We consider a logic which is semantically dual (in some precise sense of the term) to intuitionistic. This logic can be labeled as “falsification logic”: it embodies the Popperian methodology of scientific discovery. Whereas intuitionistic logic deals with constructive truth and non-constructive falsity, and Nelson’s logic takes both truth and falsity as constructive notions, in the falsificati...

In this paper I dispute the current view that intuitionistic logic is the common basis for the three main trends of constructivism in the philosophy of mathematics: intuitionism, Russian constructivism and Bishop’s constructivism. The point is that the so-called ‘Markov’s principle’, which is accepted by Russian constructivists and rejected by the other two, is expressible in intuitionistic fir...

Bi-intuitionistic logic is the extension of intuitionistic logic with exclusion, a connective dual to implication. Cut-elimination in biintuitionistic logic is complicated due to the interaction between these two connectives, and various extended sequent calculi, including a display calculus, have been proposed to address this problem. In this paper, we present a new extended sequent calculus D...

Lano, K., The intuitionistic alternative set theory, Annals of Pure and Applied Logic 59 (1993) 141-156. The Alternative Set Theory, as defined in Vopenka and Sochor, demonstrates how a set theory which avoids actually infinite sets can serve as a framework for much of classical mathematics. This paper defines a theory which can serve as an intuitionistic analogue of AST, and examines motivatio...

There are basically two ways to view intuitionistic logic: (i) as a philosoph ical–foundational issue in mathematics, (ii) as a technical discipline within mathematical logic. Let us deal with the philosophical aspects first, they will provide the motivation for the subject. In doing so, we will follow L.E.J. Brouwer, the founding father of intuitionism. Brouwer did contribute little to intuiti...

The famous de Jongh’s theorem of 1970 stated that the intuitionistic logic captured all the logical formulas which have all arithmetical instances derivable in the Heyting Arithmetic HA. In this note we extend de Jongh’s arithmetical completeness property from IPC to the basic intuitionistic logic of proofs, which allows proof assertion statements of the sort x is a proof of F. The logic of pro...

We present a Prolog program that implements a sound and complete theorem prover for first-order intuitionistic logic. It is based on free-variable semantic tableaux extended by an additional string unification to ensure the particular restrictions in intuitionistic logic. Due to the modular treatment of the different logical connectives the implementation can easily be adapted to deal with othe...

Recently, Lincoln, Scedrov and Shankar showed that the multi-plicative fragment of second order intuitionistic linear logic is undecidable, using an encoding of second order intuitionistic logic. Their argument applies to the multiplicative-additive fragment, but it does not work in the classical case, because second order classical logic is decidable. Here we show that the multiplicative-addit...

We propose an extension of answer sets, that we call safe beliefs, that can be used to study several properties and notions of answer sets and logic programming from a more general point of view. Our definition, based on intuitionistic logic and following ideas from D. Pearce [Stable inference as intuitionistic validity, Logic Programming 38 (1999) 79–91], also provides a general approach to de...

A new tableau-based calculus for first-order intuitionistic logic is proposed. The calculus is obtained from the tableau calculus for classical logic by extending its rules by A-terms. Aterms are seen as compact representation of natural deduction proofs. The benefits from that approach are two-fold. First, proof search methods known for classical logic can be adopted: Run-time-Skolemization an...