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MEILLASSOUX’S DILEMMA:
PARADOXES OF TOTALITY AFTER THE SPECULATIVE TURN
Joshua Heller and Jon Cogburn
Quentin Meillassoux’s After Finitude is first and foremost a critique of what he calls
“correlationism,” which is comprised of two theses: (1) Verificationism, the proposition that
since we cannot coherently think of reality as unthought, truths must be in principle knowable;
and (2) Finitude, that we cannot coherently think of self-subsistent totalities/absolutes such as all
of reality. Meillassoux’s critique has had a wide-ranging influence on contemporary continental
philosophy as (analogously to the role of the demise of positivism in analytic philosophy) the
much heralded “return to metaphysics” is to some extent predicated on its acceptance. 1
In what follows, we first present an account of Meillassoux’s problem of ancestrality and
critique of correlationism. Then we show how Meillassoux’s rejection of Finitude, how he gets
“after Finitude,” involves an insight developed in Beyond the Limits of Thought, one which
Graham Priest shows to have been recapitulated over and over again in the history of thought: to
posit a limit is to transcend that limit. But this very thought about the nature of limits (associated
by Priest most clearly with the writings of Hegel and Derrida)2 ends up being Meillassoux’s
1 For an influential collection of recent writings concerning the return see Bryant, Srnicek, and
Harman (2011).
2 Priest is right as far as it goes, though Fichte and Schelling are glaring omissions in his account.
Fichte might have been the first post-Kantian to turn necessity into speculative virtue here, the
contradiction between “closure” and “transcendence” assuming a productive dimension. See the
relevant discussion of the Wissenschaftlehre on pages 208-9 of Förster (2012).
2
undoing. For when Meillassoux develops his own speculative philosophy of absolute
contingency in After Finitude and then in The Divine Inexistence,3 he himself runs afoul of the
very Priestian claim that he employs in his initial critique of correlationism. Meillassoux himself
constitutively presupposes the very thing he tries to undercut. The result is that Meillassoux’s
own speculative metaphysics is inherently unstable, reeling between a fully correlationist
finitism, on the one hand, and an anti-correlationist infinitude on the other.
Given the centrality of paradoxes of totality in the post-Kantian continental tradition,
Meillassoux’s failure in this regard ends up being an issue that must be addressed both by the
generation of continental metaphysicians moved by his critique of correlationism, as well as
many influential philosophers that predate the critique.4
I. THE CRITIQUE OF CORRELATIONISM
We have delineated Meillassoux’s presentation of ancestrality and subsequent critique of
correlationism into 10 theses; the first two of which are:
3 L'inexistence divine is Meillassoux’s unpublished 1997 dissertation. Graham Harman has
translated and included portions of it in his Quentin Meillassoux: Philosophy in the Making. In
Section IIb. below we regiment one of the key arguments that Harman attributes to Meillassoux.
4 See the discussion of Derrida, Badiou, and Deleuze in Livingston (2012), as well as Gabriel
(2013a, 2013b), and Garcia (2014).
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1. Verificationism – Since we cannot coherently think of reality as unthought, and
thinkability entails the possibility of acquiring relevant evidence,5 we must
understand propositions in terms of the evidence for or against them.
2. Finitude – We cannot coherently think of self-subsistent totalities or absolutes.
Meillassoux will eventually reject Finitude in his own attempt to get out of the correlationist
circle but he initially assumes it as a necessary thesis of correlationism. He begins After Finitude
by considering “ancestral statements,” those statements whose truth makers would have occurred
prior to the existence of anybody to know them.
3. Let A be any sentence of the form, “Event Y occurred x number of years before
the emergence of humans.” A seems to undermine Verificationism, as scientists are
thinking of truth making events existing in the absence of any thinkers.6 The
5 We are just talking about “strong correlationism” in Meillassoux’s sense. Of course Kant’s
ability to be a transcendental idealist and still muse about God, immortality, and free will only
makes sense to the extent that one could adopt a weaker version that takes our human epistemic
capacities to shape what we can talk about meaningfully while still somehow allows us to
understand propositions for which we cannot in principle gather evidence. But as Eckart Förster
(2012: 110, and 226 with respect to Schelling) trenchantly demonstrates, from Jacobi (2009)
onwards the argument that weak correlationism collapses into Verificationism became one of the
major engines of German Idealism. The positivistic and phenomenological traditions that
Meillassoux critiques have internalized Jacobi’s critique.
6 Space constraints prohibited us from exploring the relationship between this literature and that
spawned by Dummett (1978) concerning how the verificationist could have knowledge of the
past. The consensus (e.g. Miller (2008)) on this debate is that the verificationist need only appeal
to counterfactuals such as “if we were to build a time machine and travel back to time T, we
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correlationist can say that the sentence is always false, in the manner of
contemporary creationists, but they understandably don’t want to do this.
To avoid creationism the correlationist will agree that A is true for us. But they cannot claim that
A is true from an absolute or “God’s eye” perspective.
The argument Meillassoux then presents on behalf of the correlationist rests on this
supposition: “there can be no X without a givenness of X, and no theory about X without a
positing of X” (2008b, 1). The commitment to this thesis entails that there are no thinkable
objects, events, laws, or entities which are not correlated with a subject thinking them. If an
individual holds a theory about X, she speaks about that X as it is given to her and as it is posited
by her.7
would see that P.” It will be clear by the end of our article that Priest’s Domain Principle (to
which we show Meillassoux to be committed) will entail that this only makes sense by way of
reference to the totality of spaces and times over which the speaker is quantifying, which is
inconsistent with Finitude and hence non-correlationist in Meillassoux’s sense.
