Intuitionistic Logic with a “Definitely” Operator

This paper introduces a logic ILED derived from standard intuitionistic predicate logic by adding two operators Dφ for “Definitely φ” and ~φ for “Experience rejects φ”. A further negation ¬φ =def (φ→⊥) ∨ ~φ , which we call real negation, is introduced. Real negation is like intuitionistic negation when there are no D-operators but deviates when there are. We see that Dφ ↔ φ is valid but ¬Dφ → ¬φ is not and hence that contraposition fails for real negation. We give a semantics for this logic, axiomatise it and prove the axiomatisation complete. Finally we show that real negation behaves as standard intuitionistic negation within D-free contexts. The logic ILED is proposed as an extension of intuitionistic logic apt for use as a general logic.