One may avoid self reference in liar paradoxes by allowing ourselves infinite propositions with unlimited self-free reference. Consider the following infinite set of propositions:

- P
_{i}is the proposition "*all of P*"._{i+1}, P_{i+2}, ... are false

Now suppose P_{0} were true. Then P_{1}, P_{2}, ... would all be false. But P_{1} asserts that P_{2}, P_{3}, ... are all false, hence we've just shown that P_{1} is true! That's impossible.

So we have disproved P_{0}, i.e. it is false. Clearly, the same argument works for P_{1}, so P_{1} too is false; continuing, each P_{i} is false. In particular, this shows that P_{0} is true (and so are all the other P_{i}'s).

Contradiction either way.