Some Admissible Rules in Modal Systems with the Brouwerian Axiom
Timothy L. Williamson · Logic Journal of IGPL · 1996
The paper studies the admissibility of some cancellation rules in normal modal systems with the Brouwerian axiom. For example, KDB and KTB are proved to admit the following rule: if ⊢ ¬(α ∧ β) and ⊢ ⋄α ≡ ⋄β then ⊢ ¬(α ∨ β). Two notions of the preservation of validity by a rule on a frame are defined; on both, the preservation of validity by the preceding rule is shown not to be a first-order condition. A speculative connection is suggested with logics of vagueness.