Super-Strict Implications

Authors

DOI:

https://doi.org/10.18778/0138-0680.2021.02

Keywords:

strict implication, paradoxes of implication, connexive implication, sequent calculi, structural rules

Abstract

This paper introduces the logics of super-strict implications, where  a super-strict implication is  a strengthening of  C.I. Lewis' strict implication that avoids not only the paradoxes of material implication but also those of strict implication. The semantics of super-strict implications is obtained by strengthening the (normal) relational semantics for strict implication. We consider all logics of super-strict implications that are based on relational frames for modal logics in the  modal cube. it is shown that all  logics of super-strict implications are connexive logics in that they validate Aristotle's Theses and (weak) Boethius's Theses. A proof-theoretic characterisation of logics of super-strict implications is given by means of G3-style labelled calculi, and it is proved that the structural rules of inference are admissible in these calculi. It  is also shown that validity in the $$\mathsf{S5}$$-based logic of super-strict implications is equivalent to validity in  G. Priest's negation-as-cancellation-based  logic. Hence, we also   give a cut-free calculus for Priest's logic.

References

[1] A. R. Anderson, N. Belnap, Entailment. The Logic of Relevance and Necessity, vol. 1, Princeton University Press (1975).


Google Scholar

[2] R. George, Bolzano’s consequence, relevance, and enthymemes, Journal of Philosophical Logic, vol. 12 (1983), pp. 299–318, DOI: http://dx.doi.org/https://doi.org/10.1007/BF00263480.


Google Scholar

[3] J. Y. Halpern, Y. Moses, A Guide to Completeness and Complexity for Modal Logics of Knowledge and Belief, Artificial Intelligence, vol. 54(2) (1992), pp. 319–379, DOI: http://dx.doi.org/10.1016/0004-3702(92)90049-4.


Google Scholar

[4] J. Heylen, L. Horsten, Strict Conditionals: A Negative Result, The Philosophical Quarterly, vol. 56(225) (2006), pp. 536–549, DOI: http://dx.doi.org/10.1111/j.1467-9213.2006.457.x.


Google Scholar

[5] D. Hitchcock, Does the Traditional Treatment of Enthymemes Rest on a Mistake?, Argumentation, vol. 12 (1998), pp. 15–37, DOI: http://dx.doi.org/10.1007/978-3-319-53562-3_5.


Google Scholar

[6] C. I. Lewis, Survey of Symbolic Logic, University of California Press (1918).


Google Scholar

[7] D. Lewis, Counterfactuals, Harvard University Press (1975).


Google Scholar

[8] E. J. Lowe, A simplification of the logic of conditionals, Notre Dame Journal of Formal Logic, vol. 24(3) (1983), pp. 357–366, DOI: http://dx.doi.org/10.1305/ndjfl/1093870380.


Google Scholar

[9] E. J. Lowe, The Truth About Counterfactuals, The Philosophical Quarterly, vol. 45(178) (1995), pp. 41–59, DOI: http://dx.doi.org/10.2307/2219847.


Google Scholar

[10] S. Negri, Proof Theory for Non-normal Modal Logics: The Neighbourhood Formalism and Basic Results, IfCoLog Journal of Logic and its Applications, vol. 4 (2017), pp. 1241–1286.


Google Scholar

[11] S. Negri, E. Orlandelli, Proof theory for quantified monotone modal logics, Logic journal of the IGPL, vol. 27(4) (2019), p. 478–506, DOI: http://dx.doi.org/10.1093/jigpal/jzz015.


Google Scholar

[12] S. Negri, J. von Plato, Proof Analysis, Cambridge University Press (2011).


Google Scholar

[13] E. Nelson, Intensional relations, Mind, vol. 39 (1930), pp. 440–453.


Google Scholar

[14] H. Omori, H. Wansing, Connexive logics. An overview and current trends, Logic and Logical Philosophy, vol. 28(3) (2019), pp. 371–387, DOI: http://dx.doi.org/10.12775/LLP.2019.026.


Google Scholar

[15] C. Pizzi, T. Williamson, Strong Boethius’ Thesis and Consequential Implication, Journal of Philosophical Logic, vol. 26 (1997), pp. 569–588, DOI: http://dx.doi.org/10.1023/A:1004230028063.


Google Scholar

[16] G. Priest, Negation as cancellation and connexive logic, Topoi, vol. 18 (1999), pp. 141–148, DOI: http://dx.doi.org/10.1023/A:1006294205280.


Google Scholar

[17] E. Raidl, Strengthened Conditionals, [in:] B. Liao, Y. N. Wáng (eds.), Context, Conflict and Reasoning, Springer Singapore (2020), pp. 139–155, DOI: http://dx.doi.org/10.1007/978-981-15-7134-3_11.


Google Scholar

[18] H. Rasiowa, An Algebraic Approach to Non-classical Logics, Elsevier (1974).


Google Scholar

[19] R. Routley, V. Routley, Negation and contradiction, Revista Colombiana de Matemáticas, vol. 19 (1985), pp. 201–230.


Google Scholar

[20] R. C. Stalnaker, A Theory of Conditionals, [in:] N. Rescher (ed.), Studies in Logical Theory, Basil Blackwell (1968), pp. 98–112.


Google Scholar

[21] A. S. Troelstra, D. van Dalen, Constructivism in Mathematics, North-Holland (1988).


Google Scholar

[22] M. Vidal, When Conditional Logic met Connexive Logic, [in:] IWCS 2017 - 12th International Conference on Computational Semantics (2017), pp. 1–11, URL: https://www.aclweb.org/anthology/W17-6816.


Google Scholar

[23] H. Wansing, D. Skurt, Negation as Cancellation, Connexive Logic, and qLPm, The Australasian Journal of Logic, vol. 15 (2018), pp. 476–488, DOI: http://dx.doi.org/10.26686/ajl.v15i2.4869.


Google Scholar

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Published

2021-01-20

How to Cite

Orlandelli, E., & Gherardi, G. (2021). Super-Strict Implications. Bulletin of the Section of Logic, 34 pp. https://doi.org/10.18778/0138-0680.2021.02

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Research Article