We introduce a propositional logic ICL, which adds to intuitionistic logic elements of classical reasoning without collapsing it into classical logic. This logic includes a new constant for false, which augments false in intuitionistic logic and in minimal logic. The new constant requires a simple-yet-significant modification of intuitionistic logic both semantically and proof-theoretically. We...

We introduce a propositional logic ICL, which adds to intuitionistic logic elements of classical reasoning without collapsing it into classical logic. This logic includes a new constant for false, which augments false in intuitionistic logic and in minimal logic. We define Kripke models for ICL and show how they translate to several other forms of semantics. We define a sequent calculus and pro...

Nelson has presented a constructive arithmetic with a negation operation (−) different from the ordinary intuitionistic one (¬). In [5] he presents a variant of Kleene's realization semantics for intuitionistic arithmetic, and proves that relative to this interpretation the arithmetic language with – has the same expressive power as the usual intuitionistic one, and fact certain theories of ari...

In the paper we propose a kind of interpretation of partial logic in the intuitionistic logic, or rather its part. We will show that finite binary trees determine this special fragment of intuitionistic logic.

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...

We explore the proof complexity of intuitionistic propositional logic (IPL). The problem of determining whether or not an intuitionistic formula is valid is PSPACE-Complete via a reduction from QBF . In view of this reduction (due to Statman), it is natural to compare the proof-theoretic strength of a standard axiomatic system for IPL with a similar proof system for classical quantified Boolean...

We present three functional interpretations of intuitionistic linear logic and show how these correspond to well-known functional interpretations of intuitionistic logic via embeddings of IL into ILL. The main difference from previous work of the second author is that in intuitionistic linear logic the interpretations of !A are simpler (at the cost of an asymmetric interpretation of pure ILL) a...

We perform a proof-theoretical investigation of two modal predicate logics: global intuitionistic logic GI and global intuitionistic fuzzy logic GIF. These logics were introduced by Takeuti and Titani to formulate an intuitionistic set theory and an intuitionistic fuzzy set theory together with their metatheories. Here we define analytic Gentzen style calculi for GI and GIF. Among other things,...

Lincoln, Scedrov and Shankar proved undecidability of intuitionistic second order multiplicative commutative linear logic by embedding LJ2 into this logic. Emms did the same for intuitionistic second order non-commutative linear logic. Recently, Lafont and Scedrov demonstrated undecidability of classical second order multiplicative commutative linear logic. As for classical second order non-com...