Abstract
In Philosophical Logic, the Liar Paradox has been used to motivate the introduction of both truth value gaps and truth value gluts. Moreover, in the light of “revenge Liar” arguments, also higher-order combinations of generalized truth values have been suggested to account for so-called hyper-contradictions. In the present paper, Graham Priest's treatment of generalized truth values is scrutinized and compared with another strategy of generalizing the set of classical truth values and defining an entailment relation on the resulting sets of higher-order values. This method is based on the concept of a multilattice. If the method is applied to the set of truth values of Belnap's “useful four-valued logic”, one obtains a trilattice, and, more generally, structures here called Belnap-trilattices. As in Priest's case, it is shown that the generalized truth values motivated by hyper-contradictions have no effect on the logic. Whereas Priest's construction in terms of designated truth values always results in his Logic of Paradox, the present construction in terms of truth and falsity orderings always results in First Degree Entailment. However, it is observed that applying the multilattice-approach to Priest's initial set of truth values leads to an interesting algebraic structure of a “bi-and-a-half” lattice which determines seven-valued logics different from Priest's Logic of Paradox.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anderson, A.R. and Belnap, N.D., 1975, Entailment: The Logic of Relevance and Necessity, Vol. I, Princeton, NJ: Princeton University Press.
Anderson, A.R., Belnap, N.D., and Dunn, J.M., 1992, Entailment: The Logic of Relevance and Necessity, Vol. II, Princeton, NJ: Princeton University Press.
Belnap, N.D., 1977, “A useful four-valued logic,” in Modern Uses of Multiple-Valued Logic, J.M. Dunn and G. Epstein, eds., D. Reidel Publishing Company, Dordrecht, pp. 8–37.
Belnap, N.D., 1977, “How a computer should think,” in Contemporary Aspects of Philosophy, G. Ryle, ed., Stocksfield, Oriel Press Ltd., pp. 30–55.
Birkhoff, G., 1967, Lattice Theory, Rhode Island: Providence.
Dunn, J.M., 1966, The algebra of intensional logics, Doctoral Dissertation, University of Pittsburgh, Ann Arbor (University Microfilms).
Dunn, J.M., 1976, “Intuitive semantics for first-degree entailment and ‘coupled trees’,”, Philosophical Studies 29, 149–168.
Dunn, J.M., 1986, “Relevance logic and entailment”, in Handbook of Philosophical Logic, D. Gabbay and F. Guenter, eds., Vol. III, Dordrecht: D. Reidel Publishing Company, pp. 117–224.
Dunn, J.M., 1999, “A comparative study of various model-theoretic treatments of negation: A history of formal negation”, in What is Negation?, D.M. Gabbay and H. Wansing, eds., Applied Logic Series, 13, Dordrecht: Kluwer Academic Publishers, pp. 23–51.
Dunn, J.M., 2000, “Partiality and its dual,” Studia Logica 66, 5–40.
Dunn, J.M. and Hardegree, D.M., 2001, Algebraic Methods in Philosophical Logic, Oxford: Clarendon Press.
Dunn, J.M. and Zhou, C., 2005, “Negation in the context of gaggle theory,” Studia Logica 80, 235–264.
Fitting, M., “Bilattices are nice things,” in Self-Reference, T. Bolander, V.F. Hendricks, and S.A. Pedersen, eds., to appear.
Ganeri, J., 2002, “Jaina logic and the philosophical basis of pluralism,” History and Philosophy of Logic 23, 267–281.
Ginsberg, M., 1986, “Multi-valued logics,” in Proceedings of AAAI-86, Fifth National Conference on Artificial Intellegence, Los Altos: Morgan Kaufman Publishers, pp. 243–247.
Ginsberg, M., 1988, “Multivalued logics: a uniform approach to reasoning in AI,” Computer Intelligence 4, 256–316.
Jain, P., 1997, Investigating Hypercontradictions, May (Unpublished Mns), 16 pp.
Meyer, R.K., 1978, Why I am not a relevantist, Research paper, no. 1, Australian National University, Logic Group, Research School of the Social Sciences, Canberra.
Priest, G., 1979, “Logic of paradox,” Journal of Philosophical Logic 8, 219–241.
Priest, G., 1984, “Hyper-contradictions,” Logique et Analyse 27, 237–243.
Priest, G., 2001, An Introduction ot Non-Classical Logic, Cambridge: Cambridge University Press.
Shramko, Y., 2005, “Dual intuitionistic logic and a variety of negations: The logic of scientific research,” Studia Logica 80, 347–367.
Shramko, Y., Dunn, J.M., and Takenaka, T., 2001, “The trilattice of constructive truth values,” Journal of Logic and Computation 11, 761–788.
Shramko, Y. and Wansing, H., 2005, “Some useful sixteen-valued logics: How a computer network should think,” Journal of Philosophical Logic 34, 121–153.
Simmons, K., 2002, “Semantical and logical paradox,” in A Companion to Philosophical Logic, D. Jacquette, ed., Malden, MA: Blackwell Publishers, pp. 115–130.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Shramko, Y., Wansing, H. Hyper-Contradictions, Generalized Truth Values and Logics of Truth and Falsehood. JoLLI 15, 403–424 (2006). https://doi.org/10.1007/s10849-006-9015-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10849-006-9015-0