7 This is clearly an instance of the Berkeley (1988) conceivability argument that concludes to be
is to be conceived. Meillassoux (2008a) and Priest (2008a) take the argument to be valid, yet to
paradoxically explode, since the person making it must transcend the very limits of
conceivability entailed by the argument. Graham Harman (2011a) and Jon Cogburn (2011b) take
the argument to be simply invalid. Meillassoux is not entirely consistent here though, as he
argues against the idealist that the unthinkability of a proposition does not entail its falsity. We
ignore this here, as (1) on our construal of the text, it does no work for Meillassoux, and (2) it
ends up being problematic, given Meillassoux’s other commitments. For a discussion, see
Cogburn (2012).
5
Of course, scientists often posit truth-making events which occurred in the absence of
subjectivity. The origin of life on Earth (3.5 billion years ago), the accretion of Earth (4.6 billion
years ago), the birth of the Milky Way galaxy (13.2 billion years ago) are all examples of events
said to have occurred prior to human subjectivity, that is, prior to the correlate of thought and
being. But such claims must be interpreted in very particular ways by the correlationist.
4. Instead of creationism, Verificationism gives an account of such scientific
statements being part of a “founded mode,” defined over more originary human
epistemic practices and perceptions. In this manner, they actually double the
meaning of the sentence. The error is thinking the sentence is true and originary,
whereas if it is understood as founded it can be true.
This founded mode is derivative, relying upon the originary structure and constraints of
human knowledge to ground truth procedures.
Thus, returning to our considered sentence type A, “Event Y occurred x number of years
before the emergence of human subjectivity,” the correlationist will hold that A sentences are
true for us (hereafter Aus) but argue that an A sentence from an absolute or “God’s eye”
perspective (hereafter Aabsolute) is uncorrelated and thus incoherent. In this manner, scientific
statements can be seen to be true within the discourse in which they occur (that of the scientist,
contingent upon Verificationism and Finitude).
5. So according to this strategy, A sentences are true for the scientists, or more
broadly for us, but not true from an external, absolute “God’s eye” perspective that
does not involve human thinkability. We can disambiguate the two understandings in
this manner, for any A, Aus is true or false, while Aabsolute lacks a truth value.
If we attend again to Verificationism, this gives us the following.
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6. For the Verificationist, Aus presupposes the following: “Event Y occurred x
number of years before the emergence of humans” correlates with a set of
verification procedures followed by scientists that lead them to assert sentences of
type A.
This is fine as far as it goes. The problem is that it gives rise to a Euthyphro dilemma,8
made clear by considering three potential responses.
(7.1) The Platonic Response – Scientists hold Aus sentences to be true for the
following reason: The set of verification procedures followed by scientists leads
them to assert A because event Y occurred x number of years before the emergence
of humans.
Though his argument doesn’t ultimately rest on this, Meillassoux clearly takes the plausibility of
the Platonic Response to constitute in itself evidence against correlationism.
“. . . if ancestral statements derived their value solely from the current universality of
their verification they would be completely devoid of interest for the scientists who
take the trouble to validate them. One does not validate a measure just to
demonstrate that this measure is valid for all scientists; one validates it in order to
determine what is measured. It is because certain radioactive isotopes are capable of
informing us about a past event that we try to extract from them a measure of their
age: turn this age into something unthinkable and the objectivity of the measure
becomes devoid of sense and interest, indicating nothing beyond itself. Science does
not experiment with a view to validating the universality of its experiments; it carries
8 Euthyphronic arguments have seen resurgence in contemporary analytic philosophy. See
especially Wright (1994) and Roland (2005).
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out repeatable experiments with a view to external referents which endow these
experiments with meaning.” (2008a, 17).
We want to say what seems to be common sense: scientists’ measurements come out the way
they do because they measure the way things are. Meillassoux argues that the correlationist is
forced to either deny this truism or to be committed to:
(7.2) The Inverted Platonic Response – The correlationist could reverse the Platonic
order of explanation, but this would be to lapse into Idealism, holding that scientists’
verification procedures caused reality to be the way it is.
It is important to realize the extent to which the French phenomenological tradition critiqued by
Meillassoux had taken for granted Heidegger’s assertion in Division One of Being and Time that
we had somehow transcended the very debate between Idealism and Realism. So the Inverted
Platonic Response clearly does not work either as a live option for the Correlationist.
Meillassoux does not explicitly examine a third response to the Euthyphronic dilemma,
but it does function as an enthymeme in the argument. But one might argue that choosing either
the Platonic or inverted Platonic response to the Euthyphronic dilemma with respect to Aus is to
make a claim about the universe as a self-subsisting totality, which the correlationist of course
takes to violate Finitude. So perhaps the answer involves Quietism. Here, we conjecture what
Meillassoux would almost certainly respond.
(7.3) Quietism – Or the correlationist could refuse to consider the Euthyphronic
dilemma with respect to Aus, just noting the correlation between human procedures
and the truth of the sentence, but not taking a stand on any possible asymmetry. This
again is in tension with the scientist’s own Platonic understanding of the claim.
8
But the correlationist might with some justification stamp her foot here. Why should we worry if
the scientist is a Platonist? Isn’t it enough that the correlationist is neither Platonist nor Idealist
while understanding the truth of Meillassoux’s ancestral claim? But Meillassoux would still be
correct to challenge the correlationist. Why should we accept Quietism here? For the
correlationist, such Quietism might be taken to follow from Finitude and Verificationism, for
recognizably Jacobian (see footnote 4) reasons. If Finitude says we do not have epistemic access
to reality as it is in itself, and Verificationism only allows us to talk meaningfully about topics
about which we have epistemic access, then it’s perfectly rational to prescind when someone
presents a Euthyphronic dilemma.
Meillassoux considers the variety of Quietism that is motivated by differentiating
between the empirical subject, understood to be one object among many, and a transcendental
subject which is the source of the norms used to evaluate whether sentences are true or false.
