Abstract: 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.

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