The liar paradox is the oldest in the world: here's what it says

«“I'm lying”, says the liar»: an apparently harmless phrase, but insidious because it is self-denying, that is, it contradicts itself. This paradox is the oldest in history And his name is liar paradox. It even dates back to 4th century BCwhen Eubulides of Miletus he formulated it. It originates from the famous phrase of Epimenides «All Cretans are liars». Eubulides is the father of other paradoxes, such as the paradox of the heap.

The paradox consists in the fact that the sentence reported above it can be neither true nor false: if the liar tells the truth, then he is not lying, but he himself claims to be lying! On the other hand, if it is true that he is lying then he is saying something false, therefore the sentence “I am lying” is false, that is… he is telling the truth!

This paradox strongly puts in logic crisis same e To date, it has no definitive solution. However, they exist different solutions more or less approved by the scientific community – and not only!

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Because the liar paradox is a paradox

The most essential and common form of the liar paradox is concentrated in the sentence

This statement is false.

This sentence cannot be true and cannot be false, because it is true and false at the same time! By claiming that it is true, what it says must be true, namely that the sentence itself is false, just like the liar's sentence!

scheme paradox liar true false

Attempts to solve the liar paradox

Over the centuries, mathematicians, philosophers and logicians have tried to find a key to this paradox, so simple and yet so complex. From Aristotle to St. Thomas Aquinas, from Ockham – famous for his “razor” – a Bertrand Russel, one of the most important logical-mathematical figures of the 20th century and father of the “barber's paradox”, up to Gödel and Tarski. The best minds of our era, and yet, no definitive solution. Let's see some of the most relevant proposed solutions, which allow us to reflect on the nature of logic, philosophy but also linguistics.

The importance of time

One of the oldest solutions is suggested by the philosopher Giovanni Buridano in 1300: a statement can be both true and false, it simply is so at different moments. In the instant t1 I tell the truth by admitting that “I'm lying”, a phrase however that refers to a different moment t2 in which I will speak falsely.

The doctrine of cassation

This solution, always medieval, is based on Metaphysics of Aristotle: whenever someone claims to be lying, the person he is saying something meaningless, such as stating “I don't speak” or “I stay silent”. A contradiction in terms, in short.

The hierarchy of language

This is perhaps the most interesting solution. It was in fact embraced – in different ways – by multiple logicians and philosophers. Already Ockham in the Middle Ages conjectured that a sentence containing the terms True or false it cannot be explanatory of the truthfulness of the sentence itself. It's a sort of language hierarchy.

It was then Bertrand Russel to better theorize this linguistic hierarchy with the so-called branched type theory: the truth or falsity of a proposition of the type n – “I am lying” – can only be discussed in a proposition like this n+1 – “The sentence 'I'm lying' is true.”

The mathematician Alfred Tarski, in 1969, he further refined the concept of hierarchy, distinguishing between object-language And metalanguage: the first is the language that is the object of our discussion (what we talk about), while the second is the language where what “true or false” means is defined and its implications are studied. In simple words: language must be distinguished with which we speak from that Of which is spoken of.

language and meta language true false liar

Some considerations

It must be said that most natural languages ​​are inconsistent if we rely strictly on logic. To say “I'll do it now” for example, it could be inconsistent: at that precise moment I am not doing what I said, but it is true that I will do it. The problem in this case is to establish what does “now” meangiven that the concept of present continually eludes us (when we say it, it's already past!) and so in how many moments will it be “now”?

liar paradox time concept

Coherence is not only a linguistic or philosophical factor, but also a strongly one mathematical: the theorems – considered exact truths – can be contradicted if you change the postulates where we start.

There is no real conclusion to this paradox, but perhaps it is correct that there is not! We can only continue to produce elegant variations on the same fundamental themes, which lead us to play with our language and stimulate logic.

“The book of paradoxes” – Nicholas Falletta University of Florence