 Probably the master argument used by modern dialetheists invokes the logical  paradoxes of self-reference.  It is customary to distinguish between two families of such paradoxes: semantic and set-theoretic. The former family typically involves such concepts as truth, denotation, definability, etc; the latter, such notions as membership, cardinality, etc. After Gödel’s and Tarski’s well known formal procedures to obtain non-contextual self-reference in formalized languages, it is difficult to draw a sharp line between the two families, among other things, because of the fact that Tarskian semantics is itself framed in set-theoretic terms. Nevertheless, the distinction is commonly accepted within the relevant literature.