Bi-intuitionistic logic is the union of intuitionistic and dual intuitionistic logic, and was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent “cut-free” sequent calculus has recently been shown to fail cut-elimination. We present a new cut-free sequent calculus for bi-intuitionistic logic, and prove it sound and complete with respect to its Kr...

We generalize the Kolmogorov axioms for probability calculus to obtain conditions defining, for any given logic, a class of probability functions relative to that logic, coinciding with the standard probability functions in the special case of classical logic but allowing consideration of other classes of “essentially Kolmogorovian” probability functions relative to other logics. We take a broa...

In this paper we investigate the fragment of intuitionistic logic which only uses conjunction (meet) and implication, using finite duality for distributive lattices and universal models. We give a description of the finitely generated universal models of this fragment and give a complete characterization of the up-sets of Kripke models of intuitionistic logic which can be defined by meet-implic...

To provide a categorical semantics for co-intuitionistic logic, one has to face the fact, noted by Tristan Crolard, that the definition of co-exponents as adjuncts of co-products does not work in the category Set, where co-products are disjoint unions. Following the familiar construction of models of intuitionistic linear logic with exponent !, we build models of co-intuitionistic logic in symm...

We investigate the complexity of the model checking problem for propositional intuitionistic logic. We show that the model checking problem for intuitionistic logic with one variable is complete for logspace-uniform AC1, and for intuitionistic logic with two variables it is P-complete. For superintuitionistic logics with one variable, we obtain NC1-completeness for the model checking problem an...

We introduce a logic 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; it can be viewed as a merging of intuitionistic logic and multiplicative intuitionistic linear logic. The naturality of BI can be seen cate...

In this paper we propose intuitionistic logic as the minimal logic most suitable to define autoepistemic logic. We start from a formal analysis of the notion of consistency on an intuitionistic basis, via Gentzen’s sequent calculus. Then we show that the epistemic operator M interpreted as a consistency operator behaves coherently and is also an adequate interpretation of the intuitionistic con...

We generalize the Kolmogorov axioms for probability calculus to obtain conditions defining, for any given logic, a class of probability functions relative to that logic, coinciding with the standard probability functions in the special case of classical logic but allowing consideration of other classes of “essentially Kolmogorovian” probability functions relative to other logics. We take a broa...

We give Binary Decision Diagram (BDD) based methods for deciding validity and satisfiability of propositional Intuitionistic Logic Int and Bi-intuitionistic Tense Logic BiKt. We handle intuitionistic implication and bi-intuitionistic exclusion by treating them as modalities, but the move to an intuitionistic basis requires careful analysis for handling the reflexivity, transitivity and antisymm...

We combine intuitionistic logic and classical logic into a new, first-order logic called Polarized Intuitionistic Logic. This logic is based on a distinction between two dual polarities which we call red and green to distinguish them from other forms of polarization. The meaning of these polarities is defined model-theoretically by a Kripke-style semantics for the logic. Two proof systems are a...