We use the truth predicate to replace the proof predicate in the Solovay functions. What we obtain is the Solovay–liar functions, a mixture of the Solovay function and the liar paradox. The Solovay–liar functions are defined on frames. We provide a sufficient and necessary condition of frames for deciding whether a Solovay–liar function leads to a paradox. Besides, we prove that all possible paradoxes generated from the Solovay–liar functions are a weakening of a paradox including the liar paradox in the sense that they have lower degrees of paradoxicality than the latter. Among such paradoxes, some are so radically different from all the known paradoxes that they cannot be characterized by the definitional equivalences in the same way as the known paradoxes are usually characterized. We also study other similar functions obtained by mixing Solovay functions with other paradoxes.

中文翻译：

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Solovay 函数和悖论

我们使用真值谓词来代替 Solovay 函数中的证明谓词。我们得到的是索洛瓦-骗子函数，索洛瓦函数和骗子悖论的混合体。Solovay-liar 函数是在帧上定义的。我们提供了一个充分必要的框架条件来决定 Solovay-liar 函数是否导致悖论。此外，我们证明了由 Solovay-liar 函数产生的所有可能的悖论都是对包括骗子悖论在内的悖论的弱化，因为它们的悖论程度低于后者。在这些悖论中，有些悖论与所有已知的悖论完全不同，以至于它们不能像通常描述的已知悖论那样用定义等价来表征。