Paradox and Truth

Sometimes language trips over itself. We need to be careful, not redesign everything.

In The Death of Logical Certainty, I covered a few logical and mathematical results described that were all inspired by a very old puzzle called the Liar Paradox. It has been discussed and debated since the Ancient Greeks (although quite possibly before then as well). The Liar Paradox is based on simple statements that, in the right context, lead to contradiction and paradox. Importantly, there is nothing intrinsically difficult or problematic about these statements. They can arise in fairly ordinary situations.

For example, there is a regularly cited example from the Bible. In the New Testament book of Titus, one of ‘Crete’s own prophets’ is quoted as saying that "Cretans are always liars". If we take the logic of this seriously (which wasn't the intent) we are left with a puzzle. If Cretans are always liars, then the Cretan himself must be a liar. Was he lying when he said that Cretans are always liars? On the other hand, if he was telling the truth, then he himself is an example of a Cretan who isn't a liar. This undermines his claim that Cretans are always liars and there calls into question whether he was actually telling the truth.

This example isn't logically watertight,¹ but it is not difficult to create cases that are robust. Let's consider the following.

1. The sentence in this Substack post listed under point 1 is not true.

Obviously, this sentence refers to itself. If we ask the question of whether it is true or not, we run into problems. If (1) is true, then what it says must be the case, which is that (1) is not true. So (1) cannot be true. However, if it is not true, then what it says is correct, which surely means that (1) is true after all. So if it is either true or not true, then it has to be both true and not true - a contradiction. Sentences which have this property are known in the literature as Liar Sentences.

This may just seem like an example of a tricksy, but inconsequential, game with language. However, it we take it seriously, the consequences are troubling. Within almost all approaches to logic, if we have a true contradiction, then we can prove anything and everything. In that case, language and knowledge become meaningless because we can prove that every possible assertion someone can make is true.

So is human language in fact meaningless? Our daily experience tells us that it isn't, yet there is no philosophical or logical consensus on how to resolve this paradox. So we are in the odd situation that we have a very old paradox that no-one takes seriously in any practical sense, yet there is no accepted way of solving it. And this isn't due to any lack of effort or neglect as there has been a long stream of publications on the topic.

Resolving the Liar Paradox

To understand why this has been so hard to resolve, let's return to the contradiction that arose when we looked at sentence (1). In normal mathematical and logical reasoning, a contradiction (like we derived) is a sign that the assumptions we have made are wrong: classic strategies for proof include reductio ad absurdum and proof by contradiction. So what are the assumptions we have made in this argument?

This is where the difficulties become clearer. The assumptions necessary for this argument simply are that (i) what it means for a sentence to be true is that what it says is the case or correct; (ii) a couple of simple and uncontroversial steps of logical reasoning; and (iii) the sentence itself is a legitimately constructed sentence. None of these are obviously open to challenge.

While sentence (1) is a bit odd, there is nothing inherently wrong in it. Moreover, several authors have shown how it is fairly easy to find examples where sentences that we often use turn out to be Liar Sentences in odd situations and contexts.² There is nothing problematic about the sentence construction. As soon as a language includes a method for referring to other sentences (or what people say) and some way of asserting something is or isn't true, then sentences like these will always arise.³

Approaches to resolving the paradox therefore have to either focus on refining what it means for something to be true or arguing that our normal systems of logic are flawed. Both approaches have been advocated, including some who argue we need to accept there can be true contradictions. If anyone here is interested in reading about the range of different proposals, the Stanford Encyclopedia of Philosophy entry is a good place to start. Here I want to focus on one promising approach (with some variations) to understanding what is going wrong and what that means for how we understand truth and knowledge.⁴

Let's focus on what it means for a sentence to be true. If a sentence is true, what is says needs to be correct, but it also has to say something meaningful about what the world is like. For example, consider the following sentence: The recently elected President of the United Kingdom is an inspiring leader. This sentence fails to be true, not because the President is uninspiring, but because there is no such person. The sentence fails to be true, not because the substantive claim in the sentence is incorrect, but because it fails to say anything meaningful about the world.⁵

This suggests that there is a second condition on what it means for a sentence to be true that isn't acknowledged in assumption (i) above: for a sentence to be true, it has to both say something meaningful about the world and be correct in what it says. So does sentence (1) say something meaningful about the world?

