Abstract:  
In various theories of truth, people have set forth many definitions to clarify in what sense a set of sentences is paradoxical. But what, exactly, is a paradox per se? It has not yet been realized that there is a gap between ‘being paradoxical’ and ‘being a paradox’. In this talk, I first explain that a paradox is a minimally paradoxical set meeting some closure property. Then, we start from a folk notion of paradoxicality implied in Tarski’s undefinability theorem of truth and give five tentative definitions of `being a paradox'. We require that a definition of ‘being a paradox’ is acceptable if its extension must be sufficiently comprehensive to include all known paradoxes on the one hand, and its standard must also be high enough to exclude those that are evidently not paradoxes on the other hand. It turns out that only the last attempt is acceptable: a set of sentences is a paradox if it is paradoxical and self-dependent, and at the same time, it and any of its paradoxical and self-dependent subsets bisimulate each other. As for the other attempts, even none of them meets the requirement that we propose, they can pass the test of some comprehensive classes of paradoxes. It means that although these attempts are unsatisfactory, they still have positive supports.