Selmer Bringsjord

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

(s0)
for all k > 0, sk is untrue,
(s1)
for all k > 1, sk is untrue,
(s2)
for all k > 2, sk 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:

(*)

But:

(*)

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 ).

Selmer Bringsjord
1999-01-19