At first glance, it is making a meaningful assertion about the truth of a particular sentence. However, can a sentence make a meaningful assertion about its own truth or falsity? There is a coherent argument that, due to the patterns of reference involved, the assertion made in (1) never reaches out to the world to say something meaningful about it; it only goes round in circles. This circularity also arises in the Cretan example.

So it would make sense that these Liar Sentences fail to meet the precondition on a sentence being true: they do not say anything meaningful about the world by themselves. This insight, if correct, resolves the Paradox because the paradoxical arguments above never take off. Any Liar Sentence fails to be true because it doesn't say anything meaningful, not because what it says about the world isn't the case.⁶

There is one somewhat counter-intuitive consequence of this explanation - a sentence saying exactly the same thing as a Liar Sentence but in a different linguistic context can be true. We have just argued that sentence (1) fails to be true. This presumably means that the sentence in this Substack post listed under point 1 is not true. In other words, while sentence (1) fails to be true because of its context, saying the same thing as (1) in a different context can be true. This may seem counter-intuitive but lines up with our daily experience that context matters crucially for meaning and truth in language.

Language tripping over itself

On this account, the Liar Paradox doesn't reveal anything deep about the nature of logic and truth. Instead it points to unavoidable limitations on our use of language. Liar Sentences arise when perfectly acceptable grammatical constructions fail to reach out beyond the language and do not make any meaningful assertions about how the world is. This arises because the representative machinery we build into languages to refer to other sentences (and enable us express ourselves well) sometimes trips over itself and leads to odd consequences. By allowing language to refer to itself, we end up with circular chains of reference (including self-reference). This means that language cannot map directly onto, or perfectly represent, the world as it is.

One point in favour of this explanation is that it reflects our practical intuitions. We normally read something like the Liar Paradox and treat it like an amusing exception. We wouldn't imagine that it might require us to master transfinite mathematical induction to understand the concept of truth or completely redesign our systems of logic. This explanation agrees with our intuitions.

A Liar Sentence is simply an exception that arises mechanically from some of the machinery of language that we need. But it doesn't fundamentally change anything we do with language - except that we need to pay attention. Notably, those who spend their time building and writing precise languages - i.e. those writing computer code - are incredibly careful about these types of issues. Being careless about reference quickly leads to infinite loops and broken programs.

This is another reminder that all representations of the world are necessarily different from the world they are describing. I have argued elsewhere that our representations, whether they be theories, narratives or models, are always partial and incomplete, but the Liar Paradox adds another factor. Our representations of the world can create internal artefacts due to their construction that are different from how the world is. Given all human knowledge is captured in representations of the world, whether via languages, theories, stories, pictures, or otherwise, we need to remember that the representation can never faithfully match the world. In other words, knowledge is hard and we need to be humble about it.

1. In this case, the paradox resolves once we accept that someone doesn't have to lie all the time to be a liar or note that lying involves an intention to deceive rather than just saying something false.

2. My personal favourite is in chapter 51 of Don Quixote: “Señor, a large river separated two districts of one and the same lordship—will your worship please to pay attention, for the case is an important and a rather knotty one? Well then, on this river there was a bridge, and at one end of it a gallows, and a sort of tribunal, where four judges commonly sat to administer the law which the lord of river, bridge and the lordship had enacted, and which was to this effect, ‘If anyone crosses by this bridge from one side to the other he shall declare on oath where he is going to and with what object; and if he swears truly, he shall be allowed to pass, but if falsely, he shall be put to death for it by hanging on the gallows erected there, without any remission.’ Though the law and its severe penalty were known, many persons crossed, but in their declarations it was easy to see at once they were telling the truth, and the judges let them pass free. It happened, however, that one man, when they came to take his declaration, swore and said that by the oath he took he was going to die upon that gallows that stood there, and nothing else.”

3. For those who have come across some of these ideas before, this is the insight behind diagonalisation arguments and constructions.

4. This explanation is deliberately non-technical. Get in touch via comments or replying to this Substack as an email if you want more technical details.

5. One paper using this broad approach explains this dynamic as a failure of a sentence to properly target the World, or of ‘immanence’.

6. This approach is definable within formal logic, with some extra logical machinery. For those interested, it is semantically complete. See https://doi.org/10.25911/5d78dcbebe66c or reach out for more details.