Conuclear images of substructural logics

A famous result by McKinsey and Tarski ([5] and [6]) provides an interpretation of intuitionistic logic into the modal logic S4. The algebras for S4 are Boolean algebras with an interior operator. One wonders if it is possible to perform a similar construction starting from a logic which is different from classical logic, say a logic extending Full Lambek Calculus FL. The algebraic counterpart of this construction is the so-called conuclear image of a variety of (pointed) residuated lattices. Recall from [3] that a conucleus on a residuated lattice R is a monotone map σ from R into R satisfying the following identities: σ(1) = 1, σ(x) ≤ x, σ(x) ≤ σ(σ(x)) and σ(x) · σ(y) ≤ σ(x · y) for all x, y ∈ R. Since an interior operator on a Boolean algebra is a special example of conu-cleus, the following generalization of McKinsey and Tarski's interpretation comes naturally: given a substructural logic L, denote by L σ the logic L with an added unary operator σ along with the axioms: 1 (i) (σ(A) · σ(B)) → σ(A · B); (ii) σ(A) → A; (iii) σ(A) → σ(σ(A)); and the necessitation rule A σ(A). Then we can define an interpretation σ of L into L σ as follows: p σ i = σ(p i) (A ∧ B) σ = σ(A σ ∧ B σ), (A\B) σ = σ(A σ \B σ) and (A/B) σ = σ(A σ /B σ). The conuclear image of L is the logic σ(L) whose theorems are precisely those formulas A of L such that L σ A σ. Interestingly, σ(L) is always a substructural logic (i.e., an extension of FL) and L is an extension of σ(L). Moreover, if L is classical logic, σ(L) is intuitionistic logic. Also, by a result proved in [7], the logic of commutative, cancellative residuated lattices is the conuclear image of the logic of lattice-ordered abelian groups. The main problem we investigate in the paper is the following: what is the relationship between L and σ(L)? In particular: (1) Which are the theorems of L which hold also in σ(L)? (2) Which properties are excluded to hold in σ(L), whatever L is? (3) Which theorems are not necessarily preserved as in (1), nor excluded to hold as in (2)? As regards to (1), we prove that all formulas corresponding to inequations of the form f ≤ g with f …