Pure Variable Inclusion Logics
DOI:
https://doi.org/10.12775/LLP.2021.015Słowa kluczowe
logics of variable inclusion, significance logics, analytic entailment, weak Kleene logics, Płonka sumsAbstrakt
The aim of this article is to discuss pure variable inclusion logics, that is, logical systems where valid entailments require that the propositional variables occurring in the conclusion are included among those appearing in the premises, or vice versa. We study the subsystems of Classical Logic satisfying these requirements and assess the extent to which it is possible to characterise them by means of a single logical matrix. In addition, we semantically describe both of these companions to Classical Logic in terms of appropriate matrix bundles and as semilattice-based logics, showing that the notion of consequence in these logics can be interpreted in terms of truth (or non-falsity) and meaningfulness (or meaninglessness) preservation. Finally, we use Płonka sums of matrices to investigate the pure variable inclusion companions of an arbitrary finitary logic.
Bibliografia
Ackermann, W., “Begründung einer strengen implikation”, Journal of Symbolic Logic 21 (2): 113–128, 1956. DOI: https://doi.org/10.2307/2268750
Anderson, A. R., and N. Belnap, Entailment: The Logic of Relevance and Neccessity, volume 1, Princeton University Press, Princeton, 1975.
Åqvist, L., “Reflections on the logic of nonsense”, Theoria 28 (2): 138–157, 1962. DOI: https://doi.org/10.1111/j.1755-2567.1962.tb00316.x
Baylis, C., “Implication and subsumption’, The Monist 41 (3): 392–399, 1931. DOI: https://doi.org/10.5840/monist193141325
Berto, F., “Simple hyperintensional belief revision”, Erkenntnis 84 (3): 559–575, 2019. DOI: https://doi.org/10.1007/s10670-018-9971-1
Bochvar, F., “On a three-valued calculus and its application in the analysis of the paradoxes of the extended functional calculus” (in Russian), Matematicheskii Sbornik 4 (46): 287–308, 1938. DOI: https://doi.org/10.1080/01445348108837023
Bonzio, S., and M. Pra Baldi, “Containment logics: Algebraic completeness and axiomatization”, Studia Logica 109 (5): 969–994, 2021. DOI: https://doi.org/10.1007/s11225-020-09930-1
Bonzio, S., T. Moraschini, and M. Pra Baldi, “Logics of left variable inclusion and Płonka sums of matrices”, Archive for Mathematical Logic 60 (1): 49–76, 2021. DOI: https://doi.org/10.1007/s00153-020-00727-6
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 Directions in Logic and the Philosophy of Science, College Publications, London, 2016.
Ciuni,R., and M. Carrara, “Semantical analysis of weak Kleene logics”, Journal of Applied Non-Classical Logics 29 (1): 1–36, 2019. DOI: https://doi.org/10.1080/11663081.2018.1547514
Ciuni, R., T. M. Ferguson, and D. Szmuc, “Relevant logics obeying component homogeneity”, Australasian Journal of Logic 15 (2): 301–361, 2018.DOI: https://doi.org/10.26686/ajl.v15i2.4864
Ciuni, R., T. M. Ferguson, and D. Szmuc, “Logics based on linear orders of contaminating values”, Journal of Logic and Computation, 29 (5): 631–663, 2019. DOI: https://doi.org/10.1093/logcom/exz009
Deutsch, H., A Aamily of Conforming Relevant Logics, UCLA, Los Angeles, 1981.
Dunn, J. M., “A modification of Parry’s analytic implication”, Notre Dame Journal of Formal Logic 13 (2): 195–205, 1972. DOI: https://doi.org/10.1305/ndjfl/1093894715
Dunn, J. M., “Partiality and its dual”, Studia Logica 66 (1): 5–40, 2000. DOI: https://doi.org/10.1023/A:1026740726955
Epstein, R., The Semantic Foundations of Logic, volume I: Propositional Logics, Oxford University Press, New York, 2nd edition, 1995.
Ferguson, T. M., “Logics of nonsense and Parry systems”, Journal of Philosophical Logic 44 (1): 65–80, 2015. DOI: https://doi.org/10.1007/s10992-014-9321-y
Ferguson, T. M., “Rivals to Belnap–Dunn logic on interlaced trilattices”, Studia Logica, 105 (6): 1123–1148, 2017. DOI: https://doi.org/10.1007/s11225-016-9695-7
Ferguson, T. M., Meaning and Proscription in Formal Logic: Variations on the Propositional Logic of William T. Parry, Springer, Dordrecht, 2017. DOI: https://doi.org/10.1007/978-3-319-70821-8
Font, J. M., “Taking degrees of truth seriously”, Studia Logica 91: 383–406, 2009. DOI: http://dx.doi.org/10.1007/s11225-009-9180-7
Font, J. M., Abstract Algebraic Logic, College Publications, London, 2016.
