Craig interpolation theorem fails in bi-intuitionistic predicate logic

Abstract

In this article we show that bi-intuitionistic predicate logic lacks the Craig Interpolation Property. We proceed by adapting the counterexample given by Mints, Olkhovikov and Urquhart for intuitionistic predicate logic with constant domains (G. Mints, G. K. Olkhovikov and A. Urquhart. Failure of Interpolation in Constant Domain Intuitionistic Logic. Journal of Symbolic Logic, 78: 937--950 (2013)). More precisely, we show that there is a valid implication $\phi \rightarrow \psi$ with no interpolant (i.e. a formula $\theta$ in the intersection of the vocabularies of $\phi$ and $\psi$ such that both $\phi \rightarrow \theta$ and $\theta \rightarrow \psi$ are valid). Importantly, this result does not contradict the unfortunately named `Craig interpolation' theorem established by Rauszer in (Cecylia Rauszer. Craig Interpolation Theorem for an Extention of Intuitionistic Logic. Bull. Ac. Pol. Sc., 25(4), 337--341 (1977)) since that article is about the property more correctly named `deductive interpolation' (see Galatos, Jipsen, Kowalski and Ono's use of this term in N. Galatos, P. Jipsen, T. Kowalski, \& H. Ono. Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Studies in Logic and the Foundations of Mathematics, Vol. 151. Amsterdam: Elsevier B. V. (2007)) for global consequence. Given that the deduction theorem fails for bi-intuitionistic logic with global consequence, the two formulations of the property are not equivalent.