**Selmer Bringsjord**

Imagine an infinite sequence of sentences each to the effect that every subsequent sentence is untrue:

**(***s*_{0})- for all
*k*> 0,*s*_{k}is untrue, **(***s*_{1})- for all
*k*> 1,*s*_{k}is untrue, **(***s*_{2})- for all
*k*> 2,*s*_{k}is untrue,

Formalizing the sentences with a truth predicate, *T*, we
have that for all natural numbers, *n*, *s*_{n} 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*:

- (*)

But:

- (*)

Hence, *Ts*_{n} entails a contradiction, so .
But
*n* was arbitrary. Hence
,
by Universal
[Introduction]. In particular, then,
,
i.e., *s*_{0}, and so *Ts*_{0}. Contradiction (since we have already
established ).