Temporal liars

One of the most famous, and most widely discussed, paradoxes is the Liar paradox, which arises when we consider the status of the Liar sentence:

This sentence is false.

The Liar sentence is true if and only if it is false, and thus can be neither (unless it can be both).

The variants of the Liar that I want to consider in this instalment arise by taking the implicit temporal aspect of the word “is” in the Liar paradox seriously. In other words, we can understand the Liar sentence as saying of itself that it is true at this very moment. Thus, the Liar is equivalent to:

This sentence is currently, at this very moment false.

But what if we replace the present-tense “is” with future or past-tense verbs such as “will be”, “was”, and the like?

Before considering such constructions, we need to be a bit clear about how we are going to understand various tensed expressions. Informally, if I say “It will always be the case that P”, I might be claiming that P is true right now and will continue to be true at every point in time in the future, or I might merely be claiming that P will will be true at every point in time after the present one, but claiming nothing whatsoever regarding whether or not P is true at the present moment. Here we will assume the latter understanding. More generally:

“P will be true” means that P holds at some point in time after the present moment.

“P will always be true” mean that P holds at every point in time after the present moment.

“P was true” means that P holds at some point in time before the present moment.

“P was always be the case” means that P holds at every point in time before the present moment.

Similar equivalences hold for sentences of the form “P will be false.”, etc. Finally, we will assume that there is no first or last point in time: for any moment, there is at least one moment before that one and at least one moment after that one.

Now, let’s consider the following self-referential sentences about the future:

[1]:       This sentence will be false.

[2]:       This sentence will always be false.

Loosely put, [1] is true at a time if and only if it is false at some later time, and [2] is true at a time if and only if it is false at every later time. Both of these sentences are paradoxical:

Assume (for reductio) that [1] is false at some time t₁. Then it is not the case that [1] will ever be false at any point in time after t₁. So at every point in time after t₁, [1] is true. Let t₂ be any point in time after t₁. Then [1] is true at every point after t₂. So [1] is false at t₂. But t₂ is after t₁, so [1] is also true at t₂. Contradiction. So [1] cannot be false at t₁, and must be true at t₁. Moment t₁ was arbitrary, however. Hence, [1] is true at every moment in time. But then [1] is true at every time after t₁. But then [1] is false at t₁, and so once again we have a contradiction.

Assume (for reductio) that [2] is true at some time t₁. Then [2] will be false at any point in time after t₁. Let t₂ be any point in time after t₁. Then [2] is false at every point after t₂. So [2] is true at t₂. But t₂ is after t₁, so [2] is also false at t₂. Contradiction. So [2] cannot be true at t₁, and must be false at t₁. Moment t₁ was arbitrary, however. Hence, [2] is false at every moment in time. But then [2] is false at every time after t₁. But then [2] is true at t₁, and we have our contradiction.

Similar arguments (obtained by simple modifications of the arguments just given) show that both of:

[3]:        This sentence was false.

[4]:        This sentence was always false.

are paradoxical.

These variants on the Liar paradox are interesting for the following reason: although they look similar to the present-tense Liar, the reasoning to a contradiction does not look like the simple argument typically used to generate a contradiction from the Liar sentence. Instead, the argument looks much more like the reasoning typically used to show that the apparently non-circular but equally paradoxical Yablo paradox:

S₁:        For all n > 1, S_n if false.

S₂:        For all n > 2, S_n is false.

:           :

S_n:        For all n > m, S_n is false.

_Sm+1:     For all n > n+1, S_n is false.

:           :

is paradoxical. None of the temporal Liars [1] through [4] involves an infinitely descending, non-circular sequence of sentences, however, so at first glance it might be puzzling why the argument for a contradiction in each case looks more like the Yablo paradox reasoning and less like the Liar reasoning. After all, these temporal Liar paradoxes seem to be circular in exactly the same manner as is the Liar sentence.

The reason why the temporal Liars are more Yablo-like than we might have initially expected, however, is easy to identity: Although each temporal Liar paradox only involves a single sentence, bringing in (infinitely many) different points in time via the use of tensed verbs means that we must consider the truth-value of this single sentence at all past, or at all future, times rather than merely considering what (single, univocal) truth value it has in the present. The Yablo paradox involves infinitely many distinct sentences, where for each sentence we need to consider what truth-value it might have in the present. The temporal paradoxes involve a single sentence, but in assessing them we need to consider what truth-value is might have at each of infinitely many different points in time.

I’ll conclude by providing the reader with a few additional examples to explore. In particular, note that we can combine more than one temporal operator such as “will be” and “always was” and obtain more complicated temporal Liar sentences such as:

[5]:        It will be the case that this sentence was always false.

[6]:        It always was the case that this sentence will be false.

Of course, lots of other combinations are possible. I’ll leave it to the reader to determine whether some, all, or none of these more complicated constructions are paradoxical.

Featured image: Blue spheres, by Splitshire. CC0 public domain via Pexels.