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The Liar Paradox

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The paradoxicality of certain sentences involving liars has been known for a long time, and in Western philosophy it dates back to the Ancient Greek period. In this article we will concentrate on the formulation of the paradox and some of its implications for the concept of truth, rather than its solutions.

The Paradox
Perhaps the earliest written formulation of the paradox is that of Eubulides of Miletus, who is well-known for having come up with several logical paradoxes. If a person says that she is a liar, is what she’s saying true or false? Suppose it’s true: then the person is not a liar; but then it’s actually false that she is a liar. If instead what the person is saying is false, then she is a liar; but then what she is saying is true. Either way, we reach conclusions that seem – literally – incredible; that’s the mark of paradoxicality.

Lying is, in general, a complex matter. How to tell a lie? That is, under which circumstances is a sentence a lie? Does the entire cultural apparatus of a society, that system of rituals, ceremonies, manners, and styles through which we organize our life, rest on a lie?

That a concept so central for the ethical and civil foundations of a society brings about also some of the most discussed logical paradoxes is too often underestimated: the fascination of lying is precisely given by the combinations of those two factors, its centrality and its elusiveness.

Variants of the Paradox
Let’s see some more examples of the liar paradox. The sentence you are reading is false. Is the sentence you just read true? If it is, then it is false, as that’s not what it says; if instead the sentence is false, then it is true, because that’s what it says.

Or, consider this other case made up of two sentences. The next sentence is true. The previous sentence is false. Suppose that the first sentence is true; then the second sentence is also true; but this implies that the first sentence is false, contrary to the initial assumption. Suppose instead that the first sentence is false; then the second sentence is also false; but this implies that the first sentence is true, contrary to the initial assumption.

The previous version of the paradox employed two sentences: scholars have discussed also "circles" of sentences that are infinite.

Truth and the Liar
The key ingredients of a liar paradox are a truth predicate and a principle of truth. In classical logic, the truth predicate is a function that is either satisfied or it’s not. That is, we have two truth values, true and false: each and every sentence has to be assigned one or the other, and cannot be assigned both or none.

It is furthermore important how the truth predicate is applied to evaluate a sentence: typically, logicians have endorsed a principle that was captured by Alfred Tarski during the last century, which goes under the name of disquotational principle. According to it, the sentence "Napoleon was a French citizen" is true just when, indeed, Napoleon was a French citizen. That is, we ought to accept as true only those sentences that we would be willing to assert.

Now, the liar paradox pressures us to rethink the thesis that every sentence is either true or false, or how the truth predicate is applied to evaluate sentences, or both of those aspects of our conception of truth. This is why it is a central problem to think about in understanding truth and lying.

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