Abstract
This paper is a step toward showing what is achievable using non-classical metatheory—particularly, a substructural paraconsistent framework. What standard results, or analogues thereof, from the classical metatheory of first order logic(s) can be obtained? We reconstruct some of the originals proofs for Completeness, Löwenheim-Skolem and Compactness theorems in the context of a substructural logic with the naive comprehension schema. The main result is that paraconsistent metatheory can ‘re-capture’ versions of standard theorems, given suitable restrictions and background assumptions; but the shift to non-classical logic may recast the meanings of these apparently ‘absolute’ theorems.
References
Badia, G, & Weber, Z (2019). A substructural logic for inconsistent mathematics. In A Reiger G Young (Eds.) Dialetheism and its applications (pp. 155–176). Springer.
Baldwin, J. (2017). The explanatory power of a new proof: Henkin’s completeness proof. In M. Piazza G. Pulcini (Eds.) Philosophy of Mathematics: Truth, Existence and Explanation, Boston Studies in the History and Philosophy of Science (pp. 147–162). Springer.
Beall, J.C. (2009). Spandrels of truth. Oxford : Oxford University Press.
Beall, J.C. (2018). The simple argument for sublcassical logic. Philosophical Issues, 28(1), 30–54.
Beall, J.C., Glanzberg, M., & Ripley, . (2018). Formal theories of truth. Oxford: Oxford University Press.
Cantini, A. (2003). The undecidability of grisin’s set theory. Studia Logica, 74(3), 345–368.
Church, A. (1956). Introduction to mathematical logic. Princeton: Princeton University Press.
Cintula, Petr, & Paoli, Francesco (2021). Is mutliset consequence trivial? Synthese, 199, 741–765.
Drake, F. (1974). Set Theory: An introduction to large cardinals. North Holland Amsterdam.
Empiricus, S. (2005). Against the logicians. Cambridge: Cambridge University Press.
Field, H. (2020). Properties, propositions, and conditionals. Australasian Philosophical Review, 4(2), 112–146.
Font, J. M. (1997). Belnap’s four-valued logic and de morgan lattices. Logic Journal of the IGPL, 5, 1–29.
French, R. (2016). Structural reflexivity and the paradoxes of self reference. Ergo, 3(5), 113–131.
Garca-Matos, M. (2005). Abstract model theory without negation. PhD thesis, University of Helsinki.
Girard, P, & Weber, Z (2019). Modal logic without contraction in a metatheory without contraction. Review of Symbolic Logic, 12(4), 685–701.
Gödel, K. (1967). The completeness of the axioms of the functional calculus of logic. In J. van Heijenoort (Ed.) From Frege to Gödel: a source book in mathematical logic, 1879–1931 (pp. 582–591). Cambridge: Harvard University Press.
Hilbert, D., & Ackermann, W. (1928). Principles of Mathematical Logic. Chelsea Publishing Company, 1928 Reprinted 1950 from Grundzuge der Theoretischen Logik by David Hilbert and Wilhelm Ackermann.
Kleene, S. C. (1952). Introduction to metamathematics. North-Holland.
Kreisel, G. (1958). A remark on free choice sequences and the topological completeness proofs. Journal of Symbolic Logic, 23(4), 369–388.
Kreisel, G. (1962). On weak completeness of intuitionistic predicate logic. Journal of Symbolic Logic, 27(2), 139–158.
Lindstrm, P. (1969). On extensions of elementary logic. Theoria, 35(1), 1–11.
Löwenheim, L (1967). On possibilites in the calculus of relatives. In J. van Heijenoort (Ed.) From Frege to Gödel: a source book in mathematical logic, 1879–1931 (pp. 228–251). Cambridge: Harvard University Press.
McCarty, C., & Tennant, N. (1987). Skolem’s paradox and constructivism. Journal of Philosophical Logic, 16(2), 1–29.
