Intuitionistic Logic with a “ Definitely ” Operator by Peter Mott January 1997

This paper introduces a logic ILED derived from standard intuitionistic sentence 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.