follow up on paradox and totality

Meillassoux deploys a stock Cantorian argument to conclude that there cannot be a totality of possibilities, i.e. a universal set of all possible ways the world could be. This takes place in the context of his attempt to disqualify an updated rendition of the old Kantian rejoinder to Hume (“If cinnabar were now red…”), according to which, on the supposition that contingency is the only necessity, we should expect to see frequent/constant variation of the so-called ‘laws of nature’. In their discussion of this problematic, Joshua Heller and Jon Cogburn suggest that Meillassoux should follow Graham Priest in accepting as true the various contradictions that arise at the limits of thought. They assert moreover that there is no middle path between an accessible paraconsistent totality and consistent plurality without totality. This is the dilemma they construct for Meillassoux, and that I wish to follow him in resisting.

Of course I readily agree that neither wilful ignorance nor complacent dismissal are acceptable responses to the paradoxes of totality. In fact it is basically axiomatic for me that dogmatic quietism is off the table: something significant, something revisionary, must be done. I take it to be true, furthermore, that there are no straight solutions to these paradoxes; only sceptical solutions, solutions that cede something significant to the ‘sceptic’ (i.e. the imagined propagator of the paradoxes). Having said this, there is an important sense in which existing sceptical solutions cede too much. In their eagerness to acknowledge the inexistence of problematic totalities like the universal set, they cede the ability to think in absolutely general terms at all, or to quantify universally and unrestrictedly. This is a mistake: to wed the generality of thought itself to the existence of paradoxical totalities is an unnecessary concession.

Thus my conclusion is that we need to find a place for pluralism somewhere within our response to the paradoxes, but without completely abrogating the viewpoint of absolute generality.

We can call those who deny the ability to think in absolutely general terms generality relativists. This leads to a preliminary puzzle. For how is it that generality relativists manage to coherently express their viewpoint? Their view, taken straightforwardly, seems to require that, for any X, X is not an absolutely general thought. This however involves universal quantification over every X. The following, similar line of thought — taken from an interesting published correspondence between Patrick Grim and Alvin Plantinga — expresses the difficulty here clearly:

Were there a sound Cantorian argument with the conclusion that there can be no universal propositions — so the argument goes — it would require at least one universal proposition as a premise. But if sound, its conclusion would be true, and thus there could be no such proposition. If sound its premises would not all be true, and thus it would not be sound. There can then be no sound Cantorian argument with the conclusion that there can be no universal propositions.

Taken straightforwardly, generality relativism is inexpressible by its own lights. It is for this reason that sophisticated formulations of generality relativism do not straightforwardly deny absolute generality in this way. But then what exactly do they do? As far as I can see, they revert to a case by case treatment of individual concepts of totality, showing how each is paradoxical, but without ever ascending to the general conclusion that there is no such conception free of paradox.

Although we might raise further questions here, I am going to assume that the generality relativist has at least succeeded in formulating her disagreement in a way that constitutes a challenge from the viewpoint of the generality absolutist. After all, and in any case, we should want the latter to actually solve the paradoxes of totality, rather than sanguinely resting her entire case on the referential incoherence of denying totality. For this reason, I think that the dialectical subtleties surrounding generality relativism are a red herring: the generality absolutist should just focus on producing an understanding of totality that isn’t susceptible to refutation on the basis of familiar relativist strategies. At any rate, this is what I propose to do.

The paradoxes of totality are ontologically fecund: rather than locking us up in some cloistered epistemic space, the inexistence of higher totalities allows what is left behind to come clearly into focus; it enables absolute generality rather than foreclosing it — but only on the condition that what we are thus generalising about is conceived in a specific, and surprising, way.

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