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# A Gentzen- or Beth-type system, a practical decision procedure and a constructive completeness proof for the counterfactual logics VC and VCS

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In [1] and [2] D. Lewis formulates his counterfactual logic VC as follows. The language contains the connectives ∧, ∨, ⊃, ¬ and the binary connective ≤. AB is read as “A is at least as possible as B”. The following connectives are defined in terms of ≤.

A < B: = ¬(BA) (it is more possible that A than that B).

A ≔ ¬(⊥ ≤ A) (⊥ is the false formula; A is possible).

A ≔ ⊥ ≤ ¬A (A is necessary).

(if A were the case, then B would be the case).

(if A were the case, then B might be the case).

and are two counterfactual conditional operators. (AB) iff ¬(A ¬B).

The following axiom system VC is presented by D. Lewis in [1] and [2]: V: (1) Truthfunctional classical propositional calculus.

References
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[1]Lewis, D., Counterfactuals, Harvard University Press, Cambridge, Massachusetts, 1976.
[2]Lewis, D., Counterfactuals and comparative possibility, Journal of Philosophical Logic, vol. 2 (1973), pp. 418446.
[3]de Swart, H. C. M., Another intuitionistic completeness proof, this Journal, vol. 41, no. 3 (1976), pp. 644662.
[4]de Swart, H. C. M., Gentzen-type systems for C, K and several extensions of C and K: Constructive completeness proofs and practical decision procedures for these systems, Logique et Analyse, vol. 23 (1980), pp. 263284.
[5]Fitting, M., Intuitionistic Logic, Model theory and forcing, North-Holland, Amsterdam, 1969.
[6]Kripke, S. A., Semantical analysis of modal logic. I, Normal modal propositional calculi, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, Band 9, (1963), pp. 6796.
[7]Stalnaker, R. and Thomason, R., A semantic analysis of conditional logic, Theoria, vol. 36 (1970), pp. 2342.
[8]Thomason, R., A Fitch-style formulation of conditional logic, Logique et Analyse, vol. 13 (1970), pp. 397412.
[9]Thomason, R., Decidability in the logic of conditionals, The logical enterprise, (Anderson, A. R., Marcus, R. B. and Martin, R. M., Editors), Yale University Press, New Haven, 1975.
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The Journal of Symbolic Logic
• ISSN: 0022-4812
• EISSN: 1943-5886
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