Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-11T01:44:48.281Z Has data issue: false hasContentIssue false
  • English
  • Français

A FULLY CLASSICAL TRUTH THEORY CHARACTERIZED BY SUBSTRUCTURAL MEANS

Published online by Cambridge University Press:  04 January 2019

FEDERICO MATÍAS PAILOS
FEDERICO MATÍAS PAILOS*
Affiliation:
Philosophy Department, University of Buenos Aires
*
*NATIONAL SCIENTIFIC AND TECHNICAL RESEARCH COUNCIL SARMIENTO 440 C104AA7 BUENOS AIRES, ARGENTINA and DEPARTMENT OF PHILOSOPHY UNIVERSITY OF BUENOS AIRES PUÁN 480 C1406CQJ BUENOS AIRES, ARGENTINA E-mail: federico.pailos@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We will present a three-valued consequence relation for metainferences, called CM, defined through ST and TS, two well known substructural consequence relations for inferences. While ST recovers every classically valid inference, it invalidates some classically valid metainferences. While CM works as ST at the inferential level, it also recovers every classically valid metainference. Moreover, CM can be safely expanded with a transparent truth predicate. Nevertheless, CM cannot recapture every classically valid meta-metainference. We will afterwards develop a hierarchy of consequence relations CMn for metainferences of level n (for 1 ≤ n < ω). Each CMn recovers every metainference of level n or less, and can be nontrivially expanded with a transparent truth predicate, but cannot recapture every classically valid metainferences of higher levels. Finally, we will present a logic CMω, based on the hierarchy of logics CMn, that is fully classical, in the sense that every classically valid metainference of any level is valid in it. Moreover, CMω can be nontrivially expanded with a transparent truth predicate.

Keywords

03BXX03B4703B5003B6003B53logicmetainferencesmetainferential validitysubstructural logicsempty logic
Type
Research Article
Information
The Review of Symbolic Logic , Volume 13 , Issue 2 , June 2020 , pp. 249 - 268
Copyright
Copyright © Association for Symbolic Logic 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Avron, A. (1988). The semantics and proof theory of linear logic. Theoretical Computer Science, 57 (2–3), 161–184.CrossRefGoogle Scholar
Avron, A. (1991). Simple consequence relations. Information and Computation, 92(1), 105–139.CrossRefGoogle Scholar
Barrio, E., Rosenblatt, L., & Tajer, D. (2015). The logics of strict-tolerant logic. Journal of Philosophical Logic, 44(5), 551–571.CrossRefGoogle Scholar
Barrio, E., Rosenblatt, L., & Tajer, D. (2015). The logics of strict-tolerant logic. Journal of Philosophical Logic, 44(5), 551–571.CrossRefGoogle Scholar
Barwise, J. & Etchemendy, J. (1987). The Liar: An Essay on Truth and Circularity. Oxford, UK: Oxford University Press.Google Scholar
Blok, W. & Jónsson, B. (2006). Equivalence of consequence operations. Studia Logica, 83(1), 91–110.CrossRefGoogle Scholar
Chemla, E., Egré, P., & Spector, B. (2017). Characterizing logical consequence in many-valued logics. Journal of Logic and Computation, 27(7), 2193–2226.Google Scholar
Cobreros, P., Egré, P., Ripley, D., & van Rooij, R. (2012). Tolerant, classical, strict. Journal of Philosophical Logic, 41(2), 347–385.CrossRefGoogle Scholar
Cobreros, P., Egré, P., Ripley, D., & Van Rooij, R. (2014). Reaching transparent truth. Mind, 122(488), 841–866.CrossRefGoogle Scholar
De, M. & Omori, H. (2015). Classical negation and expansions of belnap–dunn logic. Studia Logica, 103(4), 825–851.CrossRefGoogle Scholar
Dicher, B. & Paoli, F. (2017). St, lp, and tolerant metainferences. Unpublished manuscript.Google Scholar
Dicher, B. & Paoli, F. (2018). The original sin of proof-theoretic semantics. Unpublished manuscript.CrossRefGoogle Scholar
Fjellstad, A. (2016). Naive modus ponens and failure of transitivity. Journal of Philosophical Logic, 45(1), 65–72.CrossRefGoogle Scholar
Frankowski, S. (2004). Formalization of a plausible inference. Bulletin of the Section of Logic, 33(1), 41–52.Google Scholar
Frankowski, S. (2004). P-consequence versus q-consequence operations. Bulletin of the Section of Logic, 33(4), 197–207.Google Scholar
French, R. (2016). Structural reflexivity and the paradoxes of self-reference. Ergo, 3(5), 113–131.Google Scholar
Gabbay, D. (1996). Labelled Deductive Systems. Oxford: Oxford University Press.Google Scholar
Girard, J.-Y. (1987). Proof Theory and Logical Complexity. Napoli: Bibliopolis.Google Scholar
Hjortland, O. Theories of truth and the maxim of minimal mutilation. Manuscript.Google Scholar
Humberstone, L. (1996). Valuational semantics of rule derivability. Journal of Philosophical Logic, 25, 451–461.CrossRefGoogle Scholar
Kremer, M. (1988). Kripke and the logic of truth. Journal of Philosophical Logic, 17, 225–278.CrossRefGoogle Scholar
Kripke, S. (1975). Outline of a theory of truth. Journal of Philosophy, 72(19), 690–716.CrossRefGoogle Scholar
Malinowski, G. (1990). Q-consequence operation. Reports on Mathematical Logic, 24(1), 49–59.Google Scholar
Malinowski, G. (2014). Kleene logic and inference. Bulletin of the Section of Logic, 43(1/2), 43–52.Google Scholar
Priest, G. & Wansing, H. (2015). External curries. Journal of Philosophical logic, 44(4), 453–471.Google Scholar
Pynko, A. (2010). Gentzen’s cut-free calculus versus the logic of paradox. Bulletin of the Section of Logic, 39(1/2), 35–42.Google Scholar
Ripley, D. (2012). Conservatively extending classical logic with transparent truth. The Review of Symbolic Logic, 5(2), 354–378.CrossRefGoogle Scholar
Ripley, D. (2012). Conservatively extending classical logic with transparent truth. Review of Symbolic Logic, 5(2), 354–378.CrossRefGoogle Scholar
Ripley, D. (2013). Paradoxes and failures of cut. Australasian Journal of Philosophy, 91(1), 139–164.CrossRefGoogle Scholar
Scott, D. (1971). On engendering an illusion of understanding. The Journal of Philosophy, 68(21), 787–807.CrossRefGoogle Scholar
Shoesmith, D. & Smiley, T. (1978). Multiple-Conclusion Logic. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Shramko, Y. & Wansing, H. (2010). Truth values. In Zalta, E., editor. The Stanford Encyclopedia of Philosophy. Stanford Uniersity, Summer 2010 edition. Available at: http://plato.stanford.edu/archives/sum2010/entries/truth-values/.Google Scholar
Shramko, Y. & Wansing, H. (2011). Truth and Falsehood: An Inquiry into Generalized Logical Values. Springer Science & Business Media.Google Scholar