Abstract
We develop a paraconsistent logic by introducing new models for conditionals with acceptive and rejective selection functions which are variants of Chellas’ conditional models. The acceptance and rejection conditions are substituted for truth conditions of conditionals. The paraconsistent conditional logic is axiomatized by a sequent system \(\mathcal {C}\) which is an extension of the Belnap-Dunn four-valued logic with a conditional operator. Some acceptive extensions of \(\mathcal {C}\) are shown to be sound and complete. We also show the finite acceptive model property and decidability of these logics.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Belnap, N.D. (1977). G.Epstein A useful four-valued logic. In Dunn, J.M. (Ed.) Modern uses of multiple-valued logic (pp. 5–37). Pittsburgh: University of Pittsburgh Press.
Bobenrieth, A.M. (2007). Paraconsistency and the consistency or inconsistency of the world. In Beziau, J.-Y., Carnielli, W., Gabbay, D. (Eds.) Handbook of paraconsistency (pp. 493–512). London: College Publications.
Chellas, B.F. (1975). Basic conditional logic. Journal of Philosophical Logic, 4, 133–153.
Dunn, J.M. (1986). Relevance logic and entailment. In Gabbay, D., & Guenthner, F. (Eds.) Handbook of philosophical logic volume III: alternatives in classical logic (pp. 117–224). Dordrecht: Springer Science+Business Media.
Dunn, J.M. (1999). 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): Kluwer Academic Publishers.
Dunn, J.M. (2000). Partiality and its dual. Studia Logica, 65, 5–40.
Edgington, D. (2005). Ramsey’s legacies on conditionals and truth. In Lillehammer, H., & Mellor, D. H. (Eds.) Ramsey’s legacy (pp. 37–52). New York: Oxford University Press.
Lewis, D. (1973). Counterfactuals. Oxford: Blackwell.
Ma, M., & Lin, Y. (2018). Countably many weakenings of Belnap–Dunn logic. Studia Logica. https://doi.org/10.1007/s11225-018-9840-6.
Ma, M., & Lin, Y. (2019). A deterministic weakening of Belnap–Dunn logic. Studia Logica, 107, 283–312.
Fitting, M. (1988). Logic programming on a topological bilattice. Fundamenta Informatica, 11, 209–218.
Ginsberg, M.L. (1988). Multivalued logics: a uniform approach to inference in artificial intelligence. Computational Intelligence, 4, 265–316.
Muskens, R. (1999). On Partial and Paraconsistent Logics. Notre Dame Journal of Formal Logic, 40, 352–374.
Priest, G. (2007). Paraconsistency and dialetheism. In Gabbay, D.M., & Woods, J. (Eds.) Handbook of the history of logic vol. 8: the many valued and nonmonotonic turn in logic (pp. 129–204). Amsterdam: Elsevier.
Ramsey, F. (2013). Truth and probability. In Ramsey, F. (Ed.) The foundations of mathematics and other logical essays (pp. 156–198). Routledge.
Ramsey, F. (2013). Truth and propositions. In Ramsey, F. (Ed.) The foundations of mathematics and other logical essays (pp. 138–155). Routledge.
Ramsey, F. (1929). General propositions and causality. In Ramsey, F. (Ed.) The foundations of mathematics (pp. 237–255): VDM Verlag.
Stalnaker, R.C. (2012). A theory of conditionals. In Rescher, N. (Ed.) Ifs: conditionals, belief, decision, chance and time (pp. 98–112). Oxford: Basil Blackwell Publishers.
Acknowledgements
Thanks are given to the referee for the helpful comments on the manuscript of this paper. The first author of this work was supported by Chinese National Funding of Social Sciences (Grant No. 18ZDA033).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Ma, M., Wong, CT. A Paraconsistent Conditional Logic. J Philos Logic 49, 883–903 (2020). https://doi.org/10.1007/s10992-019-09540-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10992-019-09540-w