We prove a generalization of Maehara’s lemma to show that the extensions of classical and intuitionistic first-order logic with a special type of geometric axioms, called singular geometric axioms, have Craig’s interpolation property. As a corollary, we obtain a direct proof of interpolation for (classical and intuitionistic) first-order logic with identity, as well as interpolation for several mathematical theories, including the theory of equivalence relations, (strict) partial and linear orders, and various intuitionistic order theories such as apartness and positive partial and linear orders.

中文翻译：

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一阶逻辑扩展中的插值

我们证明了 Maehara 引理的推广，以表明经典和直觉主义一阶逻辑的扩展具有特殊类型的几何公理，称为奇异几何公理，具有 Craig 插值性质。作为推论，我们获得了（经典的和直觉的）具有恒等的一阶逻辑的插值的直接证明，以及一些数学理论的插值，包括等价关系理论、（严格的）偏序和线性阶数以及各种直觉顺序理论，例如分离性和正偏序和线性顺序。