Usually, the Liar Paradox can be formed of one sentence, where the paradox arises from self-reference; essentially, the sentence "This sentence is not true" says of itself that it is not true.
Take a sentence D such that D = ~Tr(D), where Tr(x) is our truth predicate.
~Tr(D) says that D is not true, and since D = ~Tr(D), it says of itself that it is not true.
This shows the paradox to be an issue of impredicativity, where the content of the sentence depends on the content of a set of sentences of which that sentences is a member. The very truth conditions for the sentence's being true derive a contradiction:
D is true if and only iff its not true that D; formally: Tr(D) ⟷ ~Tr(D)
Tarski uses this to show that an object language cannot contain a truth-predicate that can be used to talk about the language itself. Either the truth-predicate exists as part of a meta-language above the object and talks about sentences of the object language, or the object language contains a truth-predicate but that truth-predicate can only be used to talk about the truth of sentences in lower-level languages (According to Tarski).
The reason for this is that when a sentence of a given language is uttered, it is assumed that when we utter that sentences, we are also making the equivalent claim that the sentence is true: I.e. to say "snow is white" is also to say "the sentence "snow is white" is true". Terence Parsons (1984) calls this principle [T], where for any sentence, X, expressed in a derivation, we can assume Tr(X). This is how we form the biconditional from earlier: Tr(D) ⟷ ~Tr(D).
When this is done, we can derive a contradiction very easily. Parsons gives the following intuitive derivation:
As this derives a contradiction, if we still want the starting assumption to be assertible, we better deny one of the steps in the derivation. Hence, Tarski looks to remove principle [T], such that no object language can express propositions about the semantic values of its own sentences.
