I've had a couple of fantastic discussions with Levi Bryant about Graham Priest's connection to a slew of continental philosophers that Priest doesn't even mention in Beyond the Limits of Thought, and a bunch of weird things are popping out with respect to my current reading of Meillassoux.
The most important connection is that Meillassoux several times makes the kind of argument Priest diagnoses and logically regiments, where it is shown that just articulating a limit (Closure) forces that very limit to be contradicted (Transcendence). Given my teaching rotation, I should be able to teach a class on Meillassoux and Priest two Spring semesters from now, and the course notes for that class might be the genesis of something collaborative with Bryant (Insha'allah).
Weirdly, in Chapter 3 of After Finitude. Meillassoux does have two discussions of the possibility of true contradictions, first in arguing that a contradictory being can't exist, since it would have all properties it would nto be contingent in the way Meillassoux has argued that objects are, and second in admitting that his argument uses ex falso quodlibet, something not valid in paraconsistent logics. It's pretty clear Meillassoux hadn't read Priest at the time he wrote After Finitude, because much of Priest's work can be read as picking up the challenge Meillassoux lays down in his discussion of paraconsistency.
Tonight I initially thought that I could prove that Meillassoux actually may need it to be the case that truths that escape the correlationist circle are all true contradictions, but as I dug deeper into the modalities, I actually came up with even more formal logic derivations of Meillassoux's key arguments. At the end of this post though, I do show how actual dialetheist worries arise, but then raise a couple of issues concernin whether the resulting form of dialetheism may not be a problem for Meillassoux.
In THIS POST I showed that Harman's interpretation of a central Meillassouxian argument was provable in modal logic.
The key lemma is that Verificationism (Strong Correlationism), the position that all truths are knowable (P --> <>KP), entails that if it is impossible to know that something is impossible, then that fact is possible (~<>K~<>R --> <>R). This latter statement forms the core of Harman's reconstruction of Meillassoux's argument to absolutize facticity.
The key point is that the for the correlationist it is not possible to know that truths describing things outside the correlationist circle are impossible. But then it follows (from Verificationism) that they are in fact possible. This leads to a very wide notion of contingency since so many truths are for the correlationist such that we can neither know that they nor their negations are impossible (so both they and their negations are impossible).
This too is formalizable and valid. For Meillassoux's outside-the-correlate sentences, for correlationists the following is true (~<>K~<>R) and (~<>K~<>~R). By the contrapositive to Verificationism (~<>K~<>R) entails ~~<>R and (~<>K~<>~R) entails ~~<>~R. If we allow ourselves to excluded middle, this is exactly Meillassoux's conclusion that R could be true and could be false <>R and <>~R. So if correlationism really does entail that outside the correlate sentences are such that it is impossible to know whether they are impossible, then it follows as a matter of pure logic that R is contingent!
Now here's a weird logical fact. On pain of contradiction a Verificationist cannot affirm that a sentence and its negation are both impossible to know. If P entails that it is possible to know that P, then it being impossible to know that P entails that P is false. So if I say that some sentence and its negation are impossible to know, I get that that very sentence is both false, and not false, a contradiction.
So while Meillassoux must affirm of anti correlate sentences (~<>K~<>R) and (~<>K~<>~R), he cannot (on pain of dialtheism) affirm of any sentence (~<>K~<>R) and (~<>K<>R). The contrapositive of Verificationism would then entail (~~<>R) and (~<>R). But of course Meillassoux's whole conclusion is that it *is* possible to know that R (and its negation) is possible!
Again, since Meillassoux holds that for all anti-correlate sentences we can't know that they are impossible, to avoid dialetheism, he must hold for all anti-correlate sentences ~~<>K<>R, and by classical logic <>K<>R. But this is again of a piece with his demonstration of contingency everywhere.
The possible problem is this. Forgetting about the being able to know the impossiblity or possibility of anti-correlate sentences, Meillassoux's discussion strongly suggests that anti-correlate sentences are such that both they and their negations are impossible to know. We cannot know that we will live for ever and we cannot know that we will die.
But if anti-correlate sentences (such as the one affirming our death) are such that both they and their negations are impossible to know, Meillassoux must affirm by the contrapositive of Verificationism, that anti-correlate sentences are both true and false.
I actually don't think this is a problem, precisely because if you have a stronger form of verificationism (for example all truths are knowable by a non-existent God), then sentences about my death are not anti-correlate sentences. So only a few really odd metaphysical sentences such that neither they nor their negations are knowable by the God we hope to come would end up being such that they are both true and false.
And maybe Priest has made a compelling case that such sentences should be considered to be true contradictions anyhow.
However, there is a bigger problem rising. I worry that this kind of limiting by appeal to a non-existent God's knowledge undermine's Meillassoux's case for radical contingency. If we only apply the argument to the kind of sentences that a non-existent God could not know, then there will not be many sentences that we can be sure are such that they and their negations are false. Moreover, these might be the same class of sentences that we can be sure are both true and false.
There is an immense amount of argument to go through with this in mind (Chapter 3 of After Finitude is one of the most densely argued pieces of good philosophy that I've read), so I'm just flagging it for now so that I can think about it over the next few years.