I talked about Part I on Monday. Part II just went up. My questions were the last four included. I also contributed the clarification to the next-to-last question, obviously.
There's a lot of interesting stuff here, a good bit of which I haven't really had a chance to digest yet. One thing, however, does jump out at me immediately as a problem:
His answer to my clarification on the next-to-last question would seem to fly in the face of any intuitive understanding of the notion of 'truth-preservation.'
For background, click through to the interview. The question provides a lot of detailed background on this. "ArT" means "A relates to Truth", which is a fancy way of saying "A is True." The idea here, as Priest has explained in other contexts, such as his article What is so bad about contradictions?, is that truth is conceived, not as a function, as classical logicians understand it, but as a relation, such that a proposition can be related to truth, to falsity or to both. In What is so bad about contradictions?, he includes a fourth option--A is related to neither truth nor falsity--but that option would seem to made superfluous by his arguments against the possibility of truth-value gaps in In Contradiction, and in any case the existence or non-existence of the fourth option isn't relevant to this discussion. DS is, of course, Disjunctive Syllogism, the classical inference from ~p and (pvq) to q. Since DS, plus the dialetheist's claim that p and ~p can sometimes both be true, quickly generates triviality, Priest and other dialetheists reject it. Priest's argument is, basically, that it isn't universally truth-preserving (and hence, isn't valid) because, given the assumption that some (but not all) contradictions are true, there can be cases in which ~p is true and (pvq) is true but in which q is not.
As he says in his response to me, "The DS can be show to be invalid is the semantics of LP as follows. (The semantics has many presentations. Let us use the version in which evaluations are relations, R, between formulas and the values t and f.
"Consider the inference ~p, pvq / q. Take an interpretation where pRt, pRf, qRf, and it is not the case that qRt. By the truth and falsity conditions for negation and disjunction, (~p)Rt and (pvq)Rt. Hence there is an evaluation where the premises of the inference relate to t and the conclusion does not. Hence the inference is invalid."
"Note that this argument...[is not] undercut if it turns out that there are formulas, A, such that ARt and it is not the case that ARt - even if you could show by some argument (goodness knows what), that this held when A is the p in question. Deductive reasoning is, after all, monotonic. (Valid arguments are never made invalid by the addition of extra premises.)"
So, why do I think all of this flies in the face of any intuitive notion of 'truth-preservation'?
Well, first of all, it seems to me that Professor Priest is being a bit coy when he speaks neutrally about the possibility of "ArT" and "it is not the case that ArT" being shown to be compatible--"if it turns out that there are formulas, A, such that ARt and it is not the case that ARt..." Given Priest's assumptions, *of course* there are such formulas! After all, we can always construct a sentence A such that A="It is not the case that ArT."
Secondly--and to the point--given that such formulas would seem to have to exist on Priest's account, when he's told us that there's an interpretation on which (~p)rT and (pvq)rT but it's not the case that qrT, he hasn't precluded the possibility that qrT--in other words, he hasn't precluded the possibility that, in this case, as in all other cases, true premises, fed into DS, generate a true conclusion!
Think of it this way--Priest would not claim that the mere existence of a case in which (~p)rT and (pvq)rT but in which qrF constituted a counter-example to DS, right? Given that it's as easy to generate a formula that both does and does not relate to truth as it is to generate one that relates to both truth and falsehood, why should truth-preservation be any more violated by the existence of a case in which q doesn't relate to truth than by a case in which it does relate to falsehood?