True, Untrue, Valid, Invalid, Provable, Unprovable

Zach Weber

doi:10.12775/LLP.2024.008

Autor

Zach Weber Department of Philosophy, University of Otago, New Zealand

DOI:

https://doi.org/10.12775/LLP.2024.008

Słowa kluczowe

dialetheism, revenge paradoxes, paraconsistent metatheory

Abstrakt

There are many approaches to paraconsistency, ranging from the very moderate to the more radical. In this paper I explore and extend the more radical end of the spectrum, where there are truth-value gluts. In particular I will look at paraconsistent metatheory – the machinery of truth, validity, and proof  as developed in a glut-friendly paraconsistent setting. The aim is to evaluate the philosophical and technical tenability of such an approach. I will show that there are very significant technical challenges to face on this sort of radical approach, but that there is good philosophical support for facing these challenges.

Bibliografia

Arruda, A. I., and D. Batens, 1982, “Russell’s set versus the universal set in paraconsistent set theory”, Logique et Analyse, 25 (8): 121–133.

Badia, G., 2016, “The relevant fragment of first order logic”, Review of Symbolic Logic, 9 (1): 143–166. DOI: http://dx.doi.org/10.1017/S1755020315000313

Badia, G., and Z. Weber, 2019, “A substructural logic for inconsistent mathematics”, pages 155–176 in A. Reiger and G. Young (eds.), Dialetheism and its Applications, Springer. DOI: http://dx.doi.org/10.1007/978-3-030-30221-4_9

Badia, G., Z. Weber, and P. Girard, 2022, “Paraconsistent metatheory: new proofs with old tools”, Journal of Philosophical Logic, 51 (4): 825–856. DOI: http://dx.doi.org/10.1007/s10992-022-09651-x

Batens, D., 2019, “Looting liars masking models”, pages 139–164 in C. Baskent and T. Ferguson (eds.), Graham Priest on Dialetheism and Paraconsistency, Springer. DOI: http://dx.doi.org/10.1007/978-3-030-25365-3_8

Batens, B., 2020, “Adaptive Fregean set theory”, Studia Logica, 108 (5): 903–939. DOI: http://dx.doi.org/10.1007/s11225-019-09882-1

Beall, Jc, 2009, Spandrels of Truth, Oxford University Press. DOI: http://dx.doi.org/10.1093/acprof:oso/9780199268733.001.0001

Beall, Jc, 2014, “End of inclosure”, Mind, 123 (491): 829–849. DOI: http://dx.doi.org/10.1093/mind/fzu075

Beall, JC, R. T. Brady, A. P. Hazen, G. Priest, and G. Restall, 2006, “Relevant restricted quantification”, Journal of Philosophical Logic, 35 (6): 587–598. DOI: http://dx.doi.org/10.1007/s10992-005-9008-5

Beall, JC, and M. Colyvan, 2001, “Looking for contradictions”, Australasian Journal of Philosophy, 79: 564–569.

Beall, Jc, and S. A. Logan, 2017, Logic: The Basics, Routledge. DOI: http://dx.doi.org/10.4324/9781315723655

Beall, JC, and G. Restall, 2005, Logical Pluralism, Oxford University Press. DOI: http://dx.doi.org/10.1093/acprof:oso/9780199288403.001.0001

Brady, R., 2006, Universal Logic, CSLI, Stanford.

Berto, F., 2007, How to Sell a Contradiction, vol. 6 of Studies in Logic, College Publications.

Burgis, B., and O. Bueno, 2019, “Liars with curry: Dialetheism and the prospects for a uniform solution”, in A. Reiger and G. Young (eds.), Dialetheism and its Applications, Springer.

Cantini, A., 2003, “The undecidability of Grĭsin’s set theory”, Studia Logica, 74 (3): 345–368.

Carnielli, W., and M. E. Coniglio, 2016, Paraconsistent Logic: Consistency, Contradiction and Negation, Springer. DOI: http://dx.doi.org/10.1007/978-3-319-33205-5

Ferguson, T. M., 2019, “Inconsistent models (and infinite models) for arithmetics with constructible falsity”, Logic and Logical Philosophy, 28: 389–407. DOI: http://dx.doi.org/10.12775/LLP.2018.011

Ficara, E., 2021, “The birth of dialetheism”, History and Philosophy of Logic, 42 (3): 281–296.

Field, H., 2020, “Properties, propositions, and conditionals”, Australasian Philosophical Review, 4 (2): 112–146. DOI: http://dx.doi.org/10.1080/24740500.2021.1886687

Istre, E., 2017, “Normalized naive set theory”, PhD Thesis, University of Canterbury.

Libert, T., 2003, “ZF and the axiom of choice in some paraconsistent set theories”, Logic and Logical Philosophy, 11: 91–114. DOI: http://dx.doi.org/10.12775/LLP.2003.005

Martinez, S. J., 2021, “Algebra-valued models for lp-set theory”, Australasian Journal of Logic, 18 (7).

Meadows, T., and Z. Weber, 2016, “Computation in non-classical foundations?”, Philosophers’ Imprint, 16: 1–17.

Murzi, J., and L. Rossi, 2020, “Generalized revenge”, Australasian Journal of Philosophy, 98: 153–177.

