I’ve been a away from blogging for the past week, as I’ve been trying to get back to some of the more boring bits of my thesis and get them done. This has only been partially successful, and as such is ongoing (sorry again to those who want me to write more on Deleuze). However, I’ve also been reading After Finitude (finally). I have a number of things I could say about it, and a few issues with the argumentation (some of which Tom over at Grundlegung has tackled). I won’t go into these in detail, in part because I haven’t yet finished the book, but I will point out what appeared to be somewhat of a non-sequitur in one of Meillassoux’s arguments. I might be misinterpreting him, so feel free to put me right, but it seems somewhat blatant to me.
A preliminary point I would make is that Meillassoux identifies metaphysics with onto-theology. A lot of people do this, and I think its a false adequation (as I’ve tried to show here and here). He takes the mainstay of metaphysics to be the positing of a necessary entity. Through a bunch of very interesting argumentation he produces the principle of unreason, which consists in the necessity of contingency, and this disqualifies all such necessary entities (and thus all ‘metaphysics’). The problem I have is his attempt to deduce the principle of non-contradiction from the principle of unreason.
The argument is roughly that if a self-contradictory entity were possible then it would necessarily exist. We can then deduce by reductio ad absurdum that a self-contradictory entity is impossible, as it would be a necessary one. The problem I have is with his demonstration that a self-contradictory entity would necessarily exist. Meillassoux’s argument for this is that a self-contradictory entity could neither change, nor cease to exist, because it is already what it is not. If something cannot cease to be, nor become anything else, then if it exists it does so necessarily.
This itself is a difficult claim to disentangle, because it seems to indicate that existence is property. Ceasing to exist is here treated like a kind of becoming, analogous to changing colour from red to green, except what changes is that one moves from existing to not existing. Existing and not-existing are treated as part of the essence of the contradictory thing, as aspects of what it is. I think this is incredibly problematic. Not only is treating existence as a property a very problematic position (see Kant, Quine, et al), but it also threatens to invalidate the structure of the argument as such. If we accept that self-contradictory entities are essentially existent (as well as essentially non-existent) then it is a short step to saying that they are necessarily existent (the conclusion of the first step of the argument). We can equally move from their essential non-existence to their necessary non-existence (the conclusion of the second step of the argument). If we hold to the assumption that self-contradictory entities are essentially both existent and non-existent then we have simply assumed what we are trying to derive in the first step of the argument. Moreover, I don’t think there is any good reason for us to make this assumption: why is it that self-contradictory entities of their essence both exist and don’t exist?
If we move on from these problems with existence, there are further problems even if we consider the possibility of change. The idea here seems to be that if a self-contradictory entity is both determined in one way, and also determined as not that way, then it has all possible determinations at once, and cannot therefore change from one determination to another. This is equally problematic, if not more so. This is because of the relationship between contradiction and incompatibility.
Incompatibility is a relation that on the one hand holds between claims or propositions (e.g., “My sister is in Wales” and “I am an only child”), and on the other between predicates or properties (e.g., “acidic” and “alkaline”). Brandom has done some very interesting work on incompatibility and negation, showing that the negation of a claim (-P) is what he calls the minimum incompatible, which is to say that it is the claim which is implied by every claim that is incompatible with P. To put it another way, if I claim -P then I am saying that something which is incompatible with P is true, but I am not saying which of the different options is true. If someone says “Your sister is not in Wales”, this does not imply “You are an only child”, although this is one of the conceivable (compatible) options. The conjunction of two incompatible claims automatically entails a contradiction (i.e., if Q entails -P and P&Q then P&-P).
Incompatible predicates only entail contradictions when they are predicated of the same object. If I claim that “My sister is in Wales” and that “My brother is in America” there is no contradiction, but if I claim “My sister is in Wales and she is in America” then I have asserted a contradiction. There are interesting issues here about the scope of negation, but I won’t go into them.
Getting back to Meillassoux, let us assume that our self-contradictory entity is red and not red at the same time. There are three ways of interpreting this, each weaker than the last:-
1) It has the property of ‘being red’, and all other properties that are not the property of ‘being red’ at the same time, including ‘being a car’, ‘being a broadway show’, ‘being anemic’, etc.
2) It has the property of ‘being red’ and all properties that are incompatible with ‘being red’ at the same time, including ‘being green’, ‘being colourless’, ‘being a number’, etc., but not including ‘being a car’, or ‘being anemic’, etc.
3) It has the property of ‘being red’ and at least one property that is incompatible with ‘being red’, such as both being ‘red’ and ‘green’ at the same time.
Only the first of these is strong enough to produce the claim that such an entity cannot change any of its determinations, because it already has all determinations. However, the idea that not being red implies having all other determinations is somewhat silly (I’m a not a doctor, am I therefore also a jam sandwich because jam sandwiches are not doctors?). The next option only guarantees that the entity cannot change from being red to being something incompatible with red, because it is already everything that is incompatible with red. Something which is both red and everything incompatible with being red could also be something compatible with all of those determinations (e.g., it could be large for instance), and those determinations do not prevent it from ceasing to be in that way, or acquiring new determinations which are compatible with them, or both. Nonetheless, this interpretation is still unnatural (I’m not purple, am I therefore also pink, green and colourless?). The final option does not exclude any kind of change whatsoever. If we are willing to allow that something can be both red and green there seems no reason why we should not admit that it could cease being one or both. This is the natural interpretation of ‘not being red’.
