We formulate and investigate a bi-intuitionistic extension, BiBBI, of the well known bunched logic Boolean BI (BBI), obtained by combining classical logic with full intuitionistic linear logic as considered by Hyland and de Paiva (as opposed to standard multiplicative intuitionistic linear logic). Thus, in addition to the multiplicative conjunction ∗ with its adjoint implication —∗ and unit ⊤∗,...

We formulate and investigate a bi-intuitionistic extension, BiBBI, of the well known bunched logic Boolean BI (BBI), obtained by combining classical logic with full intuitionistic linear logic as considered by Hyland and de Paiva (as opposed to standard multiplicative intuitionistic linear logic). Thus, in addition to the multiplicative conjunction ∗ with its adjoint implication —∗ and unit ⊤∗,...

Paraconsistent logics are, by definition, inconsistency tolerant: In a paraconsistent logic, inconsistencies need not entail everything. However, there is more than one way a body of information can be inconsistent. In this paper I distinguish contradictions from other inconsistencies, and I show that many different logics are in an important sense, “paraconsistent” in virtue of being inconsist...

We extend Parigot's-calculus to form a system of realizers for classical logic which reeects the structure of Gentzen's cut-free, multiple-conclusioned, sequent calculus LK when used as a system for proof-search. Speciically, we add (i) a second binding operator, , which realizes classical, multiple-conclusioned disjunction, and (ii) explicit substitutions, , which provide suucient term-structu...

Bi-intuitionistic logic is the conservative extension of intuitionistic logic with a connective dual to implication. It is sometimes presented as a symmetric constructive subsystem of classical logic. In this paper, we compare three sequent calculi for bi-intuitionistic propositional logic: (1) a basic standard-style sequent calculus that restricts the premises of implication-right and exclusio...

A minimal formula of a given logic L is a formula which is provable in L and is not a non-trivial substitution instance of other provable formulas in L. In [5], Y. Komori asked whether normal proofs of minimal formulas are unique in the implicational fragments of natural deduction systems for the intuitionistic logic and the logic BCK. It was already shown that the answer is positive for BCK, w...