First-Order Interpolation of Non-classical Logics Derived from Propositional Interpolation

Abstract

This paper develops a general methodology to connect propositional and first-order interpolation. In fact, the existence of suitable skolemizations and of Herbrand expansions together with a propositional interpolant suffice to construct a first-order interpolant. This methodology is realized for lattice-based finitely-valued logics, the top element representing true. It is shown that interpolation is decidable for these logics.

An interpolant in predicate Gödel logic

“…Let L be a finite set of unary predicate symbols. 1 A model (over L) is a tuple M = (W, ≤, w 0 , A, (P W ) P ∈L ), where (W, ≤) is a quasi-order, the base point w 0 is an element of W such that w 0 ≤ w for all w ∈ W , A is a set, and for each P ∈ L, P W is an order-preserving function from W to P(A), i.e., if w ≤ w ′ in W , then P W (w) ⊆ P W (w ′ ). We will often suppress the superscript W in P W when no confusion can arise.…”

“…• w, a ∃yϕ, where y ∈ X, if and only if there exists b ∈ A such that w, a ∪ {(y, b)} ϕ; 1 We only need to consider unary predicates in this note.…”

“…In the setting of predicate logics, algebraic semantics are not as well understood and the question still remains open for many intermediate predicate logics despite the fact that the question has been actively pursued. Recent advances on the subject include the 2013 paper by Mints, Olkhovikov, and Urquhart [5] in which they show that constant domain intuitionistic logic does not have the interpolation property and the very recent contribution [1] showing, among other, that constant domain intermediate logics based on finite algebras of truth values as well as some fragments of Gödel logic do have the interpolation property. However, the question remains open for full predicate Gödel logic, the logic of all linearly ordered constant domain Kripke models.…”

Lyndon Interpolation holds for the Prenex ⊃ Prenex Fragment of Gödel Logic

“…In [5] it has been shown that first-order finitely valued logics admit first-order interpolation if they admit propositional interpolation. In this paper we extend these results to the prenex ⊃ prenex fragment of Gödel logic making use of the proof of Lyndon interpolation for propositional Gödel logic from [12].…”