On strong negation as linear duality

In [5] it is shown how a model for the logic of constructible duality is a symmetric monoidal closed category with products, that is, a model of linear logic. Remarkably, Nelson’s strong negation∼ behaves as linear duality ⊥. We compare it with some previous results from [4] regarding stable models in a linear logic setting, and suggest some further lines to explore. 1. From Ncd to linear logic We began by reviewing the steps taken by Patterson et al [5] for the construction of an algebraic model for the logic of constructible duality (named here Ncd), which is a conservative extension of the paraconsistent version of Nelson’s logic constructible falsity, N− c (also named N4 in [1, 2]). Starting with the logic of constructible falsity N3, (that is, positive intuitionistic logic with the involutory strong negation ∼), one obtains N4 by dropping the axiom Γ, φ,∼φ ` ∆ and keeping the constants for falsity 0 and truth 1. The formal system of N4 is obtained from that of intuitionistic logic plus the rules for ∼, as depicted in Figure 1. The first step consists in extending N4 by defining the implication ( as φ( ψ = (φ→ ψ) ∧ (∼ψ → ∼φ) (where→ is the intuitionistic implication). In contrast to→, the new implication forms a congruence on the equivalence class of interderivable formula. This implication provides an ordering on the corresponding algebra. A definable operator, which plays the rôle of a conjunction for ( and hence providing a deduction theorem in the algebraic setting, is given by φ⊗ ψ = ∼(φ(∼ψ). Γ, φ ` ∆ Γ,∼∼φ ` ∆ Γ ` ∆, φ Γ ` ∆,∼∼φ Γ,∼φ ` ∆ Γ,∼ψ ` ∆ Γ,∼(φ ∧ ψ) ` ∆ Γ ` ∆,∼φ,∼ψ Γ ` ∆,∼(φ ∧ ψ) Γ, φ,∼ψ ` ∆ Γ,∼(φ→ ψ) ` ∆ Γ ` ∆, φ Γ ` ∆,∼ψ Γ ` ∆,∼(φ→ ψ)