Maximality of bi-intuitionistic propositional logic

In the style of Lindström’s theorem for classical first-order logic, this article characterizes propositional bi-intuitionistic logic as the maximal (with respect to expressive power) abstract logic satisfying a certain form of compactness, the Tarski union property and preservation under bi-asimulations. Since bi-intuitionistic logic introduces new complexities in the intuitionistic setting by adding the analogue of a backwards looking modality, the present paper constitutes a non-trivial modification of the previous work done by the authors for intuitionistic logic (Badia and Olkhovikov, 2020, Notre Dame Journal of Formal Logic, 61, 11–30).

A Note on Gödel-Dummet Logic LC

Let \(A_{0},A_{1},...,A_{n}\) be (possibly) distintict wffs, \(n\) being an odd number equal to or greater than 1. Intuitionistic Propositional Logic IPC plus the axiom \((A_{0}\rightarrow A_{1})\vee ...\vee (A_{n-1}\rightarrow A_{n})\vee (A_{n}\rightarrow A_{0})\) is equivalent to Gödel-Dummett logic LC. However, if \(n\) is an even number equal to or greater than 2, IPC plus the said axiom is a sublogic of LC.

Efficient SAT-based Proof Search in Intuitionistic Propositional Logic

We present an efficient proof search procedure for Intuitionistic Propositional Logic which involves the use of an incremental SAT-solver. Basically, it is obtained by adding a restart operation to the system by Claessen and Rosén, thus we call our implementation . We gain some remarkable advantages: derivations have a simple structure; countermodels are in general small; using a standard benchmarks suite, we outperform and other state-of-the-art provers.

From intuitionism to Brouwer’s modal logic

We try to translate the intuitionistic propositional logic INT into Brouwer’s modal logic KTB. Our translation is motivated by intuitions behind Brouwer’s axiom p →☐◊p as discussed in [4]. The main idea is to interpret intuitionistic implication as modal strict implication, whereas variables and other positive sen-tences remain as they are. The proposed translation preserves fragments of the Rieger-Nishimura lattice which is the Lindenbaum algebra of monadic formulas in INT. Unfortunately, INT is not embedded by this mapping into KTB.

An ASP Approach to Generate Minimal Countermodels in Intuitionistic Propositional Logic

Intuitionistic Propositional Logic is complete w.r.t. Kripke semantics: if a formula is not intuitionistically valid, then there exists a finite Kripke model falsifying it. The problem of obtaining concise models has been scarcely investigated in the literature. We present a procedure to generate minimal models in the number of worlds relying on Answer Set Programming (ASP).

A CORRECT POLYNOMIAL TRANSLATION OF S4 INTO INTUITIONISTIC LOGIC

We show that the polynomial translation of the classical propositional normal modal logic S4 into the intuitionistic propositional logic Int from Fernández is incorrect. We give a modified translation and prove its correctness, and provide implementations of both translations to allow others to test our results.