Abstract
The concept of substructural logic was originally introduced in relation to limitations of Gentzen’s structural rules of Contraction, Weakening and Exchange. Recent years have witnessed the development of substructural logics also challenging the Tarskian properties of Reflexivity and Transitivity of logical consequence. In this introduction we explain this recent development and two aspects in which it leads to a reassessment of the bounds of classical logic. On the one hand, standard ways of defining the notion of logical consequence in classical logic naturally induce substructural logics when admitting more than two truth values; on the other hand, these substructural logics give rise to hierarchies of metainferences that can be used to approximate classical logic at different levels.
Article PDF
References
Alchourrón, C. E., Gärdenfors, P., & Makinson, D. (1985). On the logic of theory change: partial meet contraction and revision functions. The Journal of Symbolic Logic, 50(2), 510–530.
Anderson, A.R., Belnap, N. D., & Dunn, M. J. (1992). Entailment, Vol. II: the logic of relevance and necessity, vol. 5009. Princeton University Press.
Barrio, E., Pailos, F., & Szmuc, D. (2018). What is a paraconsistent logic. In W. Carnielli J. Malinowski (Eds.) Contradictions, from consistency to inconsistency, Springer. Trends in Logic, vol. 47, (pp. 89–108).
Barrio, E., Pailos, F., & Szmuc, D. (2021). (Meta)inferential levels of entailment beyond the Tarskian paradigm. Synthese, 198(22), 5265–5289.
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., & Szmuc, D. (2020). A hierarchy of classical and paraconsistent logics. Journal of Philosophical Logic, 49(1), 93–120.
Belnap, N.D. (1973). Restricted quantification and conditional assertion. In H. Leblanc (Ed.) Truth, syntax and modality, vol. 68, (pp. 48–75). Elsevier, Amsterdam.
Bennett, B. (1998). Modal semantics for knowledge bases dealing with vague concepts. In A. Cohn, L. Schubert, & S. Shapiro (Eds.) Principles of knowledge representation and reasoning: Proceedings of the 6th international conference (KR-98), (pp. 234–244). Morgan Kaufmann.
Blasio, C., Marcos, J., & Wansing, H. (2017). An inferentially many-valued two-dimensional notion of entailment. Bulletin of the Section of Logic, 46(3/4), 233–262.
Chemla, E., & Égré, P. (2019). Suszko’s problem: mixed consequence and compositionality. The Review of Symbolic Logic, 12(4), 736–767.
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., Égré, P., Ripley, D., & van Rooij, R. (2012). Tolerant, classical, strict. Journal of Philosophical Logic, 41(2), 347–385.
Cobreros, P., Égré, P., Ripley, D., & van Rooij, R. (2013). Reaching transparent truth. Mind, 122(488), 841–866.
Cobreros, P., Égré, P., Ripley, D., & van Rooij, R. (2015). Vagueness, truth and permissive consequence. In D. Achouriotti, H. Galinon, & J. Martinez (Eds.) Unifying the philosophy of truth, (pp. 409–430). Springer.
Cobreros, P., Égré, P., Ripley, D., & van Rooij, R. (2021). Tolerant reasoning: nontransitive or nonmonotonic? Synthese, 199, 681–705.
Da Ré, B (2021). Structural weakening and paradoxes. Notre Dame Journal of Formal Logic, 62(2), 369–398.
Da Ré, B., Pailos, F., Szmuc, D., & Teijeiro, P. (2020). Metainferential duality. Journal of Applied Non-Classical Logics, 30(4), 312–334.
Da Ré, B., Rubin, M., & Teijeiro, P. (2022). Metainferential paraconsistency. Logic and Logical Philosophy, 33, 235–260.
Dicher, B., & Paoli, F. (2019). ST, LP, and tolerant metainferences. In C. Başkent T.M. Ferguson (Eds.) Graham priest on dialetheism and paraconsistency, (pp. 383–407). Springer Verlag.
Došen, K. (1992). Modal translations in substructural logics. Journal of Philosophical Logic, 21(3), 283–336.
Field, H. (2008). Saving truth from paradox. OUP Oxford.
von Fintel, K. (1999). NPI-Licensing, Strawson-entailment, and context-dependencies. Journal of Semantics, 16(2), 97–148.
