Something occurred to me today and I'm curious to see what people who read this blog think about it.*
(1) Graham Priest has argued (convincingly, from my perspective) in many places that truth-value gaps are impossible. If there is no fact that makes P true, the fact that there is no such fact is sufficient to make it false, and that in any case gaps entail gluts--if a sentence is neither true nor false, then given that falsehood is best understood as truth of negation, by a single instance of the law of double negation, it follows that the sentence is both true and false, so (non-glutty) gaps are incoherent. Of course, there are many places in that extremely rough and condensed summary that a gap theorist could object and defend their position against these considerations, but for the moment, the important point is that it is Priest's position, and, since I don't tend to agree with him about much here, it seems worth noting that we're of the same mind on this.
(2) Priest once wrote a short story called Sylvan's Box in which the plot was explicitly inconsistent. (In it, the late Richard Sylvan (better known as Richard Routley) turns out to have been in a possession of a box that was simultaneously, observably empty and non-empty.) Of course, works of fiction frequently contain inconsistencies (the location of Watson's war wound is a famous stock example), but part of Priest's point in that story was to write something where the inconsistency was unambiguous and explicit, and no one could try to charitably interpret the inconsistency out, or break the story into maximal consistent chunks and look at them individually, or anything of the kind. We have to face up to the fact that there is a flat out contradiction (P & ~P) in the story, and that we are quite capable of (non-trivially) reasoning about it.** Over at Inconsistent Thoughts, Colin put this nicely, by saying that it shows that "we can navigate inconsistent bodies of information without triggering a “psychotic break."***
Now, in so far as this ability-to-maintain-our-sanity is supposed to show that the 'explosion' of inferences derivable from contradictions in classical logic is invalid, and we should instead adopt a weaker, inconsistency-tolerant (paraconsistent) account of the logical consequence relation, I think there's a tension here between that stance and Priest's position on truth-value gaps.
To see what I mean, let's think about a different possible moral you could take from Sylvan's Box (or, for that matter, from Watson's wandering war wound), which is not that P and ~P don't jointly entail any Q, but simply that F(P) and F(~P) don't jointly entail any F(Q). Just as it's false--but true in fiction--that Watson existed, it's true--but false in fiction that explosion is (vacuously) truth-preserving, since, while there are no true contradictions, it is true that works of fiction sometimes portray contradictions as being true. (I've advocated this approach here in the past.) Of course, as a couple of commenters then indicated, there are real questions about the feasibility of developing rigorous formal tools with which to reason about fictional worlds, but let's leave that aside for now, since right now, we're focussed on the narrower project of weighing the respective virtues of the two proposals on the table for how to do so.
Now, at least for anyone who agrees with Priest (and I) about the impossibility of truth-value gaps, I think the following consideration should tilt the balance of things in favor of explosion-fails-in-fiction-but-Sylvan's-box-tells-us-nothing-about-whether-it-really-fails approach:
Fictional worlds are chock-full of (non-glutty) truth-value gaps.
Let's take some obvious examples.
(1) "Sherlock Holmes' brother Mycroft secretly liked to dress up in women's clothing."
(2) "Watson's great-grandfather on his maternal grandmother's side had blue eyes."
(3) "Professor Moriarty once wrote a paper anticipating Russell's Paradox, but he never published it."
Now, since none of these people existed to do any of these things, (1)-(3) are all false, but are they true or false in the world of the stories? It seems to me that, far from our inability to answer these questions being a matter of ignorance, there is absolutely no way that there could be a fact of the matter one way or the other on these issues. The world of these stories is an incomplete one that, far from including information that Mycroft liked to dress in women's clothing *and* that he never did so, fails to include anything on the subject.**** While these issues give us no good reason to doubt [F(P) v ~F(P)] (in each case, I'd say that ~F(P) is true), [F(P) v F(~P)] simply fails to hold. As such, the logical structure of the world of the Holmes stories includes not only (non-trivial) gluts but also (non-glutty) gaps. None of that tells us much of anything about which inferences are valid in the actual case.
*(And yes, I know I promised a follow-up to the post about the metaphysics of change, and yes, I know that it's been a while since I did so, but the thought is fresh and I'm weirdly excited about the point, so this one comes first.)
**(The morals that Priest draws from the story are many and complicated, but for the moment let's leave it at that, since it's at least what many readers take the story to show.)
***(At the time, he asked me a question about Bohr's of the atom to which I never responded, since I forgot about it until now. My punctuality, in general, seems to be sucking this year. In any case, it's the kind of thing that deserves its own post, which I hope to have in the near future.)
****(Part of the reason that the case for the failure of explosion in the world of the Holmes stories is so compelling is that it seems implausible that Watson's inconsistently-placed war wound would entail, e.g. Mycroft's cross-dressing. This appealing feature would be quite lost if we stuck to our guns and started postulating that it was true--and false--in the world because of its incompleteness.)