Igor Zaitsev made a presentation titled "Classical and intuitionistic conditional logics: metatheory, formal models, and proof theory"

Research intern Igor Zaitsev made a presentation "Classical and intuitionistic conditional logics: metatheory, formal models, and proof theory" at the research seminar "From the Logical Point of View".

Abstract

The aim of the two talks is to provide a systematic survey of classical and intuitionistic approaches to conditional logics, and to discuss results recently obtained in this area.

The first part introduces conditional logic as a formal framework for analyzing reasoning with both indicative and counterfactual conditionals. We will discuss the motivation for introducing conditional operators and how they differ from conditionals formalized within other non-classical logics. Key systems developed by R. Stalnaker [8], D. Lewis [3, 4], B.F. Chellas [1], and D. Nute [5] will be examined together with their axiomatizations and various semantic frameworks: sphere models, (generalized) relational semantics, semantics of comparative similarity, and semantics of selection functions [5, 7, 9]. Proofs of several metatheorems concerning these systems will also be presented.

The second part focuses on intuitionistic and constructive variants of conditional logics that have been developed in recent years. We will discuss the motivations for abandoning classical presuppositions and the challenges of formalizing counterfactual reasoning within a constructive setting. We will provide a detailed analysis of the works of Y. Weiss [11, 12], I. Ciardelli, X. Liu [2], and G.K. Olkhovikov [6], which concern both semantic models (in particular, modified relational semantics—bi-relational models) and axiomatic systems for these classes of logic. Special attention is given to constructive conditional logics based on D. Nelson’s logic N4 [10] and on the logic C developed by H. Wansing.

The concluding part presents the author’s own results, including the definition of axiomatic systems and Fitch-style natural deduction calculi for intuitionistic counterparts of the Stalnaker–Lewis systems, for the generalization of connexive constructive conditional logic CCCL of H. Wansing and M. Unterhuber [10] augmented with the seriality axiom, and new constraints on the conditional accessibility relation in the context of bi-relational semantics. These results open prospects for further developments in intuitionistiс—and, more generally, constructive—conditional logic.

References:

[1] Chellas B.F. Basic Conditional Logic // Journal of Philosophical Logic. 1975. Vol. 5. No. 2. P. 133–153.

[2] Ciardelli I., Liu X. Intuitionistic Conditional Logics // Journal of Philosophical Logic. 2020. Vol. 49. No. 4. P. 807–832.

[3] Lewis D. Counterfactuals and Comparative Possibility // Journal of Philosophical Logic. 1973. Vol. 2. No. 4. P. 418–446.

[4] Lewis D. Counterfactuals. Oxford: Blackwell Publishing, 1973.

[5] Nute D., Cross C.B. Conditional Logic // Handbook of Philosophical Logic. Vol. 4. 2nd Edn. / Ed. by D.M. Gabbay, F. Guenthner. Dordrecht: Springer, 2002. P. 1–98.

[6] Olkhovikov G.K. An Intuitionistically Complete System of Basic Intuitionistic Conditional Logic // Journal of Philosophical Logic. 2024. Vol. 53. No. 5. P. 1199–1240.

[7] Segerberg K. Notes on Conditional Logic // Studia Logica. 1989. Vol. 48. No. 2. P. 157–168.

[8] Stalnaker R.C., Thomason R.H. A Semantic Analysis of Conditional Logic // Theoria. 1970. Vol. 36. No. 1. P. 23–42.

[9] Unterhuber M. Possible Worlds Semantics for Indicative and Counterfactual Conditionals? A Formal-Philosophical Inquiry into Chellas-Segerberg Semantics. Frankfurt: Ontos Verlag, 2013.

[10] Wansing H., Unterhuber M. Connexive Conditional Logic. Part 1 // Logic and Logical Philosophy. 2019. Vol. 28. P. 567– 610.

[11] Weiss Y. Basic Intuitionistic Conditional Logic // Journal of Philosophical Logic. 2018. Vol. 48. No. 3. P. 447–469.

[12] Weiss Y. Frontiers of Conditional Logic. PhD Thesis. New York: The City University of New York, 2019.