Meyer, RK (2021). Arithmetic formulated relevantly. Australasian Journal of Logic, 18(5), 154–288. Manuscript from circa 1976.
Mortensen, C. (1995). Inconsistent mathematics. Dordrecht: Kluwer Academic Publishers.
Mortensen, C. (2010). Inconsistent Geometry. College publications.
Øgaard, T.F. (2016). Paths to triviality. Journal of Philosophical Logic, 45(3), 237–276.
Øgaard, T.F. (2017). Skolem functions in non-classical logics. Australasian Journal of Logic, 14(1), 181–225.
Omori, H., & Wansing, H. (2017). 40 years of FDE: an introductory overview. Studia Logica, 105, 1021–1049.
Priest, G. (1979). The logic of paradox. Journal of Philosophical Logic, 8, 219–241.
Priest, G. (1982). To be and not to be: Dialectical tense logic. Studia Logica, 41, 249–268.
Priest, G. (2006). In Contradiction: A Study of the Transconsistent, 2nd edn. Oxford: Oxford University Press.
Priest, G. (2008). An introduction to Non-Classical logic, 2nd edn. Cambridge.
Priest, G. (2017). What if? the exploration of an idea. Australasian Journal of Logic, 14(1).
Priest, G. (2020). Metatheory and dialetheism. Logical Investigations, 20(1), 48–59.
Restall, G. (1994). On logics without contraction. PhD thesis, The University of Queensland.
Ripley, D. (2015). Comparing substructural theories of truth. Ergo, 2(13), 299–328.
Rosenblatt, L. (2019). Non-contractive classical logic. Notre Dame Journal of Formal Logic, 60(4), 559–585.
Rosenblatt, L. (2021). Towards a non-classical meta-theory for substructural approaches to paradox. Journal of Philosophical Logic, 50, 1007–1055.
Routley, R. (1980). Ultralogic as Universal? The Sylvan Jungle volume 4. Synthese Library, Springer, 2019. Edited by Z Weber. 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 Beyon, pp.892-962.
Routley, R., Plumwood, V., Meyer, R. K., & Brady, R.T. (1982). Relevant logics and their Rivals. Ridgeview.
Simpson, S.G. (2009). Subsystems of second-order arithmetic. Melbourne: Cambridge University Press.
Skolem, T. (1967). Some remarks on axiomatized set theory. In J. van Heijenoort (Ed.) From Frege to Gödel: a source book in mathematical logic, 1879–1931 (pp. 290–301). Cambridge: Harvard University Press.
Warren, J. (2018). Change of logic, change of meaning. Philosophy and Phenomenological Research, 96(2), 421–442.
Weber, Z. (2010). Transfinite numbers in paraconsistent set theory. Review of Symbolic Logic, 3(1), 71–92.
Weber, Z. (2021). Paradoxes and inconsistent mathematics. Cambridge: Cambridge University Press.
Weber, Z, Badia, G, & Girard, P. (2016). What is an inconsistent truth table? Australasian Journal of Philosophy, 94(3).
Woods, J. (2019). Logical partisanhood. Philosophical Studies, 176, 1203–1224.
Zardini, E. (2011). Truth without contra(di)ction. Review of Symbolic Logic, 4, 498–535.
Acknowledgments
Thanks to the (virtual) audience at the Logic Supergroup in August 2020 where an earlier version of this paper was presented. Thanks to two anonymous referees for extremely helpful and supportive comments and suggestions. G. Badia’s work was partially supported by the Australian Research Council grant DE220100544.
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Badia, G., Weber, Z. & Girard, P. Paraconsistent Metatheory: New Proofs with Old Tools. J Philos Logic 51, 825–856 (2022). https://doi.org/10.1007/s10992-022-09651-x
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DOI: https://doi.org/10.1007/s10992-022-09651-x
Keywords
- Paraconsistent logic
- Inconsistent mathematics
- Substructural logic
- Non-classical metatheory
- Completeness theorems