledge in Flux, The MIT Press, Cambridge, 1988.
• [10] Grant, J., Inconsistent and incomplete logics,'' Mathematics Magazine, vol. 48 (1975), pp. 154--9.
• [11]Gupta, A., and N. D. Belnap, Jr., The Revision Theory of Truth, The MIT Press, Cambridge, 1993.
• [12]Hyde, D., From heaps and gaps to heaps of gluts,'' Mind, vol. 106 (1997), pp. 641--60.
• [13] Jaśkowski, S.,Rachunek zdańdla systemów dedukcyjnych sprzecznych,'' Studia Societatis Scientiarum Torunensis, \S A 1, 8 (1948), pp. 55--77. (Translated as Propositional calculus for contradictory deductive systems,'' Studia Logica, vol. 24 (1965), pp. 143--57.)
• [14] Jennings, R. E., and P. K. Schotch, The preservation of coherence,'' Studia Logica, vol. 43 (1984), pp. 89--106.
• [15] Kamp, H., Two theories about adjectives,'' pp. 45--89 in Semantics and Natural Language, edited by E. L. Keenan, Cambridge University Press, Cambridge, 1975.
• [16]Kyburg, H. E., Jr., The rule of adjunction and reasonable inference,'' The Journal of Philosophy, vol. 94 (1997), pp. 109--25.
• [17]Levi, I., The Enterprise of Knowledge, The MIT Press, Cambridge, 1980.
• [18]Levi, I., The Fixation of Belief and Its Undoing, Cambridge University Press, Cambridge, 1991.
• [19]Lewis, D. K., Truth in fiction,'' American Philosophical Quarterly, vol. 15 (1978), pp. 37--46.
• [20]Lewis, D. K., Logic for Equivocators,'' Noûs, vol. 16 (1982), pp. 431--41.
• [21]Lewis, D. K., Postscripts to Anselm and Actuality','' pp. 21--25 in Philosophical Papers, vol. 1, Oxford University Press, New York, 1983.
• [22]Lewis, D. K., Postscripts to Truth in Fiction','' pp. 276--80 in Philosophical Papers, vol. 1, Oxford University Press, New York, 1983.
• [23] Lewis, D. K., On The Plurality of Worlds, Blackwell, Oxford, 1986.
• [24] McCall, S., Excluded middle, bivalence, and fatalism,'' Inquiry, vol. 4 (1966), pp. 384--6.
• [25] Mehlberg, H., The Reach of Science, Toronto University Press, Toronto, 1958.
• [26] Priest, G., Sylvan's box,'' Notre Dame Journal of Formal Logic, vol. 38 (1997), pp. 573--82.
• [27] Priest, G., and Routley, R., Systems of paraconsistent logic,'' pp. 151--86 in Paraconsistent Logic. Essays on the Inconsistent, edited by G. Priest, R. Routley, and J. Norman, Philosophia Verlag, Munich, 1989.
• [28]Rescher, N., and R. Brandom, The Logic of Inconsistency. A Study in Non-Standard Possible-World Semantics and Ontology, Basil Blackwell, Oxford, 1980.
• [29] Schotch, P. K., and R. E. Jennings, On detonating,'' pp. 306--27 in Paraconsistent Logic. Essays on the Inconsistent, edited by G. Priest, R. Routley, and J. Norman, Philosophia Verlag, Munich, 1989.
• [30]Scott, D., Continuous lattices,'' pp. 97--136 in Toposes, Algebraic Geometry and Logic, edited by F. W. Lawvere, Springer-Verlag, Berlin, 1972.
• [31] van Fraassen, B. C., Singular terms, truth-value gaps, and free logic,'' The Journal of Philosophy, vol. 63 (1966), pp. 481--95.
• [32] van Fraassen, B. C., The completeness of free logic,'' Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 12 (1966), pp. 219--34.
• [33]Varzi, A. C., Universal semantics,'' Ph.D. thesis, University of Toronto, 1994.
• [34]Varzi, A. C., Super-duper supervaluationism,'' pp. 17--40 in Logica '94. Proceedings of the 8th International Symposium, edited by T. Childers and O. Majer, Filosofia, Prague, 1995.
• [35]Visser, A., `Four-valued semantics and the liar,'' Journal of Philosophical Logic, vol. 12 (1984), pp. 181--212.