Semantic Closure and Classicality
DOI:
https://doi.org/10.22370/rhv2023iss22pp85-103Keywords:
semantic paradoxes, recovery operators, paraconsistent logics, substructural logics, classical logicAbstract
The semantic paradoxes show that semantic theories that internalize their semantic concepts, such as truth and validity, cannot validate all classical logic. That is, it is necessary to weaken some connective of the object language, taken as the guilty of the paradoxes, or give up some property of the consequence relation of the logical theory. Both strategies may distance us from classical logic, the logic commonly used in our current mathematical theories. So, a desirable solution to semantic paradoxes cannot distance us from classical logic. This paper analyzes two interesting proposals that aim to maintain classical logic as most as possible. The first strategy, the Barrio & Pailos & Szmuc-approach (2017) (BPS-approach), proposes the paraconsistent logic MSC that contains in its object language a connective capable of recovering classical inference whenever the sentences at issue are consistent. So, they show that it is possible to build a semantic theory over this logic that is immune to the semantic paradoxes. The second approach is based on the hierarchy STω of non-transitive systems STn, proposed by Pailos (2020a). This hierarchy recovers classical metainferences as many as possible in higher levels of the hierarchy. We argue in favor of the second approach by arguing that the first strategy must adopt weak self-referential procedures to avoid the paradoxes.
References
Antunes, H. (2020). Enthymematic classical recapture. Logic Journal of the IGPL, 28(5), 817-831.
Asenjo, F. G. (1966). A calculus of antinomies. Notre Dame Journal of Formal Logic, 7(1), 103-105.
Barrio, E. (2014). La lógica de la verdad. Eudeba.
Barrio, E., & Bezerra, E. (2023). The logics of a universal language. Manuscript, 1-26.
Barrio, E., Rosenblatt, L., & Tajer, D. (2016). Capturing naive validity in the cut-free approach. Synthese, 199(Suppl 3), 707-723.
Barrio, E., Pailos, F., & Calderón, J. T. (2021). Anti-exceptionalism, truth and the BA-plan. Synthese, 199, 12561–12586. https://doi.org/10.1007/s11229-021-03343-w.
Barrio, E., Pailos, F., & Szmuc, D. (2020). A hierarchy of classical and paraconsistent logics. Journal of Philosophical Logic, 49(1), 93-120.
Barrio, E., Pailos, F. M., & Szmuc, D. E. (2017). A paraconsistent route to semantic closure. Logic Journal of the IGPL, 25(4), 387-407.
Beall, J. (2013). LP+, K3+, FDE+, and their ‘classical collapse’. The Review of Symbolic Logic, 6(4), 742-754.
Beall, J., & Murzi, J. (2013). Two flavors of curry’s paradox. The Journal of Philosophy, 110(3), 143-165.
Carnielli, W., Coniglio, M.E., & Marcos, J. (2007). Logics of Formal Inconsistency. In D. Gabbay & F. Guenthner (Eds.), Handbook of Philosophical Logic, Vol. 14 (pp. 1-93). Springer. https://doi.org/10.1007/978-1-4020-6324-4_1
Carnielli, W., Coniglio, M. E., & Rodrigues, A. (2020). Recovery operators, paraconsistency and duality. Logic Journal of the IGPL, 28(5), 624-656.
Carnielli, W. A., & Coniglio, M. E. (2016). Paraconsistent logic: Consistency, contradiction and negation. Logic, Epistemology, and the Unity of Science, Vol. 40. Springer.
Chemla, E., Égré, P., & Spector, B. (2017). Characterizing logical consequence in many-valued logic. Journal of Logic and Computation, 27(7), 2193-2226.
Ciuni, R., & Carrara, M. (2020). Normality operators and classical recapture in many-valued logic. Logic Journal of the IGPL, 28(5), 657-683.
