I’ve long been interested in whether my truer apparatus can help us make progress with the semantic paradoxes. Here’s the kind of thing I have in mind. I’ll do this all with concrete truth values, but ideally it would be possible to replace them all with comparatives. (I don’t know if that really will be possible in this case, but it would be ideal.)

To start, assume the truth values are the subsets of [0,1), with [0,1) representing perfect truth, and the null set perfect falsity. Truer than means has a truth value that is a proper superset of the truth value of, and conjunction, disjunction, negation and the material conditional are defined in the obvious ways. Let f be a function on numbers that ‘rotates’ them around the circle from 0 to 1 by 180 degrees. Less pictorially, if x <0.5, f(x) = x + 0.5, otherwise f(x) = x – 0.5. Now the key definition.

V(S is true) = {f(x): x is an element of V(S)}

As far as I can tell, this avoids the obvious semantic paradoxes. Of course, it does so by dropping the T-schema, so this isn’t much of a success story. (Oddly the biconditional S is true is true iff S still holds.) But maybe it keeps as much of the T-schema as we could expect given that the paradoxes exist.

A handful of questions about this approach. (I have answers to approximately none of these, which is why this is a blog entry not a journal article.)

Is it plausible that we have to restrict the T-schema as much as is done here?

Is the fact that when S is determinately true, so is S is true, enough to save the intuitions behind the T-schema?

Does it matter than the measure of V(S) always equals the measure of V(S is true)?

Could this still matter if we drop the truth values and just use a comparative truer?

Can this method solve all the problems concerning the strengthened liar? (I think it can, but it requires being more careful about to valuate claims about valuation than I’m prepared to be on a hot afternoon.)

Is there any way to say that for most sentences, even vague sentences, V(S) = V(S’). I don’t really know how to expand the model so sentences like I am tall get a different kind of truth value to the liar sentences, so we could somehow modify the definition of V(S is true) so that in turn that when S itself is not about truth, V(S is true) will come back equal to V(S). This feels like a technical problem that should have a technical solution, but I don’t even really know where to start looking.