Paraconsistent logics are, by definition, inconsistency
tolerant: In a paraconsistent logic, inconsistencies need not
entail everything. However, there is more than one way a body
of information can be inconsistent. In this paper I distinguish
{contradictions} from {other inconsistencies}, and I show that
several different logics are, in an important sense, "paraconsistent"
in virtue of being inconsistency tolerant without thereby
being contradiction tolerant. For example, even though no
inconsistencies are tolerated by intuitionistic propositional
logic, some inconsistencies are tolerated by intuitionistic
predicate logic. In this way, intuitionistic predicate logic
is, in a mild sense, paraconsistent. So too are orthologic and quantum
propositional logic and other formal systems. Given this fact, a
widespread view—that traditional paraconsistent logics are especially
repugnant because they countenance inconsistencies—is undercut. Many well-understood nonclassical logics countenance inconsistencies
as well.
References
[1] Anderson, A. R. , and N. D. Belnap, Entailment: The Logic of Relevance and Necessity, vol. 1, Princeton University Press, Princeton, 1975.
[2] Anderson, A. R., N. D. Belnap , and J. M. Dunn, Entailment: The Logic of Relevance and Necessity, vol. 2, Princeton University Press, Princeton, 1992.
[3] Arruda, A. I., "Aspects of the historical development of paraconsistent logic", pp. 99--130 in Paraconsistent Logic: Essays on the Inconsistent, edited by G. Priest, R. Routley, and J. Norman, Philosophia Verlag, 1989.
[4] Bell, J. L., "A new approach to quantum logic", The British Journal for the Philosophy of Science, vol. 37 (1986), pp. 83--99.
[5] Bell, J. L., A Primer of Infinitesimal Analysis, Cambridge University Press, Cambridge, 1998.
[6] Dummett, M., Elements of Intuitionism, Clarendon Press, Oxford, 1977.
[7] Fitting, M., Intuitionistic Logic, Model Theory and Forcing, Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Company, Amsterdam, 1969.
[8] Goldblatt, R., "Semantic analysis of orthologic", The Journal of Philosophical Logic, vol. 3 (1974), pp. 19--35.
[9] Meyer, R. K. , and E. P. Martin, "Logic on the Australian plan", The Journal of Philosophical Logic, vol. 15 (1986), pp. 305--32.
[10] \global\editortrue Priest, G., R. Routley , and J. Norman, editors, Paraconsistent Logic. Essays on the Inconsistent, Analytica, Philosophia Verlag, Munich, 1989.
[11] Priest, G., "The logic of paradox", The Journal of Philosophical Logic, vol. 8 (1979), pp. 219--41.
[12] Priest, G., In Contradiction. A Study of the Transconsistent, vol. 39 of Nijhoff International Philosophy Series, Martinus Nijhoff Publishers, Dordrecht, 1987.
[13] Priest, G. , and R. Routley, "Introduction", pp. xix--xxi in Paraconsistent Logic: Essays on the Inconsistent, edited by G. Priest and R. Routley and J. Norman, Philosophia Verlag, Munich, 1989.
[14] Restall, G., ``Paraconsistent logics!'' Bulletin of the Section of Logic, vol. 26 (1997), pp. 156--63.
[15] Routley, R., V. Plumwood, R. K. Meyer , and R. T. Brady, Relevant Logics and their Rivals. Part 1: The Basic Philosophical and Semantical Theory, Ridgeview Publishing Company, Atascadero, 1982.
[16] Slater, B. H., ``Paraconsistent logics?'' The Journal of Philosophical Logic, vol. 24 (1995), pp. 451--54.
[17] van Dalen, D., "Intuitionistic logic", pp. 225--339 in Handbook of Philosophical Logic, vol. 3, edited by D. M. Gabbay and F. Günthner, Reidel, Dordrecht, 1986.
[18] van Dalen, D., "The intuitionistic conception of logic", pp. 45--77 in European Review of Philosophy, edited by A. Varzi, CSLI Publications, Stanford, 1999.