Fitting, M. (2021). A family of strict/tolerant logics. Journal of Philosophical Logic, 50(2), 363–394.
Frankowski, S. (2004). Formalization of a plausible inference. Bulletin of the Section of Logic, 33(1), 41–52.
French, R. (2016). Structural reflexivity and the paradoxes of self-reference. Ergo, 3(5), 113–131.
French, R., & Ripley, D. (2019). Valuations: bi, tri, and tetra. Studia Logica, 107(6), 1313–1346.
Gentzen, G. (1935). Untersuchungen über das logische Schließen. Mathematische Zeitschrift, 39(2), 176–210,405–431.
Girard, J.-Y. (1987). Linear logic. Theoretical Computer Science, 50(1), 1–101.
Girard, J.-Y. (1991). Proof Theory and Logical Complexity. Bibliopolis.
Hlobil, U. (2019). Faithfulness for naive validity. Synthese, 196 (11), 4759–4774.
Humberstone, L. (1988). Heterogeneous logic. Erkenntnis, 29(3), 395–435.
Humberstone, L. (1996). Valuational semantics of rule derivability. Journal of philosophical logic, 25(5), 451–461.
Lambek, J. (1958). The mathematics of sentence structure. The American Mathematical Monthly, 65(3), 154–170.
Lambek, J (1961). On the calculus of syntactic types. In R. Jakobson (Ed.) Structure of language and its mathematical aspects, vol. 12, (pp. 166–178). Ams Providence.
Makinson, D. (2005). Bridges from classical to nonmonotonic logic. King’s College Publication.
Malinowski, G. (1990). Q-consequence operation. Reports on Mathematical Logic, 24, 49–54.
Malinowski, J. (2004). Strawsonian presuppositions and logical entailment. Logique et Analyse, 47, 123–138.
Mares, E., & Paoli, F. (2014). Logical consequence and the paradoxes. Journal of Philosophical Logic, 43(2-3), 439–469.
Misiuna, K. (2010). A certain consequence relation for solving paradoxes of vagueness. Logique et Analyse, 53(209), 25–50.
Moortgat, M. (1997). Categorial type logics. In J. van Benthem A. ter Meulen (Eds.) Handbook of logic and language, (pp. 93–177). Elsevier.
Moot, R, & Retoré, C. (2012). The logic of categorial grammars: a deductive account of natural language syntax and semantics, vol. 6850. Springer.
Murzi, J., & Rossi, L. (2022). Non-reflexivity and revenge. Journal of Philosophical Logic, 51(1), 201–218.
Nait-Abdallah, A. (1995). The logic of partial information. Springer.
Nicolai, C., & Rossi, L. (2018). Principles for object-linguistic consequence: from logical to irreflexive. Journal of Philosophical Logic, 47(3), 549–577.
Ono, H. (1990). Structural rules and a logical hierarchy. In P. Petkov (Ed.) Mathematical logic, (pp. 95–104). Springer.
Ono, H. (2003). Substructural logics and residuated lattices—an introduction. Trends in logic, pp. 193–228.
Pailos, F.M. (2020). A fully classical truth theory characterized by substructural means. The Review of Symbolic Logic, 13(2), 249–268.
Paoli, F. (2013). Substructural logics: a primer, vol. 13. Springer Science & Business Media.
Poggiolesi, F. (2022). Bolzano, (the Appropriate) Relevant Logic, and Grounding Rules for Implications. In S. Roski and B. Schnieder (eds), Bolzano’s philosophy of grounding: Translation and Studies, Oxford University Press.
Priest, G. (1979). Logic of paradox. Journal of Philosophical Logic, 8, 219–241.
Priest, G. (2008). An introduction to non-classical logic, from if to is. Cambridge University Press.
Restall, G. (2002). An introduction to substructural logics. Routledge.
Restall, G. (2005). Multiple conclusions. In P. Hájek, L. Valdés-Villanueva, & D. Westerståhl (Eds.) Logic, methodology and philosophy of science: Proceedings of the twelfth international congress. KCL Publications, London.
