According to philosophical lore many sentences are self-evident. A self-evident sentence wears its semantic status on its sleeve: a self-evident truth is a true sentence whose truth strikes us immediately, without the need for any argument or evidence, once we understand what the sentence means (and similarly, a self-evident falsehood wears its falsity on its sleeve in a similar manner). Some paradigm examples of self-evident truths, according to those who believe in such things at least, include the law of non-contradiction:

No sentence is both true and false at the same time.

which was championed as self-evidently true by Aristotle, and:

1+1 = 2

Note that if a claim is self-evidently true, then its negation is self-evidently false.

Now, it seems like we have good reasons for the following claim to seem at least initially plausible:

No self-referential statement is self-evident (whether true, false, or otherwise).

One thing that becomes somewhat obvious once we look at self-referential sentences like the Liar paradox:

This sentence is false.

And the self-referential sentences discussed here, here, and here is that determining whether a particular self-referential sentence is true or false (or paradoxical, etc.) usually involves a lot of work, typically in the form of careful and complicated deductive reasoning.

Surprisingly, however, we can show that some self-referential sentences are self-evident. In particular, we will look at a self-referential sentence that is self-evidently false.

Of course, anyone who has read even a single installment in this series will likely guess that some sort of trick is coming up. Thus, in order to highlight exactly what is weird, and what is logically interesting, about the apparently self-evidently false self-referential sentence that we are going to construct, let’s first look at a more well-known self-referential puzzle.

The puzzle in question is the paradox of the knower (also known as the Montague paradox). Consider the following self-referential sentence:

This sentence is known to be false.

Now, we can easily prove that this sentence, which I shall call the ‘Knower’ is false: Assume that the Knower is true. Then what it says must be the case. It says that it is known to be false. So the Knower must be known to be false. But knowledge is what philosophers call factive: for any sentence P, if you know that P is the case, then P must be the case. So, since the Knower is known to be false, then the Knower must be false. But then, the Knower is both true and false. Contradiction. So the Knower is false. QED.

Further, a little trial and error will show that you can’t give a simple proof like the one above to show that the Knower is true. If you assume it is false, all you can conclude is that it is false, but we don’t know that it is false, and that is not contradictory at all.

But wait! Two paragraphs earlier we gave a proof that the Knower is false. Proofs generate knowledge, however: if you read through that paragraph and were paying attention (and I hope you were!) then you know that the Knower is false. So the Knower is known to be false, since you know it to be so. But that’s just what the Knower says. So it is true after all. Now we have a genuine paradox!

Notice, however, that the two pieces of reasoning that we used to generate the paradox – the reasoning used to conclude that the Knower is false, and the reasoning used to conclude that the Knower is true – are of very different types. The first bit of reasoning is just a straightforward deduction about the sentence we are calling the Knower (well, as straightforward as such reasoning about self-referential sentences gets). The second bit of reasoning is different, however: in order to conclude that the Knower is true, we didn’t reason directly about the sentence we are calling the Knower, but instead carried out a second bit of reasoning about the first bit of reasoning.

In other words, we have a proof that plays two roles: First, it shows that the Knower is false, since its conclusion just is that the Knower is false. Second, it shows that the Knower is true, since our recognition of the existence of such a proof is enough to ensure that we have knowledge of the truth of the Knower.

Something like this is also going on in the example to which we now turn: the paradox of self-evidence. Consider the following sentence:

This sentence is false, but not self-evidently false.

Let’s call this sentence the Self-evidencer. Now, we can prove that the Self-evidencer is self-evidently false.

First, we prove that the Self-evidencer is false: assume that the Self-evidencer is true. Then what it says must be the case. It says that it is false, but not self-evidently false. So the Self-evidencer is false, but it is not self-evidently false. But this means that the Self-evidencer is both true and false. Contradiction. So the Self-evidencer is false.

Now, we can prove that it is self-evidently false: given the previous paragraph, we know that the Self-evidencer is false. So what it says must not be the case. The Self-evidencer says that it is false, but not evidently so. So it must not be the case that the Self-evidencer is both false and not self-evidently false. So (by a basic logical law known as the DeMorgan law) either the Self-evidencer is not false, or it is self-evidently false. But the Self-evidencer is false. So it must also be self-evidently false. QED.

Again, like the Knower, there is no obvious contradiction or paradox lurking in the above argument – we have merely proven that the Self-evidencer is self-evidently false, similarly to how we might prove that the following sentence is true:

This sentence is either true or false.

But herein lies the problem. It seems like the only way that we can come to know that the Self-evidencer is self-evidently false is via a complicated bit of reasoning like the one we just gave. It seems unlikely that anyone will think that the falsity of the Self-evidencer is obvious, or forces itself on us, immediately once we understand the sentence.

Thus, we again have a proof that plays two roles. On the one hand, it seems to provide us with knowledge that the Self-evidencer is self-evidently false, since that is its conclusion. On the other hand, however, the fact that we can only come to this knowledge via this rather complicated proof (or some bit of reasoning equivalent to it) seems to be indirect evidence that the Self-evidencer is not self-evident after all. Contradiction.

Featured image: Paradox by Brett Jordan. CC BY 2.0 via Flickr.