Algebraic semantics and calculi for Nelson's logics

Abstract

The aim of this thesis is to study a family of logics, comprised of Nelson’s logic S, constructive logic with strong negation N 3, quasi-Nelson logic QN and quasi-Nelson implicative logic QN I. This is done in two ways. The first is by means of an axiomatisation via a Hilbert Calculus and the second is by studying some of the properties of the corresponding quasi-variety of algebras. The main contribution of the thesis is to prove that these logics fit within the theory of algebraisable logics. Making use of this result, the following are also proven. Regarding S, we introduced its first semantics, axiomatised by means of a finite Hilbert-style calculus, as well as established a version of the deduction theorem for it. Regarding QN and QN I, we showed that both are algebraisable with respect to the class of quasi-Nelson algebras and quasi-Nelson implication algebras, respectively; we showed that they are non-self-extensional; we showed how to obtain from them, by axiomatic extensions, other well-known logics, such as the {→, ∼}-fragment of intuitionistic propositional logic, the {→, ∼}-fragment of Nelson’s constructive logic with strong negation and classical logic; and finally, we made explicit the quaternary term that guarantees that both QN and QN I satisfy the deduction theorem. Regarding N 3, we study the role of the Nelson identity ((φ ⇒ (φ ⇒ ψ)) ∧ (∼ ψ ⇒ (∼ ψ ⇒ ∼ φ)) ≈ φ ⇒ ψ) in establishing order-theoretic properties for its algebraic semantics. Moreover, we have studied the ⟨∧, ∨, ∼, ¬, 0, 1⟩-subreducts of quasi-Nelson algebras, and by making use of their twist representation, proved that this object-level correspondence can be stated as a categorical equivalence. Lastly, it is worth noting that QN I is the {→, ∼}-fragment of QN , so some results concerning QN I may be easily extended to QN .