Journal of Philosophical Logic 34 (4):333-362 (2005)

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Abstract
It is natural to ask under what conditions negating a conditional is equivalent to negating its consequent. Given a bivalent background logic, this is equivalent to asking about the conjunction of Conditional Excluded Middle and Weak Boethius' Thesis. In the system CI.0 of consequential implication, which is intertranslatable with the modal logic KT, WBT is a theorem, so it is natural to ask which instances of CEM are derivable. We also investigate the systems CIw and CI of consequential implication, corresponding to the modal logics K and KD respectively, with occasional remarks about stronger systems. While unrestricted CEM produces modal collapse in all these systems, CEM restricted to contingent formulas yields the Alt2 axiom, which corresponds to the symmetry of consequential implication. It is proved that in all the main systems considered, a given instance of CEM is derivable if and only if the result of replacing consequential implication by the material biconditional in one or other of its disjuncts is provable. Several related results are also proved. The methods of the paper are those of propositional modal logic as applied to a special sort of conditional.
Keywords conditionals  modal logic  Boethius
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Reprint years 2005
DOI 10.1007/s10992-005-0555-6
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References found in this work BETA

Counterfactuals.David Lewis - 1973 - Blackwell.
The Seas of Language.Michael Dummett - 1993 - Oxford University Press.
Studies in Logical Theory.Robert Stalnaker - 1968 - Oxford: Blackwell.
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Citations of this work BETA

Conditional Heresies.Fabrizio Cariani & Simon Goldstein - 2018 - Philosophy and Phenomenological Research (2):251-282.
Connexive Logic.Heinrich Wansing - 2008 - Stanford Encyclopedia of Philosophy.
Conditional Excluded Middle Without the Limit Assumption.Eric Swanson - 2012 - Philosophy and Phenomenological Research 85 (2):301-321.
Bare Conditionals in the Red.Elena Herburger - 2019 - Linguistics and Philosophy 42 (2):131-175.

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