The (Greatest) Fragment of Classical Logic that Respects the Variable-Sharing Principle (in the FMLA-FMLA Framework)

Szmuc, Damian E. 

Bulletin of the Section of Logic, Tome 50 (2021) no. 4, pp. 421-453.

Résumé

We examine the set of formula-to-formula valid inferences of Classical Logic, where the premise and the conclusion share at least a propositional variable in common. We review the fact, already proved in the literature, that such a system is identical to the first-degree entailment fragment of R. Epstein's Relatedness Logic, and that it is a non-transitive logic of the sort investigated by S. Frankowski and others. Furthermore, we provide a semantics and a calculus for this logic. The semantics is defined in terms of a p-matrix built on top of a 5-valued extension of the 3-element weak Kleene algebra, whereas the calculus is defined in terms of a Gentzen-style sequent system where the left and right negation rules are subject to linguistic constraints.

Détail

Keywords: Relevant logics, non-transitive logics, p-matrix, weak Kleene algebra, infectious logics

Bibliographie

[1] A. Anderson, N. Belnap, Entailment: The Logic of Relevance and Neccessity, vol. 1, Princeton University Press, Princeton (1975).

[2] G. Badia, Variable sharing in substructural logics: an algebraic characterization, Bulletin of the Section of Logic, vol. 47(2) (2018), pp. 107–115 | DOI

[3] E. Barrio, F. Pailos, D. Szmuc, A Hierarchy of Classical and Paraconsistent Logics, Journal of Philosophical Logic, vol. 1(49) (2020), pp. 93–120 | DOI

[4] N. Belnap, How a Computer Should Think, [in:] G. Ryle (ed.), Contemporary Aspects of Philosophy, Oriel Press, Stocksfield (1977), pp. 30–55.

[5] F. Berto, Simple hyperintensional belief revision, Erkenntnis, vol. 84(3) (2019), pp. 559–575 | DOI

[6] D. Bochvar, On a Three-Valued Calculus and its Application in the Analysis of the Paradoxes of the Extended Functional Calculus, Matematicheskii Sbornik, vol. 4 (1938), pp. 287–308.

[7] W. Carnielli, Methods of proof for relatedness logic and dependence logics, Reports on Mathematical Logic, vol. 21 (1987), pp. 35–46.

[8] P. Cobreros, P. Égré, D. Ripley, R. van Rooij, Tolerance and mixed consequence in the s’valuationist setting, Studia Logica, vol. 100(4) (2012), pp. 855–877 | DOI

[9] P. Cobreros, P. Égré, D. Ripley, R. van Rooij, Tolerant, classical, strict, Journal of Philosophical Logic, vol. 41(2) (2012), pp. 347–385.

[10] P. Cobreros, P. Égré, D. Ripley, R. van Rooij, Priest’s motorbike and tolerant identity, [in:] R. Ciuni, H. Wansing, C. Wilkommen (eds.), Recent Trends in Philosophical Logic, Springer, Cham (2014), pp. 75–85 | DOI

[11] P. Cobreros, P. Égré, D. Ripley, R. van Rooij, Reaching transparent truth, Mind, vol. 122(488) (2014), p. 841–866.

[12] M. Coniglio, M. I. Corbalán, Sequent Calculi for the classical fragment of Bochvar and Halldén’s Nonsense Logics, [in:] Proceedings of the 7th Workshop on Logical and Semantic Frameworks with Applications (LSFA) (2012), pp. 125–136 | DOI

[13] F. Correia, Weak necessity on weak Kleene matrices, [in:] F. Wolter, H. Wansing, M. de Rijke, M. Zakharyashev (eds.), Advances in Modal Logic, World Scientific, River Edge, NJ (2002), pp. 73–90 | DOI

[14] B. Dicher, F. Paoli, ST, LP, and tolerant metainferences, [in:] C. Başkent, T. M. Ferguson (eds.), Graham Priest on Dialetheism and Paraconsistency, Springer, Dordrecht (2020), pp. 383–407 | DOI

[15] M. Dunn, Intuitive semantics for first-degree entailments and ‘coupled trees’, Philosophical Studies, vol. 29(3) (1976), pp. 149–168 | DOI

[16] R. Epstein, The Semantic Foundations of Logic, vol. I: Propositional Logics, Springer (1990) | DOI

[17] L. Fariñas del Cerro, V. Lugardon, Sequents for dependence logics, Logique et Analyse, vol. 133–134 (1991), pp. 57–71.