8. Perhaps a Kantian defense can save the scientist’s understanding of A sentences
by differentiating two levels of thought. (8.1) The empirical level where the subject
mentioned in A is just one object among others (here the subject mentioned in A is
understood empirically) and (8.2) the transcendental level which is a set of
conditions of cognition that render assertions of type A meaningful. 9 On this reading,
9 One of the greatest ironies in the history of thought is the extent to which logical positivism and
classical phenomenology have certain constitutive moves in common, arising out of their neo-
Kantian births. The kind of semantic doubling considered by Meillassoux reached its positivistic
height in Rudolph Carnap’s great essay “Empiricism, Semantics, and Ontology,” where it is
explicitly thematized in terms of the internal/external metaphor. In a sense, Carnap was only then
catching up with Heidegger’s own doubling in Division One of Being and Time, where the
question of Meillassoux’s arche-fossil is explicitly considered with respect to Newton’s
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A has a truth value as a claim about the empirical subject (i.e. “Event Y occurred x
number of years before the emergence of human beings (understood as empirical
objects)”). The transcendental subject provides the conditions for thinking A, such as
determining which bits of evidence would count for or against A.
Meillassoux points this out as the standard defense of Kantian idealism: the correlationist
supports Quietism about Euthyphronic dilemmas by arguing that answering such dilemmas
involves “conflating the empirical and the transcendental” (AF 2008a, 24). If we take A type
sentences as empirical claims, they are statements which concern the knowing of the emergence
of non-conscious objects,10 objects without subjectivity or the capacity for knowledge, within a
physical world. The transcendental question, on the other hand, concerns how the science of the
emergence of these objects is possible. Hopefully by keeping separate “the distinction between
the conscious organ which arose within nature and the subject of science which constructs the
knowledge of nature” (2008a, 22) we can do justice to the science while remaining correlationist.
The charge is that Meillassoux ignores the transcendental subject, and that he “cannot claim that
[the] problem is ‘ontological’ rather than empirical, since [the] problem of the arche-fossil is
empirical, and only empirical – it pertains to objects” (2008a, 23). To this Meillassoux must
respond:
equations. Like Carnap, Heidegger finds such questions meaningful in one mode and
meaningless in another.
10 Schelling’s 1800 System of Transcendental Idealism contained the first sustained defense for
the need of a system of nature philosophy to buttress transcendental philosophy. A fuller
treatment would carefully trace the respects in which Meillassoux recapitulates Schelling’s
reasoning. For how this reasoning arose out of sustained meditation on Fichte, see pages 222-231
of Förster (2012).
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9. But the empirical/transcendental distinction will only work for the correlationist if
there are good reasons not to apply the schema A to the very transcendental subject
who asserts sentences of A’s form,11 e.g. “Event Y occurred x number of years
before the emergence of human beings (understood as transcendental subjects).”
For the neo-Kantian, the transcendental subject is not an existent entity; rather, it is a set of
conditions which allows for the possibility of objective scientific knowledge. Regarding the
latter, this set of conditions cannot be treated as an object or entity in itself and, since it is only
objects which exist, this set of transcendental conditions do not in fact exist. They condition. As
Meillassoux describes it, objects can be said to exist in time and space, and thus are born and die
in time, but conditions do not, and thus cannot be objects of reflection. The problem of
ancestrality concerns the empirical world because it concerns objects, not conditions. Conditions
provide the form of scientific discourse and at the heart of this correlationist rejoinder is the
strategy “to ‘de-ontologize’ the transcendental by pulling it out of reach of any reflection about
being” (2008a, 24). Thus, the transcendental subject does not exist in the same way as empirical
objects exist.
11 Formal issues of self-referentiality with respect to contemporary continental philosophy go far
beyond the scope of this paper. We only discuss Priestian inclosure paradoxes, but self-
referentiality also occurs prominently in the proof of Gödel’s Incompleteness Theorems, and
Paul Livingston (2012) is able to use Gödelian reasoning analogously to the manner in which we
use Russellian reasoning here. As with Gödel’s Theorems and Russell’s Paradox, the proof of the
unsolvability of the halting problem also uses diagonalization, and so there are connnections
between these projects and the emergentist ontology of capacities developed by Jon Cogburn and
Mark Silcox (2005, 2011).
11
Meillassoux’s response is to deny the transcendental nature of the transcendental subject.
That is, if the transcendental subject can be said to have a point of view, that is, it can make
claims about objective reality, then it takes place in the world. If it takes place in the world, then
it must have emerged at some point in time. It then cannot exist outside space and time.
Meillassoux argues:
But how do notions such as Finitude, receptivity, horizon, regulative Idea of
knowledge, arise? They arise because, as we said above, the transcendental subject is
posited as a point of view on the world, and hence as a taking place at the heart of
the world. The subject is transcendental only insofar as it is positioned in the world,
of which it can only ever discover a finite aspect, and which it can never recollect in
its totality (2008a, 24-5, emphases in original).
This is the reason for Finitude. If the transcendental subject is necessarily embodied or
embedded as is claimed by nearly all post Kantian continental philosophers, then it is positioned
as essentially in the world. But a thinking body must still emerge within time, at which point we
must be able to think of the temporality of the emergence of the transcendental subject. If a body
is a necessary condition of the transcendental subject, then the transcendental subject can only be
said to have emerged in time, at which point there would be a time before the existence of the
transcendental subject.
10. Therefore, the transcendental subject is as susceptible to Finitude as the subject
considered as an empirical object. And thus, the correlationist can still make no sense
of sentences of type A.
Following Schelling, Meillassoux thus concludes that the problem of ancestrality therefore
cannot be thought from the transcendental viewpoint because the thinking of the problem of
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ancestrality posits a space-time wherein transcendental subjects are born and perish. Merely to
think this space-time is to think the conditions of the possibility of scientific discourse and thus
disallow the transcendental to be transcendental. Put another way, the transcendental subject
either has an absolute view from nowhere or is situated finitely. In the very process of describing
an empirical subject’s Finitude, though, we put the subject into a time series that transcends the
kind of derivative time the transcendental idealist posits as empirical.
II. MEILLASSOUX’S ARGUMENTS FOR CONTINGENCY
What we call Meillassoux’s dilemma stems from the fact that his critique of correlationism and
arguments for contingency cannot both be valid. Since we are sympathetic to the critique of
correlationism, we have been careful to present it in a way that makes it maximally invulnerable
to counterargument. Here we will be a bit less circumspect, presenting Meillassoux’s arguments
for contingency with broader brushstrokes. We are not trying to defend the argument against all
possible criticisms, but rather just to show that if it is valid, then Meillassoux’s critique of
correlationism is itself invalid.
IIa. The Critique of Aleatory Reasoning
Meillassoux’s argument for the claim that the only necessary truth is the truth that everything is
possible is astonishingly brief, given the power of its conclusion. He begins by stating that the
only way one could make sense of causal necessity is to make probability judgments over the set
of possible worlds. To say that it is unlikely that a billiard ball is going to start floating is to say
that it is unlikely that the actual world we live in is one of the possible worlds where that
happens:
13
The implicit principle governing the necessitarian inference now becomes clear: the
latter proceeds by extending the probabilistic reasoning which the gambler applied
to the event that is internal to our universe (the throw of the dice and its result), to
the universe as such. This reasoning can be reconstructed as follows: I construe our
own physical universe as one among an immense number of conceivable (i.e. non-
contradictory) universes each governed by different sets of physical laws; universes
in which the impact of two billiard-balls does not conform to the laws that govern
our own universe. . . Thus, I mentally construct a ‘dice-universe’ which I identify
with the Universe of all universes, bound globally by the principle of non-
contradiction alone, each face of which constitutes a single universe governed by a
determinate set of physical laws. . . (2008a, 97).
Physical laws spontaneously changing is taken to be improbable because it is rare over the set of
all possible universes.
We should note that Meillassoux does not provide an argument for his construal of what
would be required to affirm necessity of physical laws. Given that it is main premise in an
argument with such a strong conclusion, this is unfortunate. But since the focus of our criticism
is elsewhere and given that the kind of possible worlds explanations of modal concepts that
Meillassoux gestures at is a familiar one in a Lewisian age (e.g. Divers (2002)), we follow the
dialectic.
Five pages after his possible worlds explanation of nomic necessity, Meillassoux appeals
to Alain Badiou’s interpretation of standard set-theory to rule out of bounds the appeal to the set
of all possible worlds. Since Cantor has shown that there is no greatest infinite cardinality, it
follows on the standard reading that
14
It is possible to construct an unlimited succession of infinite sets, each of which is of
a quantity superior to that of the set whose parts it collects together. This succession
is known as the series of alephs, or the series of transfinite cardinals. But this series
itself cannot be totalized, in other word it cannot be collected together into some
‘ultimate’ quantity (2008a, 104).
But since we can conceive of different worlds for each different cardinality, just as we cannot
talk meaningfully of a set of all sets, we cannot talk meaningfully of a set of all possible worlds.
But then, since causal necessity requires talking meaningfully about such a set, there is no causal
necessity.
This is extraordinarily enthymematic! In addition to the quick supposition that causal
necessity can only be made intelligible in terms of probability distributions across possible
worlds, it is not immediately clear exactly how cardinality considerations undermine our ability
to talk about a set of all possible worlds. In section IIc. we provide textual evidence to show that
Meillassoux must have had something like Kaplan’s Paradox in mind. Our reconstruction both
renders his argument valid and shows how it is in tension with his own critique of
correlationism.
IIb. The Verificationist Argument for Contingency
But first, we must note that one who accepts Meillassoux’s critique of aleatory reasoning might
still argue that it doesn’t establish a positive metaphysics of contingency. Assuming that it is
successful, all he has shown in the critique is that we can’t claim to know that certain violations
of the laws of nature are impossible, but it doesn’t follow from this that they are actually
possible. Graham Harman realizes that this point is absolutely central to Meillassoux’s argument:
15
In other words, the Strong Correlationist thinks we cannot know which possibility
about the afterlife is true, but the Speculative materialist thinks we can know that any
of them could be true, and without reason. This apparently hair-splitting point is
actually the key to Meillassoux’s entire system, and is worthy of closer attention
(Harman 2011a, 26).
We needn’t go into the relevant discussion in Meillassoux about the afterlife,12 to see how the
implication Harman attributes to Meillassoux saves the critique of aleatory reasoning from the
attempted via negativia. Meillassoux’s opponent might claim that all he has shown is that for
claims of a certain sort, it is impossible to know that something is impossible. But that doesn’t
entail that all of these claims are actually such that it is possible they are true, as Meillassoux’s
metaphysics of contingency holds.
But Harman shows that the entailment does hold for the Verificationist. And Harman is
correct, as can be established by a quick demonstration.13 First, note that Verificationism, the
claim that for all propositions X, it is possible to know that X (X(X → <>KX)), contraposed is
12 We direct the reader to Harman’s treatment, which is excellent. Harman’s discussion of
“Meillassoux’s spectrum” (Dogmatic/Naïve Realism, Weak Correlationism, Strong
Correlationism, Very Strong Correlationism, Absolute Idealism) is philosophically significant in
its own right, but since it is not relevant to our critical argument we don’t present it here.
13 Given that the first ten lines of the proof just establish contraposition, analytically trained
readers might think “not quick enough.” We present the formal proofs explicitly here in an effort
to make this paper as accessible as possible to the widest possible community. In addition, it is
worthwhile to be as explicit as possible about the extent of the expressive resources required as
well as where strictly classical reasoning is involved. See the following footnote.
16
the claim that, for all X, if it is not possible to know that X, then X is not the case (X(<>KX →
X)).
1. X(X → <>KX) Verificationism
2. | [Q] arbitrary name introduced for introduction
3. | | <>KQ assumption for → introduction
4. | | |Q assumption for introduction
5. | | | Q → <>KQ 1 elimination
6. | | | <>KQ 4, 5 → elimination
7. | | | 3, 6 elimination
8. | | Q 4-7 introduction
9. | <>KP → P 3-8 → introduction
10. X(<>KX → X) 2-9 introduction
But then, since this applies to all propositions, it applies to the negations of all propositions.
When we combine this with the classical principle of double negation elimination, we can thus
conclude from Verificationism for any arbitrary claim that if it’s not possible to know that the
claim is false, then the claim is true:
11. | [P] arbitrary name introduced for introduction
12. | | <>KP assumption for → introduction
13. | | <>KP → P 10 elimination
14. | | P 12, 13 → elimination
15. | | P 14 double negation elimination14
14 Note that, modulo Minimal Logic, double negation elimination is equivalent to the law of
excluded middle. So an intuitionist would balk right here. Meillassoux thus might have
inadvertently discovered a new argument from Verificationism to the rejection of classical logic.
17
16. | <>KP → P 9-15 → introduction
17. X(<>KX → X) 11-16 introduction
But then we need only apply this claim to claims of the form that something is possible to get the
implication that Harman takes to be central to Meillassoux’s whole argument.
18. | [P] arbitrary name introduced for introduction
19. | | <>K<>P → <>P 17 elimination
20. X(<>K<>X → <>X) 18-19 introduction
Thus, Verificationism on its own entails the Harmanian inference. The correlationist’s negative
claim about the possibility of knowledge becomes a positive speculative proposition about what
is possible.
IIc. Kaplan’s Paradox
As noted above, in his presentation of the critique of aleatory reasoning Meillassoux is
frustratingly inexplicit about the exact connection between cardinality results in set theory and
reasoning about the set of all possible worlds. But it is clear that his argument has two moments.
He first states a version of Cantor’s Paradox.
But this series [of sets representing the increasing cardinal numbers] itself cannot be
totalized, in other words, it cannot be collected together into some ‘ultimate’
We’re not sure though, as (1) one can prove in Intuitionist Logic that Verificationism entails the
quasi-Harmanian implication X(<>K<>X → <>X), and (2) we think that it’s quite
probable that the Domain Principle (discussed in Sections III and IV below) works against any
arguments one might make for preferring the Heyting style proof-theoretic semantics (with their
intuitionist interpretation) over strictly classical ones. This is clearly an open question though.
For a formalization of Dummett’s own argument for logical revision, see Cogburn (2005).
18
quantity. For it is clear that were such a quantitative totalization to exist, then it
would also have to allow itself to be surpassed in accordance with the procedure of
the grouping of parts [by taking the set of its subsets]. Thus, the set T (for Totality)
of all quantities cannot ‘contain’ the quantity obtained by the set of parts of T.
(Meillassoux 2008a, 104).
Following Zermelo-Fraenkel set theory, he concludes that we should refuse to assert that the set
of all sets exists.
Consequently, this ‘quantity of all quantities’ is not construed as being ‘too big’ to be
grasped by thought - it is simply construed as not existing (ibid).
If we add to this the premise that,
If a set is thinkable, then its powerset is thinkable,
we can get to Meillassoux’s conclusion that the totality of the thinkable cannot be said to exist
either.
Within the standard set-theoretical axiomatic, that which is quantifiable, and even
more generally, that which is thinkable – which is to say, sets in general, or whatever
can be constructed or demonstrated in accordance with the requirement of
consistency – does not constitute a totality. For this totality of the thinkable is itself
logically inconceivable, since it gives rise to a contradiction. We will retain the
following translation of Cantor’s transfinite: the (quantifiable) totality of the
thinkable is unthinkable (ibid.).
Unfortunately though, Meillassoux never gives an explicit reason for thinking that this should
entail that there is no set of all possible worlds.
19
Moreover, Meillassoux realizes that some set theories, such as Quine’s NF,15 are probably
consistent and do quantify over sets of all sets. In making this admission, he weakens his
argument considerably,
What the set-theoretical axiomatic demonstrates is at the very least a fundamental
uncertainty regarding the totalizability of the possible. But this uncertainty alone
enables us to carry out a decisive critique of the necessitarian inference by destroying
one of the latter’s fundamental postulates: we can only move immediately from the
stability of laws to their necessity so long as we do not question the notion that the
possible is a priori totalizable (Meillassoux 2008a, 105).
He seems to be arguing that affirming the necessity of laws of nature requires our talk of the set
of all possible worlds to have some form of a priori status. But the fact that there are so many
different set theoretic responses to antinomies shows that it does not. Put thusly, this is a
remarkably weak argument. A prioricity does not mean that there cannot be disagreement.
This problem is not dispositive though, as Meillassoux’s reasoning is remarkably similar
to an argument discovered by David Kaplan, one that, like Meillassoux’s, trades on cardinality
considerations, but unlike Meillassoux’s: (a) explicitly connects the issue of the totality of the
thinkable with the totality of possible worlds, and (b) trades in logical necessity rather than a
prioricity. Given the way Kaplan’s argument patches the holes in Meillassoux’s, we do not think
it is an exaggeration to say that Meillassoux and Alain Badiou (2006; originally 1988), who
Meillassoux cites in this context, came close to independently discovering Kaplan’s Paradox.
15 See Oksanen (1999) for an example of just this. Oksanen also shows that the Kaplan’s
cardinality argument was first discussed in print by Davies (1981), who credits Kaplan.
20
Kaplan gives two arguments to show that (p)<>(q)(Tq (p=q)) must be considered
to be logically false by standard modal semantics. His first16 argument tracks the
Badiou/Meillassoux argument closely. (p)<>(q)(Tq (p=q)) says that any proposition is
such that there is a possible world where it uniquely has the property T in that world. So let T be
the property of being the only proposition being thought. Then Kaplan’s principle instantiated on
T would say that any proposition is such that there is a possible world where that and only that
proposition is being thought.
Kaplan is not committed to the set of all possible worlds allowing this. Rather, he holds
that logic alone should not rule it out. This much more reasonable view ends up doing the work
of Meillassoux’s claim that our knowledge of the existence (or not) of the set of all possible
worlds should be a priori. And Kaplan is able to appeal to cardinality considerations in the same
way Meillassoux and Badiou do.
In standard modal semantics, propositions are identified with the set of worlds where
those propositions are true. So if we assume that the set of all possible worlds exists, then the set
of propositions is identical to the powerset (set of all subsets) of the set of possible worlds. But
since, by Kaplan’s principle, for each one of those propositions there is a world where it is
uniquely thought, the number of propositions must be less than or equal to the number of
possible worlds. But from Cantor we know that a powerset must contain more objects than the
16 His second argument involves applying the above proposition to (p)(Tp → p) (everything
I'm now thinking is false). By the above proposition there should be a unique world where this
one proposition is being thought, but this gives rise to a liar type paradox. We think that
commentators (e.g. Bueno, Menzel, and Zalta (2013)) don’t mention this second argument when
talking about Kaplan’s Paradox because whatever one says about the liar paradox will probably
take care of this one.
21
set it is formed from. So we simultaneously have that the set of propositions is larger than the set
of possible worlds (since the former is the powerset of the latter) and that the set of propositions
is smaller or the same size as the set of possible worlds (since each proposition maps onto a
unique world). Contradiction!
Note how the gap in Meillassoux’s argument about the totalization of the thinkable and
the totalization of the set of possible worlds is patched by Kaplan’s taking as a premise the
identification of propositions with sets of possible worlds.17 Once this is done Meillassoux’s
reasoning converges with Kaplan’s.
Of course there are still many places one could, with some justification, cavil. One
moved by Kaplan’s Paradox might see it as impetus for developing an account of causal
necessity that did not involve totalizing the thinkable.18 Or one might utilize a theory of
propositional content that does not tie it to possible worlds, in the hope that one could then still
talk about causal necessity in terms of the set of all possible worlds.19 We set these responses
17 “Actualist Realists” tend to do just the opposite, taking propositions as primitive and
construing possible worlds in terms of them. But Meillassoux’s concern about the totality of all
possible worlds cannot be avoided by adopting a variant of that view. One can argue that every
variety of Actualist Realism faces a version of Russell’s Paradox. Since our morals with respect
to Kaplan’s Paradox in this paper clearly apply to Russell’s Paradox, we needn’t go into this. See
Divers (2002), Chapter 15 for a discussion.
18 For example, Hiddleston (2005) reverses the Lewisian order of explaining causality in terms of
counterfactuals which are explained by possible worlds. For Hiddleston facts about causality are
fundamental with respect to the counterfactuals.
19 As far as we can tell, Brandomian inferentialists can do something just like this. For the
inferentialist propositional content is a function of inferential role, which proof theory makes
22
aside, as responding to our tu quoque argument is far more critical to Meillassoux’s entire
project.
III. THE DOMAIN PRINCIPLE
Making explicit the tension between Meillassoux’s arguments against correlationism and for
contingency requires seeing how Kaplan’s paradox is analogous to the set of paradoxes that
Graham Priest (2002) calls “inclosure paradoxes,” which include those of Russell, Burali-Forti,
Mirimanoff, Kant (according to Priest’s reconstructed “fifth antinomy”), König, Berry, Richard,
Berkeley, Weyl, Montague, as well as the traditional liar paradox.
IIIa. Kaplan’s Paradox as Quasi-Priestian
Kaplan’s inflationary principle (p)<>(q)(Tq (p=q)) allows him to take any set of possible
worlds x and generate another set that can’t possibly be a subset of x. Thus, where M is the
multiverse, P(x) is the powerset of x, and k(y) takes a set of sets of worlds and generates a new
world for each set of worlds (the world where the proposition corresponding to that set is being
thought) in that set, we have:
Quasi-Priestian Inclosure Schema for Kaplan’s Paradox:
explicit. At least since Simpson (1994) we have had fully normalizable systems of modal logic
(for a Fitch style presentation see Cogburn (2011b)). Such systems do utilize eigenvariables that
seem to refer to possible worlds and the accessibility relation on them. The inferentialist could
follow Hiddleston and account for these eigenvariables without possible worlds, or she could be
a realist about them. In neither case is propositional content tied to possible worlds in the way
that could get the Kaplan/Russell paradoxes off the ground.
23
(1) M exists Existence
(2) if x M (a) (k(P(x)) x) Transcendence
(b) k(P(x)) M Closure
This is only quasi-Priestian because in Priest’s inclosure paradoxes the relation asserted in
Transcence and Closure is one of set membership () and not subsethood, as it is in Kaplan’s.
Nonetheless, Priest’s reasoning about inclosure paradoxes applies straightforwardly to quasi-
Priestian ones such as Kaplan’s.
Let us examine the inclosure schema for the Kaplan paradox. The k function generates a
set that is equinumerous with the set it is applied to. But the powerset of a set is always bigger
than the original set. Moreover, since we know that a subset is never larger than its superset, the
Kaplan function applied to the powerset of a set x will yield a set that is not a subset of that set x
[Transcendence]. And since the output of the Kaplan function is a set of worlds, it is a subset of
the set of all worlds. Then if we apply Transcendence to M itself (which is the reflexive move in
common to all such paradoxes), we get (k(P(M)) M) and k(P(M)) M, resulting in a
contradiction.
IIIb. Priest on the Domain Principle
As Priest shows, the typical response to inclosure paradoxes is to refuse to assert the Existence of
the totality in question. Kant’s original response to his mathematical antinomies (discussed in
Chapters Five and Six in Priest (2002)) is the model of how to attempt such a thing, and ZF set
theory can be seen as a modern day version of Kant’s gambit. But there are other possibilities.
