Constructive Logic with Strong Negation as a Substructural Logic

Gentzen systems are introduced for Spinks and Veroff’s substructural logic corresponding to constructive logic with strong negation, and some logics in its vicinity. It has been shown by Spinks and Veroff in [9], [10] that the variety of Nelson algebras, the algebras of constructive logic with strong negation N, is term-equivalent to a certain variety of bounded commutative residuated lattices called Nelson residuated lattices. An algebraic proof of this result, simplifying some aspects of the presentation, has been given by Busaniche and Cignoli in [4]. In this short note a sequent calculus is defined for Nelson residuated lattices by extending a sequent calculus (essentially CFLew or AMALL, see e.g. [6]) for involutive bounded integral commutative residuated lattices with a single structural rule. Using the translation of [9], [10] this is also a calculus for the logic N, providing an alternative to systems in the literature that make use either of decomposition rules acting on more than one connective at a time (e.g. [1], [7]) or more complicated structures for display calculi (e.g. [11]). The calculus can be used to show very easily some known results for N such as the disjunction property, decidability, and interpolation, and extended to obtain calculi for logics such as nilpotent minimum logic NM [5] and Lukasiewicz three-valued logic L3. A bounded integral commutative residuated lattice (BICRL for short) is an algebra A = 〈A,∧,∨, ,→,>,⊥〉 with binary operations ∧, ∨, , →, and constants>, ⊥ such that: 〈A,∧,∨,>,⊥〉 is a bounded lattice; 〈A, ,>〉 is a commutative monoid; and x y ≤ z iff x ≤ y → z for all x, y, z ∈ A. We also define ¬x =def x → ⊥; x ⊕ y =def ¬x → y; x =def >; and x =def x x for n ∈ N. A BICRL A is said to be involutive if ¬¬x = x