Yesterday I provided a preliminary formulation of a key lemma in Meillassoux's argument that correlationism absolutizes. I'd wanted to do this in a full natural deduction form today, but my thinking got both ahead and behind the argument. Let me get behind the argument first.
I. Correlationism, and the contrast between Meillassoux and Harman
From the position of an analytic philosopher (and I pray that this is not too distorting), Meillassoux's correlationism is best presented in terms of the following three positions.
(1) Verificationism- We cannot coherently think of reality as unthought (from the British empiricists originally, though Berkeley actually argued for it). Note that this arguably entails that if P is true, then it is possible for someone to know that P is true, but that in itself it places no restriction upon who is doing the knowing, it could be "knowable by an infinite mind." Only arguments concerning finitude force the verificationism to be knowable by something human-like.
(2) Embodiment/Embeddedness-We cannot coherently think of humans without thinking of them as embedded in a reality ( Schopenhauer and then later Heidegger developing Kant's claim that concepts without intuitions are empty, Schopenhauer with respect to the body and Heidegger with respect to a reality experienced as in some sense pre-existing, modal (involving possibilities), and valuative).
(3) Finitude- We cannot coherently think of self-subsistent totalities/absolutes (from Kant’s dialectic, but Graham Priest has discovered the true nature of this argument).
Before I go any further I must make absolutely clear the contrast between Harman and Meillassoux. Harman rejects (1) the Verificationism and maintains (3) the Finitude (if I could write an aria I'd write one in praise of this insight!) by radically externalizing the manner in which Finitude is expressed by Heidegger. Meillassoux rejects (3) the Finitude while keeping the (1) Verificationism. This is in some sense the titanic strugle at the heart of Speculative Realism.
II. Meilllassoux Needs (some form of) Verificationism
As I briefly (much more is needed exegetically) argued yesterday, the way Meillassoux gets to his rejection of Finitude actually uses Verificationism as a premise.
III. Meillassoux Rejects Verificationism
Here is a big problem. As far as I can make out, Meillassoux's argument that correlationism need not entail Berkeleyan Idealism is inconsistent with the very Verificationism he uses. Meillassoux's worry is that correlationism renders the thing in itself unthinkable/unconceivable, but then we might think that it is impossible, which is the position of Berkeleyan idealism.
So Meillassoux argues, persuasively to me (and this actually has powerful resonances with Lovecraft that are in common to all of the first generation Speculative Realists, and many of the second generation ones such as myself), that unthinkability does not entail impossibility.
But I'm not sure he can argue this. First, notice that Meillassoux is arguing against a strawman. To stop Berkeleyan Idealism, he must argue against the proposition that unthinkability does not entail falsity. For the Berkeleyan Idealist need only be committed to the claim that it is false that things in themselves exist, not that it is impossible that they do so. But the proposition that unthinkability does not entail falsity is much harder to argue against than the proposition that unthinkability does not entail impossibility. In fact, it is not a proposition that I think Meillassoux can argue against (though Harman and myself, following him, can).
No Verificationist of the sort we've been considering can argue against the calim that unthinkability entails falsity! For surely unthinkability entails unknowability. But this claim, plus Verificationism, is provably inconsistent with the claim that unthinkability does not entail falsity! Let me explain the formalism before giving the proof.