Cobreros, P., Egré, P., Ripley, D., & van Rooij, R. (2012). Tolerant, classical, strict. Journal of Philosophical Logic, 41(2), 347-385.
Corbalán, M. I. (2012). Conectivos de restauração local. Dissertação de mestrado. Universidade Estadual de Campinas.
Da Costa, N. C. (1974). On the theory of inconsistent formal systems. Notre dame journal of formal logic, 15(4), 497-510.
Ferguson, T. M., & Ramírez-Cámara, E. (2022). Deep ST. Journal of Philosophical Logic, 51, 1261-1293. https://doi.org/10.1007/s10992-021-09630-8
Field, H. (2008). Saving truth from paradox. Oxford University Press.
French, R. (2016). Structural reflexivity and the paradoxes of self-reference. Ergo, an Open Access Journal of Philosophy, 3(5). https://doi.org/10.3998/ergo.12405314.0003.005
Golan, R. (2023). On the Metainferential Solution to the Semantic Paradoxes. ournal of Philosophical Logic, 52, 797-820. https://doi.org/10.1007/s10992-022-09688-y
Goodship, L. (1996). On dialethism. Australasian Journal of Philosophy, 74(1), 153-161.
Kleene, S. C. (1938). On notation for ordinal numbers. The Journal of Symbolic Logic, 3(4), 150-155.
Kripke, S. (1976). Outline of a theory of truth. The journal of philosophy, 72(19), 690-716.
Marcos, J. (2005). Nearly every normal modal logic is paranormal. Logique et Analyse, 48(189/192), 279-300.
Meadows, T. (2014). Fixed points for consequence relations. Logique et Analyse, 57(227), 333-357. https://www.jstor.org/stable/44085292
Murzi, J., & Rossi, L. (2021). Naïve validity. Synthese, 199(Suppl 3), 819-841. https://doi.org/10.1007/s11229-017-1541-6
Pailos, F. M. (2020a). A fully classical truth theory characterized by substructural means. The Review of Symbolic Logic, 13(2), 249-268.
Pailos, F. M. (2020b). Validity, dialetheism and self-reference. Synthese, 197(2), 773-792.
Picollo, L. (2020). Truth in a logic of formal inconsistency: How classical can it get? Logic Journal of the IGPL, 28(5), 771-806.
Porter, B. (2023). Three Essays on Substructural Approaches to Semantic Paradoxes. PhD thesis, City University of New York.
Priest, G. (1979). The logic of paradox. Journal of Philosophical logic, 8(1), 219-241.
Priest, G. (2006). In Contradiction. Oxford University Press.
Ripley, D. (2012). Conservatively extending classical logic with transparent truth. The Review of Symbolic Logic, 5(2), 354-378.
Ripley, D. (2013). Paradoxes and failures of cut. Australasian Journal of Philosophy, 91(1), 139-164.
Roffe, A. J., & Pailos, F. (2021). Translating metainferences into formulae: satisfaction operators and sequent calculi. The Australasian Journal of Logic, 18(7), 724-743.
Rosenblatt, L. (2021). Expressing consistency consistently. Thought: A Journal of Philosophy, 10(1), 33-41.
Scambler, C. (2020). Classical logic and the strict tolerant hierarchy. Journal of Philosophical Logic, 49(2), 351-370.
Tajer, D. (2020). and methods of classical recapture. Logic Journal of the IGPL, 28(5), 807-816.
Tarski, A. (1956). Logic, semantics, metamathematics: papers from 1923 to 1938. Oxford Clarendom Press.
Weber, Z. (2014). Naive validity. The Philosophical Quarterly, 64(254), 99-114.
Zardini, E. (2014). Naive truth and naive logical properties. The Review of Symbolic Logic, 7(2), 351-384.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Revista de Humanidades de Valparaíso y Universidad de Valparaíso
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication, with the work after publication simultaneously licensed under a Creative Commons Attribution License (CC BY-NC-ND 4.0 International) that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
