Some Admissible Rules in Modal Systems with the Brouwerian Axiom

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.