1

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).

3

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

4

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.

6

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).

7

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

9

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).

10

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

12

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

References:

Alber, Jan, Stefan Iversen, Henrik Skov Nielsen, and Brian Richardson. 2010. “Unnatural

Narratives: Beyond Mimetic Models.” NARRATIVE 18.2: 113-36.

Askin, Ridvan. 2013. Narrative and Becoming: Differential Narratology. Ph.D. Dissertation.

University of Basel.

Badiou, Alain. 2006. Being and Event. Translated by Oliver Feltham. London: Continuum.

Berkeley, George. 1988. Principles of Human Knowledge and Three Dialogues Between Hylas

and Philanous. New York: Penguin Classics.

Bryant, Levi, Nick Srnicek and Graham Harman, ed. 2011. The Speculative Turn: Continental

Materialism and Realism. re.press.

Bueno, Otávio, Christopher Menzel, and Edward Zalta. “Worlds and Propositions Set Free.”

Erkenntnis, 165 (2013). Accessed March 23, 2014. doi: 10.1007/s10670-013-9565.

24 We would like to thank Paul Livingston, whose book The Politics of Logic: Badiou,

Wittgenstein, and the Consequences of Formalism makes a clear distinction between

philosophers of consistent plurality and philosophers of inconsistent totalities. Our entire

argument comes down to the conclusion that Meillassoux is in an unstable position between the

two. Interestingly, a striking recent result from Luca Incurvati and Julien Murzi (forthcoming in

Mind) verifies Livingston’s distinction. Incurvati and Murzi show that any maximally consistent

subset of naïve set theory is complete and unaxiomatizable. It’s hard to imagine a stronger

demonstration of Livingston’s contention that the moral of Russell’s Paradox is that we face a

decision between an inconsistent totality or consistent pluralities.

32

Carnap, Rudolph. 1950. “Empiricism, Semantics, and Ontology.” Revue Internationale de

Philosophