Meillassoux and Contradiction [Updated]

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.


10 Responses to “Meillassoux and Contradiction [Updated]”

  1. I’ll need to flick through After Finitude again, but I think Meillassoux’s weakness here is his hankering for a certain level of generality. The results of his inquiry would look, I think, a bit less impressive if he ended up with the principle of unreason, and a few exceptions to what kind of thing can exist. The latter would seem a bit ad hoc, and so he seems to go all the way and propose that no entity can be contradictory at all. I think if he wants to go down that path eventually an engagement with the work of Graham Priest will become quite important, though there’s not much evidence of that thus far.

  2. deontologistics Says:

    I’ve just updated the post to go into a bit more detail on some of his other considerations. Overall, I’m quite disappointed, as although I disagree with other parts of his argument, it has up until this point felt fairly rigorous. This particular issue just seems full of conceptual muddles.

  3. Right, I’ve had a look back at After Finitude now and also digged out my copy of Priest’s rather great ‘In Contradiction’. It seems Meillassoux has more work to do than he thinks.

    Seemingly, he is mistaken in thinking that he might be able to cast his arguments in terms of inconsistency rather than contradiction in order to avoid the objection that, in fact, contradiction (if not maximal contradiction) is logically conceivable. In ‘In Contradiction’ Priest discusses various reactions to Russell’s paradox, and says the following: “There are various ways one may do this [reject a logical principle involved in generating Russell’s Paradox]…Allow for set theory to entail contradictions but reject…Explosion…and hence contain a theory that is inconsistent but non-trivial”. So even inconsistency is apparently held to be thinkable by at least one logician.

    I wonder whether Meillasoux is aware of these finer details though – I have no idea whether he knows Priest, the only book referenced in regard to paraconsistent logic is a french book I haven’t heard of. It seems to prop up this section of the argument he’d need to really get into the nitty gritty details of paraconsistent logics…

  4. deontologistics Says:

    I think it might be possible for Meillassoux to argue against paraconsistent logics. My real issue is that I think that this has little to do with the argument he initially presented. It looks as if when he talks about reinterpreting his argument in terms of inconsistency (and then again moving it on to contradiction) he’s somewhat confused, because his original argument does not talk about inconsistent propositions, or contradictions, but contradictory entities, and his notion of a contradictory entity is very confused.

    If you ignore my attempt to reinterpret the argument above in terms of the principle of explosion, we can roughly rework how Meillassoux’s initial argument is meant to work.

    The crucial point (which I might have missed above) is that he takes a contradictory entity to _of its essence_ be everything that it is not. This idea is incredibly strong, and there is no real reason for us to think that this is the only kind of contradictory entity there could be, but we can still try to follow through the argument from the assumption that such an entity exists.

    In order to show that this kind of contradictory cannot cease to exist, he has to treat existence as something like a property. The idea is that if a contradictory entity happens to be any given way, then it is also not that way. So if it happens to exist, then it also doesn’t exist, and vice versa. It thus can’t change from one to the other, because it occupies both states at once. This assumption that existence is a property is really very awkward, but Meillassoux has at least guaranteed that via the principle of unreason no entity can have existence of its essence.

    The demonstration that such an entity cannot change is perhaps easier on this reading however. Because if something happens to have a property, then it also doesn’t have it, but, if it happens not to have a property, then it also must have it. This is what guarantees that the entity has all properties at once (and also none of them). This genuinely does preclude change.

    So, Meillassoux’s argument can genuinely prove that there cannot be a maximally contradictory entity, or an entity with a contradictory essence, but only on the assumption that existence is a determination (or property) that is subject to its contradictory essence.

    I would suggest that we should reject the latter assumption, but even if we do, the most Meillassoux can do is prove the necessary non-existence of entities with such contradictory essences, but not the more garden variety of contradictory entities (such as my entity which is both green and red), or even of plane old contradictory facts.

  5. I think you’ve correctly spotted the problems here. It is strange that he doesn’t really comment on his treatment of existence as a predicate given the controversial nature of that decision. I also agree that the idea of a contradictory entity being contradictory of essence is remarkably strong. Though in fact I’m tempted go further, I’m tempted to say it just doesn’t make any sense – but I’m not in a position to argue fully for this yet.

    If you haven’t already read it, I strongly reccomend Harman’s discussion of Meillassoux towards the end of Prince Of Networks (available as a free pdf from the publisher’s website).

  6. deontologistics Says:

    Thanks for the recommendation, I’ve read the first chapter, but I want to finish After Finitude (and a certain frustrating section of my thesis) before I go any further into it.

    However, I do agree that the idea of a contradictory or paradoxical essence makes no sense. More importantly though, the introducing of the notion just seems to raise far more questions than it helps answer.

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