Ripley, D. (2012). Conservatively extending classical logic with transparent truth. The Review of Symbolic Logic, 5(02), 354–378.
Ripley, D. (2013). Paradoxes and failures of cut. Australasian Journal of Philosophy, 91(1), 139–164.
van Rooij, R. (2011). Vagueness, tolerance and nontransitive entailment. In P. Cintula, C. Fermüller, L. Godo, & P. Hàjek (Eds.) Understanding vagueness. College publications.
Rumfitt, I. (2000). Yes and no. Mind, 109(436), 781–823.
Scambler, C. (2020). Classical logic and the strict tolerant hierarchy. Journal of Philosophical Logic, 49(2), 351–370.
P. Schroeder-Heister, K. Došen, (Eds.) (1993). Substructural logics. Oxford: Studies in Logic and Computation, Oxford University Press.
Sharvit, Y. (2017). A note on (Strawson) entailment. Semantics and Pragmatics, 10, 1–1.
Slaney, J. (2011). A logic for vagueness. The Australasian Journal of Logic, vol. 8.
Smith, N.J.J. (2008). Vagueness and Degrees of Truth. Oxford: Oxford University Press.
Strawson, P. (1952). Introduction to logical theory, Methuen: London.
Suszko, R. (1975). Remarks on Lukasiewicz’s three-valued logic. Bulletin of the Section of Logic, 4(3), 87–90.
Suszko, R. (1977). The Fregean axiom and Polish mathematical logic in the 1920s. Studia Logica: An International Journal for Symbolic Logic, 36 (4), 377–380.
Tarski, A. (1930). On some fundamental concepts of metamathematics. In Logic, Semantics, Metamathematics, Papers from 1923 to 1938 by Alfred Tarski, translated by J-H. Woodger, (pp. 30–37). Hackett Publishing Company.
Tarski, A. (1936). The concept of logical consequence. In Logic, Semantics, Metamathematics, Papers from 1923 to 1938 by Alfred Tarski, translated by J-H. Woodger, (pp. 409–420). Hackett publishing company.
Teijeiro, P. (2021). Strength and stability. Análisis Filosófico, 41(2), 337–349.
Tennant, N. (1982). Proof and paradox. Dialectica, 36(2–3), 265–296.
Tennant, N (1984). Perfect validity, entailment and paraconsistency. Studia Logica, 43(1), 181–200.
Weir, A (2005). Naive truth and sophisticated logic. In J. Beall B. Amour-Garb (Eds.) Deflationism and paradox, (pp. 218–249). Oxford University Press Oxford.
Wintein, S. (2014). On the strict–tolerant conception of truth. Australasian Journal of Philosophy, 92(1), 71–90.
Wittgenstein, L. (1922). Tractatus Logico-Philosophicus. Paul Kegan. English translation by C. K. Ogden and F. P. Ramsey.
Zardini, E. (2008). A model of tolerance. Studia Logica, 90(3), 337–368.
Zardini, E. (2011). Truth without contra(di)ction. The Review of Symbolic Logic, 4(4), 498–535.
Zardini, E. (2021). Substructural approaches to paradox: an introduction to the special issue. Synthese, 199(3), 493–525.
Acknowledgements
We thank Reinhard Muskens and Quentin Blomet for comments on the first version of this text, as well as Bruno Da Ré, Bogdan Dicher, Luca Incurvati, Sergi Oms, Federico Pailos, and David Ripley for feedback on specific points. We are most grateful to the École normale supérieure and to the Philosophy Department of ENS for making possible the visit of Eduardo Barrio to Paris in June 2022, as well as to the INALCO and the CNRS (Second summer school “Conditionals in Paris”). Special thanks go to Dimitri El Murr, Patrick Caudal, and Ghanshyam Sharma for their support in relation to this collaboration. PE also acknowledges grants ANR-19-CE28-0019-01 (AMBISENSE) and ANR-17-EURE0017 (FRONTCOG).
Author information
Authors and Affiliations
Corresponding authors
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Barrio, E., Égré, P. Editorial Introduction: Substructural Logics and Metainferences. J Philos Logic 51, 1215–1231 (2022). https://doi.org/10.1007/s10992-022-09693-1
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10992-022-09693-1