An Epistemic Interpretation of Paraconsistent Weak Kleene Logic

• Authors: Szmuc Damian E.
• Languages of publication: EN

Abstracts

EN

This paper extends Fitting’s epistemic interpretation of some Kleene logics to also account for Paraconsistent Weak Kleene logic. To achieve this goal, a dualization of Fitting’s “cut-down” operator is discussed, leading to the definition of a “track-down” operator later used to represent the idea that no consistent opinion can arise from a set including an inconsistent opinion. It is shown that, if some reasonable assumptions are made, the truth-functions of Paraconsistent Weak Kleene coincide with certain operations defined in this track-down fashion. Finally, further reflections on conjunction and disjunction in the weak Kleene logics accompany this paper, particularly concerning their relation with containment logics. These considerations motivate a special approach to defining sound and complete Gentzen-style sequent calculi for some of their four-valued generalizations.

Keywords

EN

weak Kleene logic; infectious logic; containment logic; sequent calculus

• Publisher: Uniwersytet Mikołaja Kopernika (Nicolaus Copernicus University in Toruń)
• Journal: Logic and Logical Philosophy
• Year: 2019
• Volume: 28
• Issue: 2
• Pages: 277-330

Dates

published

2019-06-15

Contributors

author

Szmuc Damian E.

• Department of Philosophy University of Buenos Aires
• IIF-SADAF CONICET Buenos Aires, Argentina

