A note on Humberstone's constant Ω

Abstract

We investigate an expansion of positive intuitionistic logic obtained by adding a constant Ω introduced by Lloyd Humberstone. Our main results include a sound and strongly complete axiomatization, some comparisons to other expansions of intuitionistic logic obtained by adding actuality and empirical negation, and an algebraic semantics. We also brie y discuss its connection to classical logic.

Another Combination of Classical and Intuitionistic Conditionals

“…Future directions to be pursued, beside fully addressing the question (Q2) based on our new system, include (i) investigations into combination of subintuitionistic logic and classical logic, (ii) investigations into other ways of adding classical conditional on top of positive intuitionistic logic, such as those making use of Humberstone's constant Ω (cf. [10,19]), (iii) investigations into the combination via Beth semantics instead of Kripke semantics, and (iv) investigations into other ways to combine two conditionals, via other routes outlined in the introduction.…”

Semantic Incompleteness of Hilbert system for a Combination of Classical and Intuitionistic Propositional Logic

“…The second one is to add classical propositional variables to intuitionistic logic, an approach that is taken in [32,33,34,43,53]. 1 The third one is to introduce classical and intuitionistic operators, an approach that is taken in [13,14,20,21,22,23,28,29,37,41,42,47,48,49,50,54]. 2 The combination C + J, which is studied in [22,23,28,37,54], is based on the third way and is one of the most studied combinations of classical and intuitionistic logic.…”