[18] M. Fitting, The Strict/Tolerant Idea and Bilattices, [in:] O. Arieli, A. Zamansky (eds.), Arnon Avron on Semantics and Proof Theory of Non-Classical Logics, Springer (2021).

[19] J. M. Font, Abstract Algebraic Logic, College Publications, London (2016).

[20] S. Frankowski, Formalization of a Plausible Inference, Bulletin of the Section of Logic, vol. 33(1) (2004), pp. 41–52.

[21] R. French, A Simple Sequent Calculus for Angell’s Logic of Analytic Containment, Studia Logica, vol. 105(5) (2017), pp. 971–994 | DOI

[22] N. Galatos, P. Jipsen, T. Kowalski, H. Ono, Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Elsevier, San Diego, CA, USA (2007).

[23] J.-Y. Girard, Proof theory and logical complexity, Bibliopolis, Napoli (1987).

[24] S. Halldén, The Logic of Nonsense, Uppsala Universitets Arsskrift, Uppsala (1949).

[25] L. Humberstone, The Connectives, MIT Press, Cambridge, MA (2011).

[26] S. C. Kleene, Introduction to metamathematics, North-Holland, Amsterdam (1952).

[27] S. Kripke, Outline of a theory of truth, Journal of Philosophy, vol. 72(19) (1975), pp. 690–716 | DOI

[28] E. Mares, Relevance Logic, [in:] E. N. Zalta (ed.), The Stanford Encyclopedia of Philosophy, spring 2014 ed., Metaphysics Research Lab, Stanford University (2014), URL: https://plato.stanford.edu/archives/spr2014/entries/logic-relevance/

[29] E. J. Nelson, Intensional relations, Mind, vol. 39(156) (1930), pp. 440–453.

[30] F. Paoli, Semantics for first-degree relatedness logic, Reports on Mathematical Logic, vol. 27 (1993), pp. 81–94.

[31] F. Paoli, Tautological entailments and their rivals, [in:] J. Béziau, W. Carnielli, D. Gabbay (eds.), Handbook of Paraconsistency, College Publications, London (2007), pp. 153–175.

[32] F. Paoli, M. P. Baldi, Proof Theory of Paraconsistent Weak Kleene Logic, Studia Logica, vol. 108(4) (2020), pp. 779–802 | DOI

[33] W. T. Parry, Ein Axiomensystem für eine neue Art von Implikation (analytische Implikation), Ergebnisse eines mathematischen Kolloquiums, vol. 4 (1933), pp. 5–6.

[34] D. Ripley, Conservatively extending classical logic with transparent truth, Review of Symbolic Logic, vol. 5(2) (2012), pp. 354–378 | DOI

[35] D. Ripley, Paradoxes and failures of cut, Australasian Journal of Philosophy, vol. 91(1) (2013), pp. 139–164 | DOI

[36] G. Robles, J. M. Méndez, A Class of Simpler Logical Matrices for the Variable-Sharing Property, Logic and Logical Philosophy, vol. 20(3) (2011), pp. 241–249 | DOI

[37] G. Robles, J. M. Méndez, A general characterization of the variable-sharing property by means of logical matrices, Notre Dame Journal of Formal Logic, vol. 53(2) (2012), pp. 223–244 | DOI

[38] G. Robles, J. M. Méndez, F. Salto, A Modal Restriction of R-Mingle with the Variable-Sharing Property, Logic and Logical Philosophy, vol. 19(4) (2010), pp. 341–351 | DOI

[39] C. Scambler, Classical logic and the strict tolerant hierarchy, Journal of Philosophical Logic, vol. 2(49) (2020), p. 351–370 | DOI

[40] K. Schütte, Proof Theory, Springer, Berlin (1977).

[41] D. Szmuc, A simple matrix semantics and sequent calculus for Parry’s logic of Analytic Implication, Studia Logica, (2021) | DOI

[42] D. Szmuc, T. M. Ferguson, Meaningless Divisions, forthcoming in Notre Dame Journal of Formal Logic.

[43] G. Takeuti, Proof Theory, 2nd ed., Elsevier, Amsterdam (1987).

[44] R. Wójcicki, Some Remarks on the Consequence Operation in Sentential Logics, Fundamenta Mathematicae, vol. 68(1) (1970), pp. 269–279.