We obtain an effective embedding of the classical predicate logic into the logic of partial quasiary predicates. The embedding has the property that an image of a non-theorem of the classical logic is refutable in a model of the logic of partial quasiary predicates that has the same cardinality as the classical countermodel of the non-theorem. Therefore, we also obtain an embedding of the classical predicate logic of finite models into the logic of partial quasiary predicates over finite structures. As a consequence, we prove that the logic of partial quasiary predicates is undecidable—more precisely, $\varSigma ^0_1$-complete—over arbitrary structures and not recursively enumerable—more precisely, $\varPi ^0_1$-complete—over finite structures.

中文翻译：

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偏偏谓词逻辑的不可判定性

我们将经典谓词逻辑有效地嵌入到部分准谓词的逻辑中。嵌入具有这样的性质，即经典逻辑的非定理的图像在与非定理的经典反模型具有相同基数的偏拟谓词的逻辑模型中是可反驳的。因此，我们还获得了将有限模型的经典谓词逻辑嵌入到有限结构上的偏拟谓词逻辑中。因此，我们证明了部分准谓词的逻辑是不可判定的——更准确地说，$\varSigma ^0_1$-complete——在任意结构上，而不是递归可枚举——更准确地说，$\varPi ^0_1$-complete——在有限的结构。