Imagine an infinite sequence of sentences each to the effect that every subsequent sentence is untrue:
Formalizing the sentences with a truth predicate, T, we have that for all natural numbers, n, sn is the sentence . Note that each sentence refers to (quantifies over) only sentence later in the sequence. No sentence, therefore, refers to itself, even in an indirect, loop-like, fashion. There seems to be no circularity.
Given this set-up, the argument to contradiction goes as follows. For any n:
Hence, Tsn entails a contradiction, so . But n was arbitrary. Hence , by Universal [Introduction]. In particular, then, , i.e., s0, and so Ts0. Contradiction (since we have already established ).