Abstract
A metainference is usually understood as a pair consisting of a collection of inferences, called premises, and a single inference, called conclusion. In the last few years, much attention has been paid to the study of metainferences—and, in particular, to the question of what are the valid metainferences of a given logic. So far, however, this study has been done in quite a poor language. Our usual sequent calculi have no way to represent, e.g. negations, disjunctions or conjunctions of inferences. In this paper we tackle this expressive issue. We assume some background sentential language as given and define what we call an inferential language, that is, a language whose atomic formulas are inferences. We provide a model-theoretic characterization of validity for this language—relative to some given characterization of validity for the background sentential language—and provide a proof-theoretic analysis of validity. We argue that our novel language has fruitful philosophical applications. Lastly, we generalize some of our definitions and results to arbitrary metainferential levels.
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References
Badia, G., Girard, P., & Weber, Z. (2016). What is an inconsistent truth table? Australasian Journal of Philosophy, 94(3), 533–548.
Barrio, E., & Pailos, F. (2021). Validities, antivalidities and contingencies: A multi-standard approach. Journal of Philosophical Logic, 1–24.
Barrio, E., Rosenblatt, L., & Tajer, D. (2015). The logics of strict-tolerant logic. Journal of Philosophical Logic, 44(5), 551–571.
Barrio, E.A., Pailos, F., & Calderón, J.T. (2021). Anti-exceptionalism, truth and the ba-plan. Synthese, 1–26.
Barrio, E.A., Pailos, F., & Szmuc, D. (2020). A hierarchy of classical and paraconsistent logics. Journal of Philosophical Logic, 49(1).
Barrio, E.A., Pailos, F., & Szmuc, D. (2021). Substructural logics, pluralism and collapse. Synthese, 198(20), 4991–5007.
Burgess, J.P. (2014). No requirement of relevance. In S. Shapiro (Ed.) The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford University Press.
Chemla, E., Égré, P., & Spector, B. (2017). Characterizing logical consequence in many-valued logic. Journal of Logic and Computation, 27(7), 2193–2226.
Cobreros, P., Egré, P., Ripley, D., & Rooij, R.V. (2013). Reaching transparent truth. Mind, 122(488), 841–866.
Cobreros, P., Egré, P., Ripley, D., & van Rooij, R. (2012). Tolerance and mixed consequence in the s’valuationist setting. Studia logica, 100(4), 855–877.
Cobreros, P., Egré, P., Ripley, D., & van Rooij, R. (2012). Tolerant, classical, strict. Journal of Philosophical Logic, 41(2), 347–385.
Cobreros, P., Egré, P., Ripley, D., & van Rooij, R. (2015). Vagueness, truth and permissive consequence. In Unifying the philosophy of truth (pp. 409–430). Springer.
Cobreros, P., Egré, P., Ripley, D., & van Rooij, R. (2020). Inferences and metainferences in st. Journal of Philosophical Logic, 49(6), 1057–1077.
Cobreros, P., La Rosa, E., & Tranchini, L. (2020). (i can’t get no) antisatisfaction. Synthese, 1–15.
Da Ré, B., & Pailos, F. (2022). Sequent-calculi for metainferential logics. Studia Logica, 110(2), 319–353.
Da Re, B., Pailos, F., Szmuc, D., & Teijeiro, P. (2020). Metainferential duality. Journal of Applied Non-Classical Logics, 0(0), 1–23.
Da Ré, B., Szmuc, D., & Teijeiro, P. (2021). Derivability and metainferential validity. Journal of Philosophical Logic, 1–27.
Da Re, B., Teijeiro, P., & Rubin, M. What really is a paraconsistent logic. Forthcoming in Logic and Logical Philosophy.
De, M., & Omori, H. (2018). There is more to negation than modality. Journal of Philosophical logic, 47(2), 281–299.
Dicher, B., & Paoli, F. (2019). St, lp and tolerant metainferences. In Graham Priest on dialetheism and paraconsistency (pp. 383–407). Springer.
Dummett, M. (1977). Elements of Intuitionism. Oxford: Clarendon Press. 2nd edition 2000.
Field, H. (2020). The power of naïve truth. The Review of Symbolic Logic, 1–34.
French, R. (2016). Structural reflexivity and the paradoxes of self-reference. Ergo, an Open Access Journal of Philosophy, 3.
Gentzen, G. (1934). Untersuchungen iiber das logische schliessen. Mathematische Zeitschrift, 39, 176–210 and 405–431.
Golan, R. (2021). Metainferences from a proof-theoretic perspective, and a hierarchy of validity predicates. Journal of Philosophical Logic, 1–31.
