We focus on the persistence principle over weak interpretability logic. Our object of study is the logic obtained by adding the persistence principle to weak interpretability logic from several perspectives. Firstly, we prove that this logic enjoys a weak version of the fixed point property. Secondly, we introduce a system of sequent calculus and prove the cut-elimination theorem for it. As a consequence, we prove that the logic enjoys the Craig interpolation property. Thirdly, we show that the logic is the natural basis of a generalization of simplified Veltman semantics, and prove that it has the finite frame property with respect to that semantics. Finally, we prove that it is sound and complete with respect to some appropriate arithmetical semantics.

中文翻译：

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弱可解释性逻辑的持久性原则

我们关注持久性原则而不是弱可解释性逻辑。我们的研究对象是从多个角度将持久性原则添加到弱可解释性逻辑中得到的逻辑。首先，我们证明该逻辑具有弱版本的定点属性。其次，我们介绍了一个序列微积分系统并证明了它的割消除定理。因此，我们证明该逻辑具有克雷格插值性质。第三，我们证明该逻辑是简化 Veltman 语义推广的自然基础，并证明它具有相对于该语义的有限框架属性。最后，我们证明它在某些适当的算术语义方面是合理且完整的。