The problematic open sentences in ZF set theory are in von Neumann’s set theory expressible as
proper classes, which are like sets but prohibited from being members of anything. Thus Closure
24
is blocked, since it cannot apply to the proper class V. And at least with respect to inclosure
paradoxes proper (and one would guess quasi-Priestian ones as well) the proof to Transcendence
can be blocked by adopting weaker logics, such as the De Morgan logic developed by Hartry
Field (2008).
Priest notoriously accepts Existence, Closure, and Transcendence, holding that inclosure
paradoxes reflect true contradictions. When critiquing alternative approaches Priest appeals to
two principles: (1) the Principle of Uniformed Solutions (PUS) which states that similar
paradoxes should have similar solutions, and (2) the Domain Principle, which in its simplest
formulation states Cantor’s principle that “For every potential infinity there is a corresponding
actual infinity (Priest 2006, 124).”
When Priest actually uses the Domain Principle he intends something stronger than
Cantor’s principle, for example,
. . .for any claim of the form ‘all sets are so and so’ to have determinate sense there
must be a determinate totality over which the quantifier ranges. It would clearly be
wrong to suppose that this totality is a set satisfying the axioms of Zermelo-Fraenkel
set theory, or of some other theory of sets; but that there is a well defined totality
seems to me to be undeniable. Moreover, it is clearly a totality that we can think of
as a single thing, since we can legitimately refer to it as that totality: the totality of
sets (Priest 2002, 281).
Of course Priest doesn’t just restrict himself to set theoretic paradoxes in the book, and from his
discussion it is clear that he takes the Domain Principle to apply to quantifiers ranging over all
sorts of entities.
Priest’s Domain Principle:
25
For any claim of the form “all xs are so and so” to have determinate sense, there
must be a determinate totality over which the quantifier ranges. We can refer to this
totality and add it to the group of objects we quantify over.20
If this is correct then standard ways of blocking Existence are prohibited. The collection of all
sets exist since we make claims about all sets.21
Priest takes Kant to have first argued for the Domain Principle and also shows how its
application allowed Cantor to construct the ordinal numbers. Priest extends Cantor’s own
argument (Priest 2002, 123-127). Consider a sentence with a variable in it. The determinate
sense of the sentence will depend upon the collection of things the variable ranges over.
For example, consider the claim ‘Let z be a root of the equation ax2 + bx + c = 0.
Then z has at least one value.’ This is true if z may be complex; false if z must be
real (Priest 2002, 125).
Since the truth value of a sentence is a function of its sense, given the way the world is, the sense
of a sentence containing a variable is only fixed by determining what the variable ranges over.
Clearly, then, if there is no determinate totality over which the variable ranges, the sentence will
not have a determinate sense!
This is why Priest holds that Aristotle’s doctrine that there is only a potential infinity
makes no sense. The sentence that every finite number is such that we can construct a greater
20 Put this clearly, the principle is suspiciously similar to Frege’s Comprehension Axiom, which
entails that the set of all sets exists. See also Priest’s (2005) discussion of the Characterization
Principle that holds that for any property there is some possible object that this property is true
of.
21 Priest (2002, 164) argues that the standard Von Neumann method of blocking Closure actually
ends up blocking Existence for the paradox run on classes.
26
number is itself a sentence about the infinite collection of numbers. The Domain Principle should
thus be understood as entailing the Hegelian idea that one cannot talk about parts as parts without
referencing the whole of which these parts are parts. For if a different whole were referenced
then the meaning of phrases referring to all of the parts would be different.
IIIc. Meillassoux’s use of the Domain Principle in the argument against correlationism
Meillassoux’s argument against correlationism crucially involved a Priestian move, taking a
sentence uttered by the transcendental subject to be about the transcendental subject. We gave
this as step 9 in our exegesis of Meillassoux:
9. But the empirical/transcendental distinction will only work for the correlationist if
there are good reasons not to apply A to the very transcendental subject who is
asserting A, “Event Y occurred x number of years before the emergence of human
beings (understood as transcendental subjects).”
And Meillassoux’s reasoning here was again thoroughly Priestian. The transcendental subject’s
Finitude is motivated by taking into account the manner in which the transcendental subject is
bounded. But we can only determinately refer to such boundaries if we refer to the whole in
which the subject is situated. Thus is the transcendental subject placed as a finite being in the
absolute time line that provides the horizon of her Finitude. As argued above, it is precisely this
move that leads Meillassoux to title his book After Finitude.
Meillassoux is either here tacitly assuming the Domain Principle, or his argument against
correlationism is fallacious. For the correlationist could clearly respond to him analogously to the
way Kant responds to the mathematical antinomies. A subject’s Finitude is made sense of
relative to a greater Finitude. My n number of years on earth make sense in part because of the n
27
+ 2 years that sandwich them. But a greater finite understanding can make sense of those n + 2
years. And this can clearly be iterated to any moment in space and time.
Such a response only fails if Meillassoux intends to invoke the Domain Principle. In
talking about finite amounts of time and space, the correlationist is quantifying over all times and
places. For her claims to have determinate sense then this object, the objective universe of space
and time, must also exist and be such that we can meaningfully refer to it.
IIId. The Domain Principle and Kaplan’s Paradox
And we can now see how the Domain Principle applies to attempts to block Kaplan’s Paradox
(by denying Existence). First, note that Meillassoux can only present his own philosophy of
contingency by quantifying over possibilities. For Meillassoux, many states of affairs that the
rest of us regard as impossible are actually possible. But Priest would argue that if Meillassoux is
going to talk about possibilities he must talk about the totality of possibilities. Otherwise, as with
Priest’s example of equations where we do not specify if they range over complex or real
numbers, Meillassoux’s quantifications over possibilities lack determinate sense.
