A Liar cycle is a finite sequence of sentences where each sentence in the sequence except the last says that the next sentence is false, and where the final sentence in the sequence says that the first sentence is false. Thus, the 2-Liar cycle (also known as the No-No paradox or the Open Pair) is:

C1 : Sentence C 2 is false.

C2 : Sentence C 1 is false.

And the 3-Liar cycle is:

C1 : Sentence C 2 is false.

C2 : Sentence C 3 is false.

C3 : Sentence C 1 is false.

The Liar paradox itself is just the 1-Liar cyle (where the Liar sentence plays the role of both the first sentence and the last sentence in the sequence of length one):

C1 : Sentence C 1 is false.

We can prove, for any finite number n, that if n is odd then there is no stable assignment of truth and falsity to each sentence – that is, that the sequence is paradoxical, and if n is even then there are exactly two distinct stable assignments of truth and falsity (the trick is noticing that any stable assignment will alternate between true sentences and false sentences).

The closely related Curry paradox arises by considering a conditional statement (an “if… then…” statement) that says that its own truth implies that some completely unrelated sentence holds. Here we are assuming that the conditional in question is what logicians call a material conditional: an “if… then…” statement that is false if and only if the antecedent (the “if…” bit) is true and the consequent (the “then…” bit) is false, and is true otherwise. Here is a typical Curry conditional:

C 1 : If C 1 is true, then Santa Claus exists.

We can use the Curry conditional above, plus straightforward platitudes about truth (i.e. that a sentence is true if and only if what it says is the case) to prove that Santa Claus exists:

Proof: Assume (for reductio ad absurdum) that the Curry conditional is false. Then the antecedent of the Curry conditional is true (and the consequent false). The antecedent of the Curry conditional says that the Curry conditional is true. Since the antecedent is true, what it says must be the case. Hence the Curry conditional is true, contradicting the assumption with which we began.

Thus, the Curry conditional cannot be false, so it must be true. But if the Curry conditional is true, then what it says must be the case. The Curry conditional says that, if the Curry conditional is true, then Santa Claus exists. So if the Curry conditional is true, then Santa Claus exists. But we already established that the Curry Conditional is true. Hence Santa Claus exists. QED.

Interestingly, Curry cycles have not, to my knowledge, been investigated until now. A Curry cycle is a finite sequence of conditionals where each conditional in the sequence except the last says that if the next conditional is true, then some clearly false sentence holds, and where the final conditional in the sequence says that if the first conditional is true, then some clearly false sentence holds. The following is an example of the 2-Curry cycle:

“Either Santa Claus exists, or the Easter Bunny exists, or the Great Pumpkin exists.”

C 1 : If conditional C 2 is true then Santa Claus exists.

C 2 : if conditional C 1 is true then the Easter Bunny exists.

And the following is a 3-Curry cycle:

C 1 : If conditional C 2 is true then Santa Claus exists.

C 2 : If conditional C 3 is true then the Easter Bunny exists.

C 3 : If conditional C 1 is true then the Great Pumpkin exists.

The Curry paradox itself is of course just the 1-Curry.

Now, if n is a finite even number, then (similar to Liar cycles) the n-Curry cycle is not paradoxical (where here a paradox arises if we are forced to accept as true one of the clearly false consequents). In fact, each such cycle has two distinct stable truth value assignments where all the consequents are false (hint: every other conditional is true).

Things get more interesting when we look at Curry cycles of odd length, however. These are paradoxical, but in a certain sense not as paradoxical as one might think. One might guess that the 3-Curry cycle above would allow us to prove that Santa Claus exists, and prove that the Easter Bunny exists, and prove that the Great Pumpkin exists. But we can’t prove any of these. What we can prove, however, is:

Either Santa Claus exists, or the Easter Bunny exists, or the Great Pumpkin exists.

Proof: Assume that the offset claim above is false. So “Santa Claus exists” is false, and “The Easter Bunny exists is false”, and “The Great Pumpkin exists” is false. We will show that this assumption leads to a contradiction (and hence that the offset claim above must be true after all). Now, either the conditional C 1 is true, or it is false.

Case 1: The conditional C 1 is true. The antecedent of conditional C 3 says that conditional C 1 is true, so the antecedent of conditional C 3 is true. Thus, the conditional C 3 has a true antecedent and false consequent, so the conditional C 3 is false. The antecedent of conditional C 2 says that conditional C 3 is true, so the antecedent of conditional C 2 is false. Thus, the conditional C 2 has a false antecedent and false consequent, so the conditional C 2 is true. The antecedent of conditional C 1 says that conditional C 2 is true, so the antecedent of conditional C 1 is true. Thus, the conditional C 1 has a true antecedent and false consequent, so the conditional C 1 is false. This contradicts our initial assumption that C 1 was true.

Case 2: Similar to Case 1, and left to the reader (it helps to draw a little 3 x 3 grid, to keep track of the truth values of antecedents, consequents, and conditionals). QED.

We can’t do better than this, though, and similar results hold for longer odd-length Curry cycles. In short, odd-length Curry cycles are paradoxical in that they entail that some clearly false claim is true, but if the cycle contains three or more conditionals (with three or more distinct consequents) then we can’t tell which of the clearly false claims is the one that, according to the paradox, must be true.

Featured image credit: ‘Squares, circles, and lines, oh my!’ Photo by kennymatic, CC BY 2.0 via Flickr.