on the Liar.

Let us assume written and spoken declarative sentences, as opposed to propositions, express propositions.  Let us assume that, in normal circumstances, when I write or speak a declarative sentence I am asserting the truth of the proposition which the sentence expresses. Thus, when I say “the cat is on the mat”, I am affirming that the proposition ‘the cat is on the mat’ is true.  Likewise, when I say “the cat is not on the mat”, I affirming that the proposition ‘the cat is not on the mat’ is true.

With these considerations, let us suppose that up until now the proposition ‘all Cretans are liars’ is true.  Right after this moment, Crete, at Cretan, says “All Cretans are liars”.  What is the proposition which Crete’s speech act expresses?  Which proposition is Crete affirming?  Well, I suppose that which proposition Crete is affirming depends upon what Crete believes.  Since he’s not here to tell us, let us explore the possibilities. Here’s one: Crete believes that

(1) ‘all Cretans are liars, except for me.’

If Crete affirms (1), there’s no paradox.

Here’s another: Crete affirms

(2)  All Cretans are liars.

But Crete cannot affirm (2), for Crete would be affirming that (2) and (2)’s negation are true.  But Crete cannot possibly believe this—it’s psychologically impossible for one to affirm a proposition and its negation at once.  Since Crete cannot believe (2), Crete cannot affirm (2), and if (2) is not affirmed, there is no true contradiction here.