Halbach, V., & Nicolai, C. (2018). On the costs of nonclassical logic. Journal of Philosophical Logic, 47, 227–257.
Horn, L.R., & Wansing, H. (2020). Negation. In E.N. Zalta (Ed.) The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University Spring 2020 edition.
Humberstone, L. (2011). The connectives. MIT Press.
Kleene, S.C. (1952). Introduction to metamathematics. Amsterdam: North-Holland.
Murzi, J., & Rossi, L. (2020). Generalized revenge. Australasian Journal of Philosophy, 98(1), 153–177.
Pailos, F. (2019). A family of metainferential logics. Journal of Applied Non-Classical Logics, 29(1), 97–120.
Pailos, F. (2021). Empty logics. Journal of Philosophical Logic.
Pailos, F.M. (2020). A fully classical truth theory characterized by substructural means. The Review of Symbolic Logic, 13(2), 249–268.
Porter, B. (2021). Supervaluations and the strict-tolerant hierarchy. Journal of Philosophical Logic, 1–20.
Priest, G. (1979). The logic of paradox. Journal of Philosophical logic, 8(1), 219–241.
Priest, G., Routley, R., & Norman, J. (1989). Paraconsistent Logic: Essays on the Inconsistent. Philosophia Verlag.
Pynko, A. (2010). Gentzen’s cut-free calculus versus the logic of paradox. Bulletin of the Section of Logic, 39(1/2), 35–42.
Ripley, D. (2012). Conservatively extending classical logic with transparent truth. Review of Symbolic Logic, 5(02), 354–378.
Ripley, D. (2013). Paradoxes and failures of cut. Australasian Journal of Philosophy, 91(1), 139–164.
Ripley, D. (2013). Revising up: Strengthening classical logic in the face of paradox. Philosophers, 13.
Ripley, D. (2015). Comparing substructural theories of truth. Ergo, an Open Access Journal of Philosophy, 2.
Ripley, D. (2021). One step is enough. Journal of Philosophical Logic, 1–27.
Roffé, A. J., & Pailos, F. (2021). Translating metainferences into formulae: Satisfaction operators and sequent calculi. Australasian Journal of Logic, 3.
Rosenblatt, L. (2021). Paradoxicality without paradox. Erkenntnis, 1–20.
Rosenblatt, L. (2021). Should the non-classical logician be embarrassed? Philosophy and Phenomenological Research.
Rosenblatt, L., & Fiore, C. Recapture results and classical logic. Unpublished.
Scambler, C. (2020). Classical logic and the strict tolerant hierarchy. Journal of Philosophical Logic, 49(2), 351–370.
Teijeiro, P. (2021). Strength and stability. Análisis Filosófico, 41(2), 337–349.
Woods, J. (2019). Logical Partisanhood. Philosophical Studies, 176, 1203–1224.
Acknowledgements
The ideas included in this article were presented to the audiences of the X Workshop on Philosophical Logic (Buenos Aires, 2021), the Oberseminar Logik und Sprachtheorie (Tuebingen, 2021) and the Buenos Aires Logic Group WIP Seminar (Buenos Aires, 2021), to which we are grateful for their feedback. We are specially thankful to Eduardo Barrio, Bruno Da Ré, Bodgan Dicher, Thomas Ferguson, Andreas Fjellstad, Rea Golan, Elio La Rosa, Isabella McAllister, Dave Ripley, Ariel Roffé, Lucas Rosenblatt, Peter Schroeder-Heister, Damián Szmuc, Paula Teijeiro, Joaquín Toranzo Calderón, Luca Tranchini and the members of the Buenos Aires Logic Group. Also, we would like to thank an anonymous reviewer of this journal for their valuable comments. While writing this paper, Federico Pailos enjoyed a Humboldt Research Fellowship for experienced researchers (March 2020 to July 2021)
Funding
During the preparation of this paper, Federico Pailos enjoyed a Humboldt Research Fellowship for Experienced Researchers (March 2020 to July 2021); also, Federico Pailos, Mariela Rubin and Camillo Fiore were supported by CONICET and the University of Buenos Aires.
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The author report the work was divided as follows: 33% Camillo Fiore, 33% Federico Pailos 33% Mariela Rubin. Availability of data and materials: Not applicable.
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Fiore, C., Pailos, F. & Rubin, M. Inferential Constants. J Philos Logic 52, 767–796 (2023). https://doi.org/10.1007/s10992-022-09687-z
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DOI: https://doi.org/10.1007/s10992-022-09687-z