Conditional Excluded Middle in Systems of Consequential Implication
Pizzi (Claudio) & Williamson (Timothy)
Source: Journal of Philosophical Logic 34, 4 (2005): 333-362
Paper - Abstract

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Author’s Abstract

  1. It is natural to ask under what conditions negating a conditional is equivalent to negating its consequent.
  2. 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 (WBT1, opposite conditionals are not both true).
  3. In the system CI.0 of consequential implication, which is intertranslatable with the modal logic2 KT, WBT3 is a theorem, so it is natural to ask which instances of CEM are derivable.
  4. We also investigate the systems CIw and CI of consequential implication, corresponding to the modal logics4 K and KD respectively, with occasional remarks about stronger systems.
  5. While unrestricted CEM produces modal5 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.
  6. 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.
  7. The methods of the paper are those of propositional modal logic6 as applied to a special sort of conditional.

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