The Liar paradox is often informally described in terms of someone uttering the sentence:

I am lying right now.

If we equate lying with merely uttering a falsehood, then this is (roughly speaking) equivalent to a somewhat more formal, more precise version of the paradox that arises by considering a sentence like:

This sentence is false.

If we accuse someone of lying, however, we don’t typically mean that someone merely told a falsehood. For example, if someone tells you that the Earth is hollow because they truly believe that to be the case, we wouldn’t typically call that person a liar. Instead, we would be more likely to accuse them merely of getting things wrong. In short, what seems important about lying is not the falsity of the utterance, but rather the intent to deceive.

Thus, we might adopt the following definition of lying (the subscript is to distinguish this understanding from a second understanding of lying I will introduce below):

S is lying 1 when she utters P if and only if:

P is false. S believes that P is false. S intends the listener to believe that P is true.

Notice that condition (3) is required since there might be all sorts of reasons other than deception for uttering a claim that we believe is true. We might be reasoning hypothetically, or discussing a fiction, or reading a mistaken historical text aloud, or engaging in a multitude of other uses of language. But, for the remainder of this post, let’s restrict our attention to situations where the speaker is making an utterance in order to convince the audience that the utterance in question is true. In such cases, we can simplify our definition to:

S is lying 1 when she utters P if and only if:

P is false. S believes that P is false.

Given this somewhat more sophisticated account of lying, we can now ask the obvious question: is a straightforward utterance of the lying 1 variant of the familiar Liar paradox sentence paradoxical?:

L 1 : I am lying 1 right now.

First, note that L 1 can’t be true: If L 1 were true, then the speaker would have to be lying 1 when uttering L 1 But the speaker can only be lying 1 when uttering L 1 if L 1 is false. So if L 1 is true then L 1 is false. Contradiction. So L 1 can’t be true.

Thus, L 1 must be false. If L 1 is false, then the speaker must not be lying 1 when uttering L 1 . The speaker is lying 1 if and only if L 1 is false and she believes that L 1 is false. But L 1 is false, so the speaker must not believe that L 1 is false.

We arrive at the following interesting conclusion: If the speaker believes that L 1 is false, then any utterance of L 1 by the speaker is paradoxical – it reduces to a variant of more familiar versions of the Liar paradox. If, however, the speaker does not believe that L 1 is false then L 1 is not paradoxical but false. Notice that the non-paradoxicality of L 1 does not require that the speaker get things wrong. She does not have to mistakenly believe that L 1 is true. Instead, she might merely have no opinion about the truth-value of L 1 . Finally, if the speaker herself carries out the first piece of reasoning, and comes to believe that L 1 is false (e.g. if the speaker is taken to be logically omniscient), then L 1 is no longer false, but is instead a genuine paradox. So an assertion of “I am lying right now” – where lying is understood as asserting a falsehood that one believes is a falsehood – is not prima facie paradoxical (although it becomes paradoxical if we assume the utterer believes everything provable).

It is worth noting, however, that, we sometimes accuse someone of lying even if they say something true. For example, if someone truly believes that the Earth is hollow, but tells you that it is not in order to hide the existence of some (mistakenly believed-in) underground society from you, it seems right to say that the person has lied even though what they said is true. In short, lying might not require uttering a falsehood, but might intend be merely an utterance that is intended to deceive. Given this idea, we can consider another, somewhat simpler definition of lying:

S is lying 2 when she utters P if and only if:

S believes that P is false. S intends the listener to believe that P is true.

Condition (3) is required for the same reasons as before, but as before we can restrict our attention to situations where the speaker is making an utterance in order to convince the audience that the utterance in question is true, arriving at the following simplified account:

S is lying 2 when she utters P if and only if:

S believes that P is false.

Again, the obvious question: is a straightforward utterance of the lying 2 variant of the familiar Liar paradox sentence paradoxical?:

L 2 : I am lying 2 right now.

The answer is easy, since L 2 is just equivalent, given our definition of “lying 2 ”, to a familiar puzzle regarding self-reference and belief:

I believe that this sentence is false.

This sentence is not paradoxical regardless of the speaker’s beliefs: if the speaker believes that L 2 is false, then L 2 is true, and the speaker is lying 2 (but not lying 1 ), and if the speaker does not believe that L 2 is false, then L 2 is false, and hence the speaker is not lying 2 (nor is she lying 1 ). In short, any utterance of L 2 is either a case where L 2 is false, or a case where the speaker believes L 2 is false, but not both.

Another way of putting all of this is as follows: when uttering L 2 , the speaker believes they are not lying 2 (i.e. believes the negation of L 2 ) if and only if they are in fact lying 2 . Hence, on the lying 2 understanding, whether or not one is lying is not something one can always know, even though whether one is lying depends solely on what one believes. This is not quite a paradox, but is puzzling nonetheless.

Featured image credit: Smoke, by Carsten Schertzer. CC-BY-2.0 via Flickr.