Journal of Philosophical Logic

, Volume 26, Issue 5, pp 569-588

First online:

Strong Boethius’ Thesis and Consequential Implication

  • Claudio PizziAffiliated withDipartimento di Filosofia e Scienze Sociali, Università di Siena
  • , Timothy WilliamsonAffiliated withPhilosophy Department, The University of Edinburgh

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The paper studies the relation between systems of modal logic and systems of consequential implication, a non-material form of implication satisfying “Aristotle's Thesis” (p does not imply not p ) and “Weak Boethius' Thesis” (if p implies q, then p does not imply not q ). Definitions are given of consequential implication in terms of modal operators and of modal operators in terms of consequential implication. The modal equivalent of “Strong Boethius' Thesis” (that p implies q implies that p does not imply not q) is identified.