References

• Angell, R.B., “Deducibility, entailment and analytic containment”, pages 119–143 in J. Norman and R. Sylvan (eds.), Directions in Relevant Logic, Springer, Dordrecht, 1989. DOI: http://dx.doi.org/10.1007/978-94-009-1005-8_8
• Avron, A., “Natural 3-valued logicscharacterization and proof theory”, The Journal of Symbolic Logic 56, 1 (1991): 276–294. DOI: http://dx.doi.org/10.2307/2274919
• Avron, A., and A. Zamansky, “Non-deterministic semantics for logical systems”, pages 227–304 in D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, volume 16, Springer, Dodrecht, 2011. DOI: http://dx.doi.org/10.1007/978-94-007-0479-4_4
• Beall, J., “Off-topic: A new interpretation of weak-Kleene logic”, The Australasian Journal of Logic 13, 6 (2016): 136–142.
• Beall, J.C., “Multiple-conclusion LP and default classicality”, The Review of Symbolic Logic 4, 2 (2011): 326–336. DOI: http://dx.doi.org/10.1017/S1755020311000074
• Belnap, N., “A useful four-valued logic”, pages 5–37 in M. Dunn and G. Epstein (eds.), Modern uses of Multiple-Valued Logic, volume 2, Springer, Dodrecht, 1977. DOI: http://dx.doi.org/10.1007/978-94-010-1161-7_2
• Bochvar, D.A., “On a three-valued logical calculus and its application to the analysis of the paradoxes of the classical extended functional calculus”, History and Philosophy of Logic 2, 1–2 (1981): 87–112. DOI: http://dx.doi.org/10.1080/01445348108837023
• Bolzano, B., Wissenschaftslehre, Seidel, Sulzbach, 1837.
• Bonzio, S., J. Gil-Ferez, F. Paoli and L. Peruzzi, “On paraconsistent weak Kleene logic: Axiomatization and algebraic analysis”, Studia Logica 105, 2 (2017): 253–297. DOI: http://dx.doi.org/10.1007/s11225-016-9689-5
• Ciucci, D., D. Dubois and J. Lawry, “Borderline vs. unknown: Comparing three-valued representations of imperfect information”, International Journal of Approximate Reasoning 55, 9 (2014): 1866–1889. DOI: http://dx.doi.org/10.1016/j.ijar.2014.07.004
• Ciuni, R., ‘Conjunction in paraconsistent weak Kleene Logic”, pages 61–76 in P. Arazim and M. Dančák (eds.), The Logica Yearbook 2014, College Publications, London, 2015.
• Ciuni, R., and M. Carrara, “Characterizing logical consequence in paraconsistent weak Kleene”, pages 165–176 in L. Felline, A. Ledda, F. Paoli and E. Rossanese (eds.), New Developments in Logic and the Philosophy of Science, College Publications, London, 2016.
• Coniglio, M., and M.I. Corbalán, “Sequent calculi for the classical fragment of Bochvar and Halldén’s nonsense logics”, pages 125–136 in D. Kesner and P. Viana (eds.), Proceedings of the Seventh Workshop on Logical and Semantic Frameworks with Applications (LSFA 2012), volume 113 of EPTCS, 2012.
• Correia, F., “Semantics for analytic containment”, Studia Logica 77, 1 (2004): 87–104. DOI: http://dx.doi.org/10.1023/B:STUD.0000034187.37935.24
• Deutsch, H., “Relevant analytic entailment”, The Relevance Logic Newsletter 2, 1 (1977): 26–44.
• Došen. K., “A note on the law of identity and the converse Parry property”, Notre Dame Journal of Formal Logic 19, 1 (1978): 174–176. DOI: http://dx.doi.org/10.1305/ndjfl/1093888223
• Dubois, D., “On ignorance and contradiction considered as truth-values”, Logic Journal of the IGPL 16, 2 (2008): 195–216. DOI: http://dx.doi.org/10.1093/jigpal/jzn003
• Dubois, D., “Reasoning about ignorance and contradiction: Many-valued logics versus epistemic logic”, Soft Computing 16, 11 (2012): 1817–1831. DOI: http://dx.doi.org/10.1007/s00500-012-0833-5
• Dunn, M., “Intuitive semantics for first-degree entailments and ‘coupled trees’”, Philosophical Studies 29, 3 (1976): 149–168. DOI: http://dx.doi.org/10.1007/BF00373152
• Epstein, R.L., The Semantic Foundations of Logic, volume I: Propositional Logics, Oxford University Press, New York, 2nd edition, 1995.
• Ferguson. T.M., “A computational interpretation of conceptivism”, Journal of Applied Non-Classical Logics 24, 4 (2014): 333–367. DOI: http://dx.doi.org/10.1080/11663081.2014.980116
• Ferguson, T.M., “Logics of nonsense and Parry systems”, Journal of Philosophical Logic 44, 1 (2015): 65–80. DOI: http://dx.doi.org/10.1007/s10992-014-9321-y
• Ferguson, T.M., “Cut-down operations on bilattices”, pages 24–29 in Proceedings of the 45th International Symposium on Multiple-Valued Logic (ISMVL 2015), IEEE Computer Society, 2015. DOI: http://dx.doi.org/10.1109/ISMVL.2015.14
• Ferguson, T.M., “Faulty Belnap computers and subsystems of FDE”, Journal of Logic and Computation 26, 5 (2016): 1617–1636. DOI: http://dx.doi.org/10.1093/logcom/exu048
• Ferguson, T.M., “Rivals to Belnap-Dunn logic on interlaced trilattices”, Studia Logica 105, 6 (2017): 1123–1148. DOI: http://dx.doi.org/10.1007/s11225-016-9695-7
• Ferguson, T.M. “The proscriptive principle and logics of analytic implication”, PhD thesis, City University of New York, 2017.
• Fine, K., “Angellic content”, Journal of Philosophical Logic 45, 2 (2016): 199–226. DOI: http://dx.doi.org/10.1007/s10992-015-9371-9
