Two interpolation theorems
Abstract
In this paper we present two proofs of the interpolation theorem: One for propositional logic and one for first
order logic ($\ell_{\aleph_0\aleph_0}$). Both are performed in the context of model theory. The interpolation theorem states that if $\varphi$ and $\psi$ are formulas, where $\varphi$ is not a contradiction, $\psi$ is not valid, and $\psi$ is a logical consequence of $\varphi$ ($\varphi \models \psi$), then there exists a formula $\delta$ which is written in a common language to that of $\varphi$ and $\psi$, such that $\varphi \models \delta$ and $\delta \models \psi$. The interpolation theorem was first proved for $\ell_{\aleph_0\aleph_0}$ by William Craig in 1957, and since then the possibility of generalizing or applying it has been investigated. This
theorem has generalizations or applications in proof theory, abstract model theory, computer science, modal logic, intuitionistic logic, etc. Examples of applications or generalizations of the interpolation property are presented related to infinitary logics, generalized quantifiers, second order, non-classical, abstract, etc, are presented. References on open problems regarding the interpolation property in the context of abstract model theory are also offered.
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