A Hierarchy of Classical and Paraconsistent Logics
Tóm tắt
In this article, we will present a number of technical results concerning Classical Logic, ST and related systems. Our main contribution consists in offering a novel identity criterion for logics in general and, therefore, for Classical Logic. In particular, we will firstly generalize the ST phenomenon, thereby obtaining a recursively defined hierarchy of strict-tolerant systems. Secondly, we will prove that the logics in this hierarchy are progressively more classical, although not entirely classical. We will claim that a logic is to be identified with an infinite sequence of consequence relations holding between increasingly complex relata: formulae, inferences, metainferences, and so on. As a result, the present proposal allows not only to differentiate Classical Logic from ST, but also from other systems sharing with it their valid metainferences. Finally, we show how these results have interesting consequences for some topics in the philosophical logic literature, among them for the debate around Logical Pluralism. The reason being that the discussion concerning this topic is usually carried out employing a rivalry criterion for logics that will need to be modified in light of the present investigation, according to which two logics can be non-identical even if they share the same valid inferences.
Tài liệu tham khảo
Barrio, E., Pailos, F., Szmuc, D. (Forthcoming). A recovery operator for non-transitive approaches. Review of Symbolic Logic. https://doi.org/10.1017/S1755020318000369.
Barrio, E., Pailos, F., Szmuc, D. (Forthcoming). Substructural logics, pluralism and collapse. Synthese. https://doi.org/10.1007/s11229-018-01963-3.
Barrio, E., Pailos, F., Szmuc, D. (2018). What is a paraconsistent logic? In Carnielli, W., & Malinowski, J. (Eds.) Between consistency and inconsistency, trends in logic (pp. 89–108). Dordrecht: Springer.
Barrio, E., Rosenblatt, L., Tajer, D. (2015). The logics of strict-tolerant logic. Journal of Philosophical Logic, 44(5), 551–571.
Blok, W., & Jónsson, B. (2006). Equivalence of consequence operations. Studia Logica, 83(1), 91–110.
Chemla, E., Egré, P., Spector, B. (2017). Characterizing logical consequence in many-valued logics. Journal of Logic and Computation, 27(7), 2193–2226.
Cobreros, P., Egré, P., Ripley, D., van Rooij, R. (2012). Tolerance and Mixed Consequence in the S’valuationist Setting. Studia logica, 100(4), 855–877.
Cobreros, P., Egré, P., Ripley, D., van Rooij, R. (2012). Tolerant, classical, strict. Journal of Philosophical Logic, 41(2), 347–385.
Cobreros, P., Egré, P., Ripley, D., van Rooij, R. (2014). Priest’s motorbike and tolerant identity. In Ciuni, R., Wansing, H., Willkommen, C. (Eds.) Recent trends in logic (pp. 75–83). Cham: Springer.
Cobreros, P., Egré, P., Ripley, D., van Rooij, R. (2014). Reaching transparent truth. Mind, 122(488), 841–866.
Dicher, B., & Paoli, F. (Forthcoming). The original sin of proof-theoretic semantics. Synthese. https://doi.org/10.1007/s11229-018-02048-x.
Dicher, B., & Paoli, F. (Forthcoming). ST, LP, and tolerant metainferences. In Başkent, C., & Ferguson, T.M. (Eds.) Graham priest on dialetheism and paraconsistency. Dordrecht: Springer.
Girard, J.-Y. (1987). Proof theory and logical complexity. Napoli: Bibliopolis.
Humberstone, L . (1996). Valuational semantics of rule derivability. Journal of Philosophical Logic, 25(5), 451–461.
Keefe, R. (2014). What logical pluralism cannot be? Synthese, 191(7), 1375–1390.
Pailos, F. (Forthcoming). A family of metainferential logics. Journal of Applied Non-Classical Logics. https://doi.org/10.1080/11663081.2018.1534486.
Pailos, F. (Forthcoming). A fully classical truth theory characterized by substructural means. The Review of Symbolic Logic. https://doi.org/10.1017/S1755020318000485.
Priest, G. (2006). Logic: one or many? In Woods, J., & Brown, B. (Eds.) Logical consequence: rival approaches. Proceedings of the 1999 conference of the society of exact philosophy. Stanmore: Hermes.
Pynko, A. (2010). Gentzen’s cut-free calculus versus the logic of paradox. Bulletin of the Section of Logic, 39(1/2), 35–42.
Read, S., & Kenyon, T. (2006). Monism: the one true logic. In de Vidi, D. (Ed.) A logical approach to philosophy: essays in memory of Graham Solomon. Dordrecht: Springer.
Ripley, D. (2012). Conservatively extending classical logic with transparent truth. Review of Symbolic Logic, 5(02), 354–378.
Ripley, D. (2013). Paradoxes and failures of cut. Australasian Journal of Philosophy, 91(1), 139–164.
Ripley, D. (2015). Comparing substructural theories of truth. Ergo, 2(13), 299–328.
Rosenblatt, L. Non-contractive classical logic. Notre Dame Journal of Formal Logic. Forthcoming.
Williamson, T. (1987). Equivocation and existence. Proceedings of the Aristotelian Society, 88, 109–127.
Wintein, S. (2016). On all strong Kleene generalizations of classical logic. Studia Logica, 104(3), 503–545.
Woods, J. (2018). Intertranslatability, theoretical equivalence, and perversion. Thought, 7(1), 58–68.
Zardini, E. (2013). Naive modus ponens. Journal of Philosophical Logic, 42(4), 575–593.