The important thing to recognize is that all of these three possible entities are self-contradictory. They all imply contradiction. What Meillassoux’s argument seems to do is to show the necessary non-existence of a maximally self-contradictory entity, one which simultaneously has all determinations (amongst which there are many incompatibilities) and which both exists and does not exist. The problem is that this does not show the necessary non-existence of self-contradictory entities in general, because there are weaker forms of self-contradiction available.
This is not to say that I believe in self-contradictory entities, I don’t. I’m happily behind the principle of non-contradiction and happily opposed to Hegel. I simply don’t think that Meillassoux’s argument for the principle is compelling as much as it is frustrating. If someone could show me how it isn’t horribly confused I’d be most grateful.
—- Update —-
Having read through that bit of the Meillassoux again, I realise that it is important to point to his admission that his initial proof does not take into account the difference between contradiction and inconsistency (these notions form a conceptual trio with the notion of incompatibility). At this point he does admit that his position only excludes an extreme form of contradictory entity, and seems to suggest that he needs to do more work on the matter. However, he is defending himself against a challenge which is somewhat different to the one I am fielding against him. He takes the challenge to be that posed by paraconsistent logics, which allow for contradictions that do not entail everything.
He seems to think that it is only under the guise of such logics that we can make sense of entities that are contradictory in specific ways, and do not thus become maximally contradictory in the way I outlined above. He rightly points out that such logics have an epistemic flavour, and are really meant to deal with what happens when we have contradictory commitments, rather than dealing with contradictory states-of-affairs or self-contradictory entities. He thus takes his task to be a matter of defending himself against paraconsistent logic by demonstrating its epistemic limitations in a rigorous way, thus showing that the world as it is in itself must abide by the vanilla principle of non-contradiction.
What is interesting about this interchange is that it shows that Meillassoux takes his proof to involve an implicit reference to the principle of explosion, or the claim that a contradiction implies everything. If we reconsider Meillasoux’s argument in this light, then we see that the reference to a contradictory entity, even a maximally contradictory entity is somewhat misplaced. If his argument is really dependent on the principle of explosion then absolutely any contradiction, including claims that do not make claims about the properties of specific existents, such as “there are things and there are not things” or “if something is green then it is not green”, implies that all entities both exist and don’t exist, and that they have all properties and have no properties at all, in virtue of making true every claim that could possibly be said about anything (what ‘possibly’ indicates here is something troublesome of itself, as it seems that we must understand it combinatorially). However, if we read the argument in this way nothing interesting is said about contradictory properties or predicates specifically, and so nothing specifically about self-contradictory entities.
Moving on, if one take this approach, one could then perhaps follow through the rest of Meillassoux’s argument, showing that if there were a true contradiction no entities could change or cease to exist, rather than some specific self-contradictory entity being unable to change or cease to exist. This could be taken to entail that all possible entities exist necessarily and we could then perform the reductio ad absurdum to prove the principle of non-contradiction. However, this is unnecessary, since due to the principle of explosion any contradiction implies the negation of the principle of unreason. Indeed, it also implies the principle of unreason itself. It implies all contradictions. If one allows both the principle of explosion and the inference rule for reductio ad absurdum then one gives oneself a free pass to discharge any contradictory assumption. Indeed, arguably, one necessitates it. The principle of explosion and the inference rule is all that is required to refute any contradictory claim, including a claim to the existence of a contradictory entity. The principle of unreason is not required here.
However, the appeal to the principle of unreason is meant to move us from a mere epistemic level – where we talk about what is thinkable – to the level of the absolute – where we talk about what is possible in itself. Seemingly, the fact that the principle of unreason has been established as an absolute means that if anything contradicts it we can perform a reductio to a similarly absolute claim. And, because via the principle of explosion, any contradiction implies its negation (through implying everything), we can absolutely deny the truth of all contradictions, i.e., we can uphold the principle of non-contradiction as an absolute. I’m not entirely certain if this works, but lets assume it does for now.
However, all of this comes down to the question of the status of the principle of explosion itself. In essence, if Meillassoux can vindicate explosion then he can vindicate non-contradiction, and to do so he needs make no reference to self-contradictory entities at all. The whole discussion of such entities seems entirely redundant. It seems then that the crucial point is trying to establish that the principle of explosion is required in thinking the absolute, and that paraconsistent limitations of this principle do not touch on the absolute, but are merely epistemic. There is a certain problem with this however, in that as I’ve noted, if you allow for both explosion and the reductio rule, you get the principle of non-contradiction for free. If Meillassoux can honestly demonstrate the fact that only the principle of explosion is applicable in thinking the absolute, then he doesn’t even need the principle of unreason to establish the absolute nature of the principle of non-contradiction.
In short, I think Meillassoux’s whole argumentative approach here is very confused. He seems to confuse the issue of contradictory entities and contradictions in general, and if his argument is to make any sense, it seems to depend on a further implicit principle which he needs to establish independently, and if indeed he does establish this principle independently then he has no need of his argument.
Again I’m happy to be shown the error of my ways here if I’ve gotten anything wrong. Logic (and paraconsistent logic) is not my primary area of expertise.