Priest (2009) formulates a propositional logic which, by employing the worldsemantics for intuitionist logic, has the same positive part but dualises the negation, to produce a paraconsistent logic which it calls 'Da Costa Logic'. This paper extends matters to the first-order case. The paper establishes various connections between first order da Costa logic, da Costa's own Cω, and classical logic. Tableau and natural deductions systems are provided and proved sound and complete.

One of Da Costa’s motives when he constructed the paraconsistent logic C! was to dualise the negation of intuitionistic logic. In this paper I explore a different way of going about this task. A logic is defined by taking the Kripke semantics for intuitionistic logic, and dualising the truth conditions for negation. Various properties of the logic are established, including its relation to C!. Tableau and natural deduction systems for the logic are produced, as are appropriate algebraic structures. The paper (...) then investigates dualising the intuitionistic conditional in the same way. This establishes various connections between the logic, and a logic called in the literature ‘Brouwerian logic’ or ‘closed-set logic’. (shrink)

In this paper we will study the properties of the least extension n(Λ) of a given intermediate logic Λ by a strong negation. It is shown that the mapping from Λ to n(Λ) is a homomorphism of complete lattices, preserving and reflecting finite model property, frame-completeness, interpolation and decidability. A general characterization of those constructive logics is given which are of the form n(Λ). This summarizes results that can be found already in [13, 14] and [4]. Furthermore, we determine the (...) structure of the lattice of extensions of n(LC). (shrink)

This paper considers Kripke completeness of Nelson's constructive predicate logic N3 and its several variants. N3 is an extension of intuitionistic predicate logic Int by an contructive negation operator ∼ called strong negation. The variants of N3 in consideration are by omitting the axiom A → , by adding the axiom of constant domain ∀x ∨ B) → ∀xA ∨ B, by adding ∨ , and by adding ¬¬; the last one we would like to call the axiom of potential (...) omniscience and can be interpreted that we can eventually verify or falsify a statement, with proper additional information. The proofs of completeness are by the widely-applicable tree-sequent method; however, for those logics with the axiom of potential omniscience we can hardly go through with a simple application of it. For them we present two different proofs: one is by an embedding of classical logic, and the other by the TSg method, which is an extension of the tree-sequent method. (shrink)