Omori, H., and Z. Weber, 2019, “Just true? On the metatheory for paraconsistent truth”, Logique et Analyse, 248: 415–433.

Omori, H., and Z. Weber, 2023, “A note on the expressive completeness of LP in a metatheory without negation”, manuscript.

Ono, H., and Y. Komori, 1985, “Logic without the contraction rule”, Journal of Symbolic Logic, 50 (1): 169–201. DOI: http://dx.doi.org/10.2307/2273798

Petersen, U, 2000, “Logic without contraction as based on inclusion and unrestriced abstraction”, Studia Logica, 64: 365–403.

Priest, G., 1979, “The logic of paradox”, Journal of Philosophical Logic, 8 (1): 219–241. DOI: http://dx.doi.org/10.1007/BF00258428

Priest, G., 1984, “Hypercontradictions”, Logique et Analyse, 107: 237–243.

Priest, G., 2002b, Beyond the Limits of Thought, 2nd edition, Oxford University Press (first edition 1995, Cambridge University Press). DOI: http://dx.doi.org/10.1093/acprof:oso/9780199254057.001.0001

Priest, G., 2002b, “Paraconsistent logic”, pages 287–394 in D. M. Gabbay and F. Günthner (eds.), Handbook of Philosophical Logic, 2nd edition, Vol. 6, Kluwer. DOI: http://dx.doi.org/10.1007/978-94-017-0460-1_4

Priest, G., 2006, In Contradiction: A Study of the Transconsistent, 2nd edition, Oxford University Press, Oxford.

Priest, G., 2008, An Introduction to Non-Classical Logic, 2nd edition, Cambridge. DOI: http://dx.doi.org/10.1017/CBO9780511801174

Priest, G., 2020, “Metatheory and dialetheism”, Logical Investigations, 26 (1): 48–59.. DOI: http://dx.doi.org/10.21146/2074-1472-2020-26-1-48-59

Priest, G., and R. Routley, 1983, On Paraconsistency, Research School of Social Sciences, Australian National University. Reprinted as introductory chapters in (Priest et al., 1989).

Priest, G., R. Routley, and J. Norman (eds.), 1989, Paraconsistent Logic: Essays on the Inconsistent, Philosophia Verlag, Munich.

Restall, G., 1992, “A note on naïve set theory in LP”, Notre Dame Journal of Formal Logic, 33 (3): 422–432. DOI: http://dx.doi.org/10.1305/ndjfl/1093634406

Ripley, D., 2015, “Naive set theory and nontransitive logic”, Review of Symbolic Logic, 8 (3): 553–571. DOI: http://dx.doi.org/10.1017/S1755020314000501

Routley, R., 1977, Ultralogic as Universal?. First appeared in two parts in The Relevance Logic Newsletter 2(1): 51-90, January 1977 and 2(2):138-175, May 1977. Reprinted as appendix to Exploring Meinong’s Jungle and Beyond 1980, pp. 892–962. New edition as The Sylvan Jungle, Vol. 4, edited by Z. Weber, Synthese Library, 2019.

Routley, R., 1979, “Dialectical logic, semantics and metamathematics”, Erkenntnis, 14 (3): 301–331. DOI: http://dx.doi.org/10.1007/BF00174897

Routley, R., and R. K. Meyer, 1976, “Dialectical logic, classical logic and the consistency of the world”, Studies in Soviet Thought, 16: 1–25.

Shapiro, S., 2002, “Incompleteness and inconsistency”, Mind, 111 (444): 817–832. DOI: http://dx.doi.org/10.1093/mind/111.444.817

Sylvan, R., and J. Copeland, 2000, “Computability is logic relative”, pages 189–199 in D. Hyde and G. Priest (eds.), Sociative Logics and their Applications, Ashgate.

Tedder, A., 2021, “On consistency and decidability in some paraconsistent arithmetics”, Australasian Journal of Logic, 18 (5): 473–502. DOI: http://dx.doi.org/10.26686/ajl.v18i5.6921

Terui, K., 2004, “Light affine set theory: A naive set theory of polynomial time”, Studia Logica, 77: 9–40.

Weber, Z., 2010, “Transfinite numbers in paraconsistent set theory”, Review of Symbolic Logic, 3 (1): 71–92. DOI: http://dx.doi.org/10.1017/S1755020309990281

Weber, Z., 2020, “Property identity and relevant conditionals”, Australasian Philosophical Review, 4 (2): 147–155. DOI: http://dx.doi.org/10.1080/24740500.2021.1886688

Weber, Z., 2021, Paradoxes and Inconsistent Mathematics, Cambridge University Press. DOI: http://dx.doi.org/10.1017/9781108993135

Weber, Z., 2022, Paraconsistency in Mathematics, Cambridge Elements, Cambridge University Press. DOI: http://dx.doi.org/10.1017/9781108993968

Weber, Z., G. Badia, and P. Girard, 2016, “What is an inconsistent truth table?”, Australasian Journal of Philosophy, 94 (3): 533–548. DOI: http://dx.doi.org/10.1080/00048402.2015.1093010

Young, G., 2019, “A revenge problem for dialetheism”, pages 21–46 in A. Reiger and G. Young (eds.), Dialetheism and its Applications, Springer. DOI: http://dx.doi.org/10.1007/978-3-030-30221-4_2