Abstract
We show that intuitionistic logic is deductively equivalent to Connexive Heyting Logic (\(\textrm{CHL}\)), hereby introduced as an example of a strongly connexive logic with an intuitive semantics. We use the reverse algebraisation paradigm: \(\textrm{CHL}\) is presented as the assertional logic of a point regular variety (whose structure theory is examined in detail) that turns out to be term equivalent to the variety of Heyting algebras. We provide Hilbert-style and Gentzen-style proof systems for \(\textrm{CHL}\); moreover, we suggest a possible computational interpretation of its connexive conditional, and we revisit Kapsner’s idea of superconnexivity.
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Acknowledgements
A preliminary version of this material has been presented at Trends in Logic 21 − Frontiers of Connexive Logic, Bochum, December 6–8, 2021. Thanks are due to the organisers of that conference and to all participants for their insightful comments. In particular, we are grateful to Luis Estrada González, Andi Kapsner, Jacek Malinowski, Hitoshi Omori, and Heinrich Wansing, to whom we are indebted for several stimulating discussions. We acknowledge the precious contribution of two SL reviewers, who helped us enormously to improve a first draft of this paper. We also gratefully acknowledge the support of Fondazione di Sardegna within the projects “Resource sensitive reasoning and logic”, Cagliari, CUP: F72F20000410007 and “Ubiquitous Quantum Reality (UQR): Understanding the natural processes under the light of quantum-like structures”, Cagliari, CUP: F73C22001360007; and of MIUR within the projects PRIN 2017: “Theory and applications of resource sensitive logics”, CUP: 20173WKCM5 and “Logic and cognition. Theory, experiments, and applications”, CUP: 2013YP4N3.
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Fazio, D., Ledda, A. & Paoli, F. Intuitionistic Logic is a Connexive Logic. Stud Logica (2023). https://doi.org/10.1007/s11225-023-10044-7
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DOI: https://doi.org/10.1007/s11225-023-10044-7
Keywords
- Connexive logic
- Intuitionistic logic
- Heyting algebra
- Semi-Heyting algebra
- Algebraic logic
- Connexive Heyting algebra
- Connexive Heyting logic