Dialetheists, since they think some contradictions are true, don't think that, e.g. Disjunctive Syllogism is truth-preserving. After all, if P and ~P can both be true, then we can have a situation where P is true (and as such (P v Q) is true as well), and ~P is true, but Q is (just) false. This has the advantage for them that it gives them a principled reason to reject the inference from contradictions to triviality (which relies on DS), but it has the extremely counter-intuitive effect of forcing them to deny that an intuitively obviously-valid-seeming inferential move used a thousand times a day in ordinary reasoning is valid. To use a shop-worn example, even a dog uses Disjunctive Syllogism after a fashion when following a scent trail to a fork in the road, sniffing one fork, not finding the scent, then immediately bounding down the second without bothering to check it first.
As such, dialetheists generally attempt what's called the "classical re-capture," the idea being that they have some principled reason to continue to reason classically in at least some domains, despite not regarding some of the classical rules they're reasoning with as being technically valid.
In the past, I've mentioned the probabilistic argument for the classical re-capture, which is the main one that Priest uses in In Contradiction. A conversation over lunch at the APA helped clarify for me another formulation of classical re-capture that doesn't rely on any statistical claims, the "where they live" argument.
It goes like this:
Contradictions live in these four houses--semantic phenomena involving self-reference, mathematics (where we have the set-theoretic paradoxes and the dialetheic take on arithemtic incompleteness), instants of change, and the law (where we can both have and not have certain legal rights), etc. Or, actually, these days, for Priest at least, I guess they live in five houses, since he's recently moved to embrace vagueness paradoxes as instances of Inclosure. We haven't discovered contradictions yet in any of the other houses, so we can probably assume we're in a contradiction-free zone when we visit them.
An analogy someone gave me is this:
When driving around the back roads of rural Australia, you need to watch out for kangaroos that hop onto the road and cause messy accidents. When driving around Los Angeles, you don't need to worry about that. It's not logically impossible that a kangaroo could somehow make it to L.A., but it would be silly to alter your driving practices in any special way because of kangaroo danger there, whereas there are bits of Australia where that would be far from irrational.
Similarly, when reasoning in contexts known to contain true contradictions, you need to reason paraconsistently, since you might run into counter-examples to some of the classical laws. When reasoning in contexts not known to contain true contradictions, you don't need to worry about it.
I'm deeply skeptical about this analogy. In the case of kangaroos, after all, we can tell a causal story about why they're present in Australia but they aren't likely to make it to L.A. It's not clear what kind of story about why true contradictions are present in these domains but its not likely that they'd crop up in others we could tell on the Priestian account, given the variety and heterogeneous nature of the 4 to 5 "houses" listed above.
Moreover, they're hard to miss. To combine the analogies for a moment, if there's a kangaroo jumping around your house, you'll notice it in short order. Maybe instead, we should think about cockroaches. Cockroaches like to live in dark, damp, out of the way places, and you can have cockroaches in your house for a long time before you notice one scurrying around in the periphery of your vision. This automatically makes cockroaches a better analogy if there are such things as true contradictions, given that In Contradiction wasn't published until 1982, following a few thousand years of western philosophy pretty much systematically failing to notice the existence of true contradictions.
Now, imagine that three houses are all laid out in similar ways and inhabited by three very similar people with very similar habits and possessions. Imagine that cockroaches are discovered in the house on the left and in the house on the right, and the inhabitants of those houses start taking various defensive measures--being extra-careful about leaving food out where the cockroaches could get at it, applying bug spray to dark corners, etc. Now, if there's some sort of principled causal story about the differences between the houses that makes the house in the middle less likely to have cockroaches--i.e., as in the classic civil defense video, the house in the middle is kept much cleaner than the house on the left or the house on the right--then it might make perfect sense for the owner of the middle house not to bother taking these extra measures. If, on the other hand, we have no such story to tell--the house in the middle is no cleaner than the ones on the left and the right, the owner is no less likely to leave crumbs on the counter, there are no fewer dark and damp spaces, and so on--then the mere fact that we haven't discovered any cockroaches in the middle house yet seems utterly irrelevant to the question of whether we should take such measures.
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