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.

Abstract We associate an elementary cellular automaton with a set of self-referential sentences, whose revision process is exactly the evolution process of that automaton. A simple but useful result of this connection is that a set of self-referential sentences is paradoxical, iff (the evolution process for) the cellular automaton in question has no fixed points. We sort out several distinct kinds of paradoxes by the existence and features of the fixed points of their corresponding automata. They are finite homogeneous paradoxes and infinite homogeneous paradoxes. In some weaker sense, we will also introduce no-no-sort paradoxes and virtual paradoxes. The introduction of these paradoxes, in turn, leads to a new classification of the cellular automata.