In this context it is essential to realize just how correlationist Meillassoux becomes when
he refuses to totalize the collection of possibilities. For example, in this passage he argues that
Kant wasn’t correlationist enough!
This ignorance [of whether the possible can be totalized] suffices to expose the
illegitimacy of extending aleatory reasoning beyond a totality that is already given in
experience. Since we cannot decide a priori (i.e. through the use of logical-
mathematical procedures alone) whether or not a totality of the possible exists, then
we should restrict the claims of aleatory reasoning solely to objects of experience,
28
rather than extending it - as Kant implicitly does in his objective deduction - to the
very laws that govern our universe, as if we knew that the latter necessarily belongs
to some greater Whole (Meillassoux 2008a, 105).
But Meillassoux never defends the claim that the Kantian refusal to totalize is necessary with
respect to the totality of the possible, but impermissible with respect to the totality of space-time.
His argument against correlationism only works given the impermissibility with respect to space-
time. But his argument for absolute contingency then uses the very gesture he prohibits earlier.
Again, it is absolutely unclear why the Kantian response should be licit with respect to the set of
possibilities but not licit with respect to the set of space-times, or why the Domain Principle
should be impermissible in one domain and necessary in the other. Barring further argumentation
we should conclude that one cannot simultaneously accept Meillassoux’s argument for
contingency and his critique of correlationism. Something’s got to go.22
IV. MEILLASSOUX’S CIRCLE
We conjecture that Meillassoux’s attempted defense of the law of non-contradiction in Chapter 3
of After Finitude works precisely to secure his endorsement of a neo-Kantian response to
contemporary paradoxes of totality. For if, as with Priest and Tristan Garcia, the set of all sets
22 One way out of this dilemma is to refuse to accept the Berkley/Fichte type argument that to be
is to be conceivable. If one rejects that, as Cogburn and Harman do (see footnote 6 above) then
there is arguably no need to invoke the Domain Principle in the Schellengian manner that
Meillassoux does in premise 9 of the argument for contingency. We think that this is how
Harman manages to be a consistent anti-Kantian philosopher of finitude. One way to think of our
argument in this paper is to see it as a demonstration that Meillassoux occupies an unstable
position between Harman’s finitude and Priest’s infinitude.
29
that do not contain themselves (the Russell set) is allowed to be an inconsistent totality, then
paradoxes of totality homologous to Russell’s Paradox present no problem with our ability to
meaningfully talk about objects such as the set of all possible worlds.23
But unfortunately, Meillassoux’s criticism of contradictory entities such as the Russell set
already presupposes the conclusion of his argument to contingency:
Here is the first thesis: a contradictory entity is absolutely impossible, because if an
entity were contradictory, it would be necessary. But a necessary entity is absolutely
impossible; consequently so too is a contradiction (Meillassoux 2008a, 67).
Meillassoux holds that since a contradiction entails every proposition it would make every
proposition necessarily true. We needn’t evaluate this argument (which a defender of
paraconsistency such as Priest would take to be invalid) though, since it is a non-sequitur. Even
if we grant the italicized part, this is only a problem if one has already established the necessity
of contingency. But in the very argument for the necessity of contingency, Meillassoux assumes
that the set of all possible worlds must be prohibited because it is a contradictory entity.
Not only is this a small circle, but the critique of correlationism at least prima facie
suggests that Meillassoux should break out of it by following Priest’s view of Russell’s Paradox.
We must consider totalities both as limited and as transcending those very limits. The neo-
Kantian, correlationist response is to try to further limit the reach of human understanding to
achieve a (if Priest is correct) chimerical consistency. The anti-correlationist Meillassoux should
23 To be clear, Tristan Garcia (2014) affirms both true contradictions and a neo-Kantian response
to Russell’s Paradox. But his problem with the Russell set is not that it is contradictory, but that
it is a member of itself (and hence in-itself, “compact” in Garcia’s terminology). For Garcia,
metaphysics is importantly prior to logic.
30
side with Priest, Hegel, and Derrida here. But Meillassoux as the philosopher of contingency
cannot seem to join with them, held back by the chimera of a correlationist contingency.
One moved by these concerns might wonder if Meillassoux the philosopher of
contingency really needs the argument for contingency. Perhaps there is an alternate reading,
where Meillassoux’s speculative moment simply is a version of Schelling’s 1803 declaration in
Ideas for a Philosophy of Nature: “I am nature.” For it is this assertion that licenses the
speculative externalization of phenomenology’s fruits to nature itself. The Schellingian move is a
very common trope in recent continental metaphysics. For example, a representative passage of
Tristan Garcia’s reads, “But the fact that things are present for me, or for us, is enough to assume
that presence is in the universe. Since I am a part of the universe, the fact that presence is for me
sufficiently demonstrates that presence is for the universe (Garcia 2014, 168).” Badiou (2006)
makes similar Schellingian claims, and Hamilton-Grant (2008) has produced an extended
meditation on Schellingian metaphysics. Graham Harman’s (2011b) metaphysical system is to
some extent the result of externalizing major theses in Heidegger and Husserl. Perhaps
Meillassoux could attempt something analogous with respect to a Sartrean phenomenology of
contingency.
The authors cited above would have no problems with this. We should be clear that no
one involved in these debates thinks that this kind of Schellingian inference (which should be
distinguished from the Priest’s Domain Principle, which is arguably Schelling’s other key
inference) is always deductively valid. Rather, recent speculative philosophers in the continental
tradition take instances of the inference to present enough prima facie evidence for us to proceed
to evaluate the metaphysics in question. But in a widely publicized lecture Meillassoux (2012)
actually critiques his fellow continental metaphysicians for being “subjectualists” for the very
31
reason that they proceed via the Schellingian inference. Unfortunately then, reading Meillassoux
as a more complete Schellingian saves him from one tu quoque at the cost of another.24
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