Abstract

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 ⊢ ¬ and ⊢ ⋄α ≡ ⋄β then ⊢ ¬. 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