Conditional Excluded Middle in Systems of Consequential Implication |
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Authors: | Email author" target="_blank">C?PizziEmail author T?Williamson |
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Institution: | (1) Dipartimento di Filosofia e Scienze Sociali, Università di Siena, Via Roma 47, 53100 Siena, Italy;(2) New College, Oxford, OX1 3BN, United Kingdom |
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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 (CEM, opposite conditionals
are not both false) and Weak Boethius' Thesis (WBT, opposite conditionals are not both true). 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 (semantically, each world can see at most
two worlds), 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. |
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Keywords: | conditionals modal logic Boethius |
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