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ON THREE-VALUED PRESENTATIONS OF CLASSICAL LOGIC

Part of: General logic

Published online by Cambridge University Press:  11 May 2023

BRUNO DA RÉ Open the ORCID record for BRUNO DA RÉ [Opens in a new window] ,
DAMIAN SZMUC Open the ORCID record for DAMIAN SZMUC [Opens in a new window] ,
EMMANUEL CHEMLA Open the ORCID record for EMMANUEL CHEMLA [Opens in a new window]  and
PAUL ÉGRÉ Open the ORCID record for PAUL ÉGRÉ [Opens in a new window]
BRUNO DA RÉ*
Affiliation:
IIF (CONICET-SADAF) UNIVERSITY OF BUENOS AIRES BUENOS AIRES ARGENTINA E-mail: [email protected]
DAMIAN SZMUC
Affiliation:
IIF (CONICET-SADAF) UNIVERSITY OF BUENOS AIRES BUENOS AIRES ARGENTINA E-mail: [email protected]
EMMANUEL CHEMLA
Affiliation:
LABORATOIRE DE SCIENCES COGNITIVES ET DE PSYCHOLINGUISTIQUE DÉPARTEMENT D’ÉTUDES COGNITIVES CNRS, ENS, PSL UNIVERSITY, EHESS PARIS, FRANCE E-mail: [email protected]
PAUL ÉGRÉ
Affiliation:
INSTITUT JEAN-NICOD DÉPARTEMENT D’ÉTUDES COGNITIVES & DÉPARTEMENT DE PHILOSOPHIE CNRS, ENS, PSL UNIVERSITY, EHESS PARIS, FRANCE E-mail: [email protected]
*
E-mail: [email protected]
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Abstract

Given a three-valued definition of validity, which choice of three-valued truth tables for the connectives can ensure that the resulting logic coincides exactly with classical logic? We give an answer to this question for the five monotonic consequence relations $st$, $ss$, $tt$, $ss\cap tt$, and $ts$, when the connectives are negation, conjunction, and disjunction. For $ts$ and $ss\cap tt$ the answer is trivial (no scheme works), and for $ss$ and $tt$ it is straightforward (they are the collapsible schemes, in which the middle value acts like one of the classical values). For $st$, the schemes in question are the Boolean normal schemes that are either monotonic or collapsible.

Keywords

classical logicstrict-tolerant logicsthree-valued logicslogical consequence

MSC classification

Primary: 03B05: Classical propositional logic 03B47: Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) 03B50: Many-valued logic
Type
Research Article
Information
The Review of Symbolic Logic , Volume 17 , Issue 3 , September 2024 , pp. 682 - 704
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

