A refinement of the Craig-Lyndon Interpolation Theorem for classical first-order logic (with identity)

Abstract

We refine the interpolation property of the {&, v, ~, A, E}-fragment of classical first-order logic, showing that if [G is satisfiable] and [D is ot logically true] and G|- D then there is an interpolant c, constructed using only non-logical vocabulary common to both members of G and members of D, such that (i) G entails c in the first-order version of Kleene’s strong three-valued logic (K3), and (ii) c entails D in the first-order version of Priest’s Logic of Paradox (LP). The proof proceeds via a careful analysis of derivations in a cut-free sequent calculus for first-order classical logic. Lyndon’s strengthening falls out of an observation regarding such derivations and the steps involved in the construction of interpolants. The proof is then extended to cover the {&, v, ~, A, E}-fragment of classical first-order logic with identity. Keywords: Craig–Lyndon Interpolation Theorem (for classical first-order logic); Kleene’s strong 3-valued logic;Priest’s Logic of Paradox; Belnap’s four-valued logic