My apologies for the weeks between this and the last post. I'm back in Miami, slugging through my reading list, and I should be posting at a much more frequent clip from now on.
Meanwhile, I have a thought (not really a full thought, but at least the beginning of one), not about dialetheism per se but about what Graham Priest calls the "second grade of paraconsistent involvement."
Just for future reference, his "grades" are:
1st: "Gentle-strength paraconsistency" (you reject the principle that anything follows from a contradiction)
2nd: "Full-strength paraconsistency" (you think there some inconsistent but interesting, non-trivial theories)
3rd: "Industrial-strength paraconsistency" (some of those theories may be true)
4th: "Dialetheism" (some of those theories *are* true)
Standard apologetics for the usefulness of paraconsistent logic often include historical examples of inconsistent but non-silly theories. It generally goes something like this. "Impressive Scientist X believed P and he also believed Theory Q, and he knew that Theory Q entails that not-P, but he didn't derive just any claim R, so the underlying logic he was using was clearly not classical."
Now, on the face of it the only sort of explosive logical rule that this situation would be any kind of challenge to would be a claim in the language of epistemic logic that Bxp & Bx~p entailed Bxq for any q. [Of course, (Bxp & Bx~p) is not a contradiction. Only (Bxp & ~Bxp) would be a contradiction.]
Sadly, this has nothing to do with what the historical examples are there for. The historical claim here is not that according to any sort of classical logic, it should follow from Scientist X believing P and ~P that Scientist X will believe that Q for any Q. Rather, the point is that if Scientist X believes that P and he believes that ~P, then classical logic would give him permission to draw the conclusion that Q. The fact that he never exercised this privilege is then seen as evidence that he was (albeit unconsciously, when we're talking about figures who predated the development of paraconsistent logic) operating according to paraconsistent rules of inference in which there are strict limits on what you can derive from a contradiction.
Now, I have severe doubts about the very idea that either scientific practices (or, worse yet, as is sometimes claimed, natural languages) have "underlying logics," but for the moment I'm going to put that to one side.
Instead, let's go for a simple analogy.
Einstein believed in the Special Theory of Relativity.
Einstein never drew the conclusion that "if it is not the case that it is not the case that it is not the case that it is not the case that it is not the case that the Special Theory of Relativity is true, then it is not the case that it is not the case that it is not the case that it is not the case either that Hitler won World War II or that the national flag of Australia is a pirate flag containing a jewel, that, under magnification, is revealed to be a detailed map of the surface of Mars with an X marking the spot where Elvis has high tea every afternoon at four with the Mad Hatter and a ghostly semi-physical representative of the Hegelian world-spirit."
According to classical logic, the entire quoted claim is entailed by the truth of the Special Theory of Relativity. (Unless, of course, I slipped up while counting the number of negations.) If the STR is true then it follows that if you put an odd number of negation signs in front of the proposition that it is true, you have a false claim, and any conditional statement with a false antecedent is true, regardless of whether the consequent is true or false.
...so Einstein never drew that conclusion, or any of the infinite number of other similar conclusions classical logic would have given him permission to draw. Does it therefore follow that his "underlying logic" must have been some alternative non-classical logic, where strict rules are in place to reign in the sorts of consequents that can be put on these conditionals?