• Fitting, M., “Bilattices and the semantics of logic programming”, The Journal of Logic Programming 11, 2 (1991): 91–116. DOI: http://dx.doi.org/10.1016/0743-1066(91)90014-G
• Fitting, M., “Kleene’s three valued logics and their children”, Fundamenta Informaticae 20, 1–3 (1994): 113–131. DOI: http://dx.doi.org/10.3233/FI-1994-201234
• Fitting, M., “Bilattices are nice things”, pages 53–78 in T. Bolander, V. Hendricks and S.A. Pedersen (eds.), Self-Reference, CSLI Publications, 2006.
• French, R., and S. Standefer, “Non-triviality done proof-theoretically”, pages 438–450 in A. Baltag, J. Seligman, and T. Yamada (eds.), Logic, Rationality, and Interaction: 6th International Workshop (LORI 2017), Springer, Berlin, 2017. DOI: http://dx.doi.org/10.1007/978-3-662-55665-8_30
• Halldén, S., The Logic of Nonsense, Uppsala Universitets Arsskrift, Uppsala, 1949.
• Hintikka, J., Logic, Language-Games and Information: Kantian Themes in the Philosophy of Logic, Clarendon Press, Oxford, 1973.
• Humberstone, L., “Power matrices and Dunn-Belnap semantics: Reflections on a remark of Graham Priest”, The Australasian Journal of Logic 11, 1 (2014): 14–45.
• Humberstone, L., and R. Meyer, “The relevant equivalence property”, Logic Journal of the IGPL 15, 2 (2007): 165–181.
• Kleene, S.C., Introduction to Metamathematics, North-Holland, Amsterdam, 1952.
• Lawry, J., and D. Dubois, “A bipolar framework for combining beliefs about vague propositions”, pages 530–540 in G. Brewka, T. Eiter and S. McIlraith (eds.), Principles of Knowledge Representation and Reasoning: Proceedings of the Thirteenth International Conference, AAAI Press, 2012.
• Oller, C.A., “Paraconsistency and analyticity”, Logic and Logical Philosophy 7 (1999): 91–99. DOI: http://dx.doi.org/10.12775/LLP.1999.008
• Omori, H., “Halldén’s logic of nonsense and its expansions in view of logics of formal inconsistency”, pages 129–133 in Proceedings of the 27th International Workshop on Database and Expert Systems Applications (DEXA 2016), IEEE Computer Society, 2016. DOI: http://dx.doi.org/10.1109/DEXA.2016.039
• Omori, H., and D. Szmuc, “Conjunction and disjunction in infectious logics”, pages 268–283 in A. Baltag, J. Seligman, and T. Yamada (eds.), Logic, Rationality, and Interaction: 6th International Workshop (LORI 2017), Springer, Berlin, 2017. DOI: http://dx.doi.org/10.1007/978-3-662-55665-8_19
• Paoli, F., “Regressive analytical entailments”, Technical Report number 33, Konstanzer Berichte zur Logik und Wissenschaftstheorie, 1992.
• Paoli. F., “Tautological entailments and their rivals”, pages 153–175 in J.-Y. Béziau, W.A. Carnielli and D.M. Gabbay (eds.), Handbook of Paraconsistency, College Publications, 2007.
• Parry, W.T., “Ein axiomensystem für eine neue art von implikation (analytische implikation)”, Ergebnisse eines mathematischen Kolloquiums 4 (1933): 5–6.
• Petrukhin, Y., “Natural deduction for three-valued regular logics”, Logic and Logical Philosophy 26, 2 (2017): 197–206. DOI: http://dx.doi.org/10.12775/LLP.2016.025
• Petrukhin, Y., “Natural deduction for Fitting’s four-valued generalizations of Kleene’s logics”, Logica Universalis 11, 4 (2017): 525–532. DOI: http://dx.doi.org/10.1007/s11787-017-0169-0
• Petrukhin, Y., “Natural deduction for four-valued both regular and monotonic logics”, Logic and Logical Philosophy 27, 1 (2018): 53–66. DOI: http://dx.doi.org/10.12775/LLP.2017.001
• Priest, G., An Introduction to Non-Classical Logic: From If to Is, Cambridge University Press, 2nd edition, 2008. DOI: http://dx.doi.org/10.1017/CBO9780511801174
• Priest, G.. “Plurivalent logics”, The Australasian Journal of Logic 11, 1 (2014): 2–13.
• Routley, R., V. Plumwood, R. Meyer and R. Brady, Relevant Logics and Their Rivals: The Basic Philosophical and Semantical Theory, Ridgeview, Atascadero, CA, 1982.
• Sylvan, R., “Relevant containment logics and certain frame problems of AI”, Logique et Analyse 31 (1988): 11–25.
• Szmuc, D., “Defining LFIs and LFUs in extensions of infectious logics”, Journal of Applied Non-Classical Logics 26, 4 (2017): 286–314. DOI: http://dx.doi.org/10.1080/11663081.2017.1290488
• Szmuc, D., “Track-down operations on bilattices”, in Proceedings of the 48th International Symposium on Multiple-Valued Logic (ISMVL 2018), IEEE Computer Society, 2018. Forthcoming.
• Tomova, N., “About four-valued regular logics”, Logical Investigations 15 (2009): 223–228.
• Urquhart, A., “Basic many-valued logic”, pages 249–295 in D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, volume 2, Springer, Dodrecht, 2001. DOI: http://dx.doi.org/10.1007/978-94-017-0452-6_4
• Wansing, H., and N. Belnap, “Generalized truth values. A reply to Dubois”, Logic Journal of the IGPL 18, 6 (2010): 921–935. DOI: http://dx.doi.org/10.1093/jigpal/jzp068
• Zimmermann, T.E., “Free choice disjunction and epistemic possibility”, Natural Language Semantics 8, 4 (2000): 255–290. DOI: http://dx.doi.org/10.1023/A: 1011255819284