BIBLIOGRAPHY

Barrio, E., Rosenblatt, L., & Tajer, D. (2015). The logics of strict-tolerant logic. Journal of Philosophical Logic, 44(5), 551–571.CrossRefGoogle Scholar
Blasio, C., Marcos, J., & Wansing, H. (2017). An inferentially many-valued two-dimensional notion of entailment. Bulletin of the Section of Logic, 46(3/4), 233–262 CrossRefGoogle Scholar
Cantwell, J. (2008). The logic of conditional negation. Notre Dame Journal of Formal Logic, 49(3), 245–260.CrossRefGoogle Scholar
Carnielli, W., Marcos, J., & De Amo, S. (2000). Formal inconsistency and evolutionary databases. Logic and Logical Philosophy, 8, 115–152.CrossRefGoogle Scholar
Chemla, E., & Égré, P. (2019). Suszko’s problem: Mixed consequence and compositionality. Review of Symbolic Logic, 12(4), 736–767.CrossRefGoogle Scholar
Chemla, E., & Égré, P. (2021). From many-valued consequence to many-valued connectives. Synthese, 198(22), 5315–5352.CrossRefGoogle Scholar
Chemla, E., Égré, P., & Spector, B. (2017). Characterizing logical consequence in many-valued logic. Journal of Logic and Computation, 27(7), 2193–2226.Google Scholar
Cobreros, P., Égré, P., Ripley, D., & van Rooij, R. (2012). Tolerant, classical, strict. Journal of Philosophical Logic, 41(2), 347–385.CrossRefGoogle Scholar
Cobreros, P., Égré, P., Ripley, D., & van Rooij, R. (2015). Vagueness, truth, and permissive consequence. In Achourioti, T., Galinon, H., Martínez Fernández, J., and Fujimoto, K., editors. Unifying the Philosophy of Truth. Dordrecht: Springer, pp. 409–430.Google Scholar
Cooper, W. S. (1968). The propositional logic of ordinary discourse. Inquiry, 11(1–4), 295–320.CrossRefGoogle Scholar
Dicher, B., & Paoli, F. (2019). ST, LP, and tolerant metainferences. In Başkent, C. and Ferguson, T. M.. editors. Graham Priest on Dialetheism and Paraconsistency. Dordrecht: Springer, pp. 383–407.CrossRefGoogle Scholar
Dunn, J. M. (1976). A Kripke-style semantics for R-mingle using a binary accessibility relation. Studia Logica, 35(2), 163–172.CrossRefGoogle Scholar
Égré, P., Rossi, L., & Sprenger, J. (2021). De Finettian logics of indicative conditionals, Part I: Trivalent semantics and validity. Journal of Philosophical Logic, 50(2), 187–213.CrossRefGoogle Scholar
Ferguson, T. M. (2022). Monstrous content and the bounds of discourse. Journal of Philosophical Logic, 52, 111–143.CrossRefGoogle Scholar
Field, H. (2008). Saving Truth from Paradox. New York: Oxford University Press.CrossRefGoogle Scholar
Fitting, M. (1994). Kleene’s three valued logics and their children. Fundamenta Informaticae, 20(1–3), 113–131.CrossRefGoogle Scholar
Fitting, M. (2021). A family of strict/tolerant logics. Journal of Philosophical Logic, 50(2), 363–394.CrossRefGoogle Scholar
Fitting, M. (2021). Strict/tolerant logics built using generalized weak Kleene logics. Australasian Journal of Logic, 18(2), 1–20.CrossRefGoogle Scholar
Frankowski, S. (2004). Formalization of a plausible inference. Bulletin of the Section of Logic, 33(1), 41–52.Google Scholar
French, R. (2016). Structural reflexivity and the paradoxes of self-reference. Ergo, 3(5), 113–131.Google Scholar
Gentzen, G. (1934). Untersuchungen über das logische Schliessen. Mathematische Zeitschrift, 39, 176–210.CrossRefGoogle Scholar
Girard, J.-Y. (1987). Proof Theory and Logical Complexity. Naples: Bibliopolis.Google Scholar
Malinowski, G. (1990). Q-consequence operation. Reports on Mathematical Logic, 24(1), 49–59.Google Scholar
McCarthy, J. (1959). A basis for a mathematical theory of computation. Studies in Logic and the Foundations of Mathematics, 26, 33–70.CrossRefGoogle Scholar
Nicolai, C., & Rossi, L. (2018). Principles for object-linguistic consequence: From logical to irreflexive. Journal of Philosophical Logic, 47(3), 549–577.CrossRefGoogle ScholarPubMed
Peters, S. (1979). A truth-conditional formulation of Karttunen’s account of presupposition. Synthese, 40(2), 301–316.CrossRefGoogle Scholar
Post, E. (1921). Introduction to a general theory of elementary propositions. American Journal of Mathematics, 43(3), 163–185.CrossRefGoogle Scholar
Priest, G. (2008). An Introduction to Non-Classical Logic: From If to Is (second edition). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Priest, G. (2023). Interpretations of the third value. In Égré, P. and Rossi, L., editors. Handbook of Trivalent Logics. MIT Press, forthcoming.Google Scholar
Rescher, N. (1969). Many-Valued Logic. New York: Mcgraw-Hill.Google Scholar
Suszko, R. (1977). The Fregean axiom and Polish mathematical logic in the 1920s. Studia Logica, 36(4), 377–380.CrossRefGoogle Scholar
Szmuc, D., & Ferguson, T. M. (2021). Meaningless divisions. Notre Dame Journal of Formal Logic, 62(3), 399–424.CrossRefGoogle Scholar
Teijeiro, P. (2023). Vague connectives. Philosophical Studies, 180: 1559–1578.Google Scholar