You might believe that this article has been published in Think, but do not be too sure of that! I am going to show that the following sentence is false:
(A) This article has been published in Think.
Consider the following auxiliary statement:
(S) Either this sentence is not true, or no article with the title ‘A proof that this article has not been published in Think’ has been published.
S is a disjunction (or-sentence) with two disjuncts. At the time of writing its second disjunct was true. Therefore, like all disjunctions with at least one true disjunct, S was true.
Let us now consider the truth value of S at the time when you are reading this. Suppose that S is not true. As we have already noted, all disjunctions with at least one true disjunct are true, so it follows from the supposed non-truth of S that its first disjunct is not true. In other words, it is not true that S is not true. It follows from this contradiction that S is true.
Since S is true, its first disjunct is not true. Since all true disjunctions have at least one true disjunct, its second disjunct is true. From this we can conclude that A is false. In other words: this article has not been published in Think.
But do not worry. I can show with a similar proof that you have not read it either.
‘I am going to show that the following sentence is false: … This article has been published in Think.’
PS
It is known since antiquity that self-reference can give rise to paradoxes. The liar paradox is attributed to the Cretan philosopher Epimenides, who is reputed to have said that all Cretans are liars. One variant of the liar paradox is the simple sentence ‘This sentence is false.’ If it is true, then it must be false. But if it is false, then it must be true.
Curry sentences, named after the American logician Haskell Curry (1900–82), are even stranger creatures. They are if-sentences that refer both to themselves and to some other statement, for instance ‘if this sentence is true, then the moon is made of cheese’. Let A be the sentence If A then B. If we assume A, then we also have If A then B, so under this assumption we can conclude that B. Thus, B follows from A. Therefore, If A then B is true, and so is the equivalent sentence A. Since both A and If A then B are true, so is B. In this way we can prove any sentence B, including the sentence that the moon is made of cheese. Our construction with an or-sentence does the same, arguably in a somewhat more convenient way.
‘One variant of the liar paradox is the simple sentence “This sentence is false.” If it is true, then it must be false. But if it is false, then it must be true.’