Font, J. M., A. Gil, A. Torrens, and V. Verdu, “On the infinite-valued Łukasiewicz logic that preserves degrees of truth”, Archive for Mathematical Logic 45: 839–868, 2006. DOI: https://doi.org/10.1007/s00153-006-0001-7
Goddard, L., and R. Routley, The Logic of Significance and Context, volume 1, Scottish Academic Press, Edimburgh, 1973.
Halldén, S., The Logic of Nonsense, Uppsala Universitets Arsskrift, Uppsala, 1949.
Humberstone, L., The Connectives, MIT Press, Cambridge, MA, 2011.
Jansana, R., “Selfextensional logics with a conjunction”, Studia Logica 84: 63–104, 2006. DOI: https://doi.org/10.1007/s11225-006-9003-z
Jarmużek, T., “Relating semantics as fine-grained semantics for intensional logics”, pages 13–30 in A. Giordani and J. Malinowski (eds.), Logic in High Definition, Springer, Berlin, 2021. DOI: https://doi.org/10.1007/978-3-030-53487-5_2
Kleene, S. C., Introduction to Metamathematics, North-Holland, Amsterdam, 1952.
Łoś, J., and R. Suszko, “Remarks on sentential logics”, Indagationes Mathematicae 61: 177–183, 1958.
Nelson, E. J., “Intensional relations”, Mind 39 (156): 440–453, 1930. DOI: https://doi.org/10.1093/mind/XXXIX.156.440
Paoli, F., “Tautological entailments and their rivals”, pages 153–175 in J. Béziau, W. Carnielli, and D. Gabbay (eds.), Handbook of Paraconsistency, College Publications, London, 2007.
Paoli, F., and M. Pra Baldi, “Proof theory of paraconsistent weak Kleene logic”, Studia Logica 4 (108): 779–802, 2020. DOI: https://doi.org/10.1007/s11225-019-09876-z
Paoli, F., and M. Pra Baldi, “Extensions of paraconsistent weak Kleene logic”, Logic Journal of the IGPL 29 (5): 798–822, 2021. DOI: https://doi.org/10.1093/jigpal/jzaa024
Parry, W. T., “Ein Axiomensystem für eine neue Art von Implikation (analytische Implikation)”, Ergebnisse eines mathematischen Kolloquiums 4: 5–6, 1933.
Parry, W. T., “Analytic implication: its history, justification, and varieties”, pages 101–118 in J. Norman and R. Sylvan (eds.), Directions in Relevant Logic, Springer, Netherlands, 1989. DOI: https://doi.org/10.1007/978-94-009-1005-8_7
Płonka, J., On a method of construction of abstract algebras”, Fundamenta Mathematicae 61 (2): 183–189, 1967.
Shoesmith, D. J., and T. J. Smiley, “Deducibility and many-valuedness”, Journal of Symbolic Logic 36 (4): 610–622, 1971. DOI: https://doi.org/10.2307/2272465
Szmuc, D., “Track-down operations on bilattices”, pages 74–79 in R. Wille and M. Lukac (eds.), Proceedings of the 48th IEEE International Symposium on Multiple-Valued Logic, IEEE Computer Society Press, Los Alamitos, CA, 2018. DOI: https://doi.org/10.1109/ISMVL.2018.00021
Szmuc, D., “A simple logical matrix and sequent calculus for Parry’s logic of Analytic Implication”, Studia Logica 109 (4): 791–828, 2021. DOI: https://doi.org/10.1007/s11225-020-09926-x
Szmuc, D., and H. Omori, “A note on Goddard and Routley’s significance logic”, Australasian Journal of Logic, 15 (2): 431–448, 2018. DOI: https://doi.org/10.26686/ajl.v15i2.4867
Urquhart, A., “Basic many-valued logic”, pages 249–295 in D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, volume 2, Springer, Berlin, 2nd edition, 2001. DOI: https://doi.org/10.1007/978-94-017-0452-6_4
Weiss, Y., “Sextus Empiricus’ fourth conditional and containment logic”, History and Philosophy of Logic 40 (4): 307–322, 2019.
Wójcicki, R., Theory of Logical Calculi: Basic Theory of Consequence Operations, volume 199 of Synthese Library, Reidel, Dordrecht, 1987. DOI: https://doi.org/10.2307/2274502
Yablo, S., Aboutness, Princeton University Press, Princeton, 2014. DOI: https://doi.org/10.1515/9781400845989
Pobrania
Opublikowane
Jak cytować
Numer
Dział
Licencja
Prawa autorskie (c) 2021 Francesco Paoli, Michele Pra Baldi, Damian Szmuc
![Creative Commons License](http://i.creativecommons.org/l/by-nd/4.0/88x31.png)
Utwór dostępny jest na licencji Creative Commons Uznanie autorstwa – Bez utworów zależnych 4.0 Międzynarodowe.
Statystyki
Liczba wyświetleń i pobrań: 868
Liczba cytowań: 0