Abstract
Does there exist any equivalence between the notions of inconsistency and consequence in paraconsistent logics as is present in the classical two valued logic? This is the key issue of this paper. Starting with a language where negation (\({\neg}\)) is the only connective, two sets of axioms for consequence and inconsistency of paraconsistent logics are presented. During this study two points have come out. The first one is that the notion of inconsistency of paraconsistent logics turns out to be a formula-dependent notion and the second one is that the characterization (i.e. equivalence) appears to be pertinent to a class of paraconsistent logics which have double negation property.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arruda I.A.: Aspects of the historical development of paraconsistent logic. In: Priest, G., Routley, R., Norman, J. (eds) Paraconsistent Logic: Essays on the Inconsistent, pp. 99–129. Philosophia, München (1989)
Avron A.: Natural 3-valued logics: characterization and proof theory. J. Symb. Log. 56(1), 276–294 (1991)
Carnielli, A.W., Marcos, J.: A taxonomny of C systems. In: Carnielli, W.A., Coniglio, M.E., D‘Ottavino, I.M.L. (eds.) Paraconsistency—the logical Way to the Inconsistent Lecture Notes in Pure Applied and Mathematics, vol. 228, pp. 1–94. Marcel Dekker, New York (2002)
Carnielli A.W., Coniglio M.E., Marcos J.: Logics of formal inconsistency. In: Gabbay, D., Guenthner, F. (eds) Handbook of Philosophical Logic, vol. 14, pp. 1–93. Kluwer Academic Publishers, Netherlands (2003)
Chakraborty M.K., Basu S.: Graded consequence and some metalogical notions generalized. Fundamenta informaticae 32, 299–311 (1997)
Dubois D., Prade H.: Possibilistic logic, a retrospective and prospective view. Fuzzy Sets Syst. 144(1), 3–23 (2004)
Michael Dunn J.: Star and perp: two treatments of negation. Philos. Perspect. Lang. Log. 7, 331–357 (1993)
Michale Dunn, J.: A comparative study of various model theoretic treatments of negation: a history of formal negation. In: Gabbay, D.M., Wansing H. (eds.) What is Negation? pp. 23–51 (1999)
Michael Dunn J., Restall G.: Relevance logic. In: Gabbay, D., Guenthner, F. (eds) Handbook of Philosophical Logic, vol. 6, 2nd edn, pp. 1–136. Kluwer Academic Publishers, Dordrecht (2002)
Gentzen G. (1969). Investigations into logical deductions. In: The Collected Papers of Gentzen G., Szabo M.E. (eds). North Holland Publications, Amsterdam, pp 68–131
Goswami, S.: Relevant logic: philosophy and applications. Ph.D thesis submitted at Jadavpur University (2009)
Restall G.: Laws of non-contradiction, laws of the excluded middle, and logics. In: Priest, G., Beall, J.C., Garb-Armour, J.C. (eds) The Law of Non-Contradiction., pp. 73–84. Oxford University Press, New York (2004)
Priest G., Routley R.: A preliminary history of paraconsistent and dialethic approaches. In: Priest, G., Routley, R., Norman, J. (eds) Paraconsistent Logic: Essays on the Inconsistent., pp. 3–75. Philosophia, München (1989)
Surma S.J.: The growth of logic out of the foundational research in mathematics. In: Agazzi, E. (eds) Modern Logic—A Survey., pp. 15–33. D. Reidel Publishing co., Dordrecht (1981)
Tarski, A.: Fundamentale begriffe der metodologie der deduktiven wissenschaften, monatshefte fuer mathematik und physik. XXXVII, 361–404 (1930)
Tarski, A.: Methodology of deductive sciences. In: Logic, Semantics, Mathematics, pp. 60–109. Clarendon Press, Oxford (1956)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dutta, S., Chakraborty, M.K. Negation and Paraconsistent Logics. Log. Univers. 5, 165–176 (2011). https://doi.org/10.1007/s11787-011-0029-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11787-011-0029-2