On Natural Deduction for Herbrand Constructive Logics III: The Strange Case of the Intuitionistic Logic of Constant Domains

Abstract

The logic of constant domains is intuitionistic logic extended with the so-called forall-shift axiom, a classically valid statement which implies the excluded middle over decidable formulas. Surprisingly, this logic is constructive and so far this has been proved by cut-elimination for ad-hoc sequent calculi. Here we use the methods of natural deduction and Curry-Howard correspondence to provide a simple computational interpretation of the logic.

Theory of constructive semigroups with apartness -- foundations, development and practice

“…where x is not a free variable of P . The intermediate logic obtained in this way, as it is pointed out in [1], further proves intuitionistically as well as classically valid theorems, yet they often possess a strong constructive flavour. From a given co-quasiorder τ , with CD as a logical background, we are able to prove the connection of its classes with sd-subsets of S. Proof.…”

“…Following [1], the intuitionistic logic of constant domains CD arises from a very natural Kripke-style semantics, which was proposed in [20] as a philosophically plausible interpretation of intuitionistic logic. CD can be formalized as intuitionistic logic extended with from the classical algebra point of view pretty strong principle, the Constant Domain Axiom, CDA,…”