Matthew Spinks

Constructive Logic with Strong Negation is a Substructural Logic. I

The goal of this two-part series of papers is to show that constructive logic with strong negation N is definitionally equivalent to a certain axiomatic extension NFLew of the substructural logic FLew. In this paper, it is shown that the equivalent variety semantics of N (namely, the variety of Nelson algebras) and the equivalent variety semantics of NFLew (namely, a certain variety of FLew-algebras) are term equivalent. This answers a longstanding question of Nelson [30]. Extensive use is made of the automated theorem-prover Prover9 in order to establish the result.The main result of this paper is exploited in Part II of this series [40] to show that the deductive systems N and NFLew are definitionally equivalent, and hence that constructive logic with strong negation is a substructural logic over FLew.

Cancellation in Skew Lattices

Distributive lattices are well known to be precisely those lattices that possess cancellation:

The Logic of Quasi-MV Algebras

x \lor y = x \lor z

Axiomatizing the Skew Boolean Propositional Calculus

and

Abelian Logic and the Logics of Pointed Lattice-Ordered Varieties

x \land y = x \land z

Skew lattices and binary operations on functions

imply y = z. Cancellation, in turn, occurs whenever a lattice has neither of the five-element lattices M3 or N5 as sublattices. In this paper we examine cancellation in skew lattices, where the involved objects are in many ways lattice-like, but the operations

Quasi-discriminator varieties

\land

Paraconsistent constructive logic with strong negation as a contraction-free relevant logic

and

Multiple-valued logic as a programming language

\lor

Algebraic Semantics for Nelson’s Logic \(\mathcal {S}\)

no longer need be commutative. In particular, we find necessary and sufficient conditions involving the nonoccurrence of potential sub-objects similar to M3 or N5 that ensure that a skew lattice is left cancellative (satisfying the above implication) right cancellative (