Socrates is supposed to have raised the question:

Is the following statement in italics true or false?

This statement in italics is false.

Suppose the statement in italics is true.

Then it is false.

That is, what it says is different from the way things actually are.

So, the sentence is not false.

Therefore it is true.

But it cannot be both.

We can more formally demonstrate this as follows:

Let $\map T P$ be a modal operator interpreted as "The proposition $P$ is true."

Assume $P \iff \map T P$, or in other words, that declaring the proposition $P$ is equivalent to declaring that $P$ is true.

Let $L$ be a proposition representing the liar sentence, and define $L$ as $\neg \map T L$.

We thus have by definition:

$\neg \map T L \iff L$

and by assumption:

$L \iff \map T L$

By Biconditional is Transitive, we have $\neg \map T L \iff \map T L$, a contradiction.

## Resolution

There are multiple proposed resolutions to this paradox, and there is no consensus as to which one to choose.

One resolution is due to Alfred Tarski's semantic theory of truth.

For any proposition $P$ in a language of truth, this theory requires that:

$P$ is true if and only if $P$.

For instance:

'Snow is white' is true if and only if snow is white.

This is analogous to the principle stated above:

$P \iff \map T P$

Tarski distinguishes object languages from metalanguages.

To talk about the truth of statements in language $A$, we must talk from the perspective of a higher language $B$.

The statement in language $A$ becomes an object of study, and not itself a vehicle with which to make truth assertions about statements in $A$.

Thus, the sentence "Snow is white is true if and only if snow is white" is composed of two languages:

"Snow is white" is an assertion in English

and:

"... is true if and only if snow is white" is an assertion in some metalanguage.

Thus, the paradox's resolution is that the sentence:

"this sentence is not true"

is meaningless, as the language within which the sentence is expressed serves as its own metalanguage.

In other words, the sentence makes an assertion of truth about itself, which is disallowed.

An issue with Tarski's resolution is that it imposes a severe limitation on our ability to coherently use the predicate "... is true".

For instance, consider the statement: "No language can talk about its own truth."

This statement expresses an assertion of truth about all languages, including itself.

This seemingly violates the very principle that Tarski uses to resolve the paradox.