Tarski gave three sufficient conditions in his 1944 paper The Semantic Conception of Truth for the Liar paradox to occur:
- The language in which the Liar sentence is stated in is semantically closed, meaning that it has a truth predicate True(x) such that it can refer to its own semantics (where x is the name of the true sentence), and it can refer to its own expressions - for example, a sentence P has the name "P".
- Laws of logic hold, including the law of excluded middle (LEM).
- Self-reference in this language is possible, e.g. by denoting a sentence with a name
But then Tarski claims that condition 3 is actually not needed, as there are ways to deduce a contradiction from the Liar with just the first 2. (I don't quite get the example Tarski gave, but I take it that something like the following also work: 'Alan: What Boris said is true.', 'Boris: What Alan said is false.')
My questions are as follows:
a) Did I understand condition 3 correctly? (i.e. Cond 3 is indeed about self reference being possible?)
b) If I understand cond 3 correctly, isn't it implied by cond 1? If a language has names for its own expressions, surely that is already sufficient for expressions to refer to themselves?