Bzzt, wrong, thank you for playing.

A schema in which all statements have truth values has this sort of algebra:

T V T = T

T V F = T

F V T = T

F V F = F

Now let N = the truth value "has no truth value". Let A be "this statement is false" and let B be "this statement has no truth value". So jliszka is looking to establish the truth value of A V B. We now have nine possibilities (T V T, T V F, T V N, N V F, etc) for the combinations of A and B, of which only four are covered in the truth table above.

And how do we decide the values of the others? Hmm...

Let C = (A V B). Suppose (A V B) has no truth value. Then A says C is false, and B says C hasnotruthvalue (let's make that a single predicate). So by assumption "C is false" isn't true: we *can't* assign the truth value "false" to C. And so...? And so C is either T or N. T would be a contradiction, but N is still consistent.

That was considering the A branch: we seem to have established that value(C) = N is the only consistent interpretation. Fine. Now let's look at the B disjunct. This says "C hasnotruthvalue". This is either T, F, or N. Well T is fine. T is consistent with what we know about A. F is bad, because it contradicts what we've just learnt about A. What about the third possibility, value(B) = N? This is claiming "'This sentence has no truth value' has no truth value" -- err, at this stage I can't make sense of that, so let's say okay, N isn't ruled out.

So A has to be N. B can be T or, possibly, N. Therefore the compound claim C = (A V B) has two possible truth values, either N V T or N V N. And those algebraically work out to...?

The fallacy is this. "X hasnotruthvalue" is true if X hasnotruthvalue. But "X hasnotruthvalue OR Y something" cannot imply anything straightforward, because the boolean algebra of OR only applies to true and false. It doesn't apply to values that are by definition outside the rules of the game. You can't invoke meta-rules which say "oh, by the way, I'm going to keep these in play for as long as it suits me, even though I agreed to exclude them".