We present a system of natural deduction and associated term calculus for intuitionistic non-commutative linear logic (INCLL) as a conservative extension of intuitionistic linear logic. We prove subject reduction and the existence of canonical forms in the implicational fragment.

The famous double negation translation [16, 17] establishes an embedding of classical into intuitionistic logic. Curiously, the reverse direction has not been covered in literature. Utilizing a normal form for logic [20], we establish small model property propositional We use this direct encoding Kripke semantics and quantified Boolean formulae. Next, transfer developed techniques to first orde...

We reconsider Dalla Pozza and Garola pragmatic interpretation of intuitionistic logic [13] where sentences and proofs formalize assertions and their justifications and revise it so that the costruction is done within an intuitionistic metatheory. We reconsider also the extension of Dalla Pozza and Garola’a approach to cointuitionistic logic, seen as a logic of hypotheses [5, 9, 4] and the duali...

We present the logic of bunched implications, BI, in which a multiplicative (or linear) and an additive (or intuitionistic) implication live side-by-side. The propositional version of BI arises from an analysis of the proof-theoretic relationship between conjunction and implication, and may be viewed as a merging of intuitionistic logic and multiplicative, intuitionistic linear logic. The predi...

In this paper we discuss Gabbay’s idea of basing nonmonotonic deduction on semantic consequence in intuitionistic logic extended by a consistency operator and Turner’s suggestion of replacing the intuitionistic base system by Kleene’s three-valued logic. It is shown that a certain counterintuitive feature of these approaches can be avoided by using Nelson’s constructive logic N instead of intui...