Combinatorics of Interpolation in Gödel Logic

We investigate the combinatorics of interpolation in Gödel logic, the propositional logic whose algebraic semantics is the variety of Heyting algebras generated by chains. 1 Motivation In recent work, Busaniche and Mundici prove that Lukasiewicz logic has the Robinson property RP [4], exploiting the geometry of prime filters in the free finitely generated MV-algebra. In this note we recast their techniques in the combinatorial setting of Gödel logic, proving the RP for that logic (Theorem 1). As the algebraic semantics of Lukasiewicz logic and Gödel logic, respectively MV-algebras and Gödel algebras, are varieties of commutative residuated lattices, the RP is equivalent to the deductive interpolation property DIP [8]. The results presented can be obtained by the literature. Indeed, the RP of Gödel logic follows from the stronger, classical result, that Gödel logic has the Craig interpolation property CIP; this has been proved nonconstructively by Maksimova in 1977 [9], and constructively by Baaz and Veith in 1999 [2]. In the commutative case, the CIP is equivalent to a very strong version of the RP, the superRP; in general, the superRP implies a strong version of the RP, the strongRP, and the strongRP implies the RP; the latter is equivalent, in the commutative case, to the DIP [8]. Therefore, a constructive proof of the RP (equivalently, of the DIP) for Gödel logic is implicit in the aforementioned work of Baaz and Veith. Nevertheless the methods we adopt, based on the finite structure of the free finitely generated Gödel algebra, are naturally related to the dual space of Gödel algebras [5], and are suitable for investigating consequence relations and interpolation properties of other substructural and many-valued logics [1]. In fact, an important motivation for us to investigate the combinatorics of the DIP in Gödel logic is related to Hájek’s Basic logic BL, as we now explain. 1 BL fails the CIP but, as Montagna proved nonconstructively in recent work, enjoys the DIP [10]. A natural development of the latter result, in fact asked for 1Hájek’s Basic logic, BL, can be equivalently introduced as the logic of all continuous triangular norms and their residua (emphasizing its foundational rôle with respect to many-valued logics), or the logic of commutative bounded integral divisible prelinear residuated lattices, namely BL-algebras (emphasizing its pretty high positioning in the lattice of substructural logics). For background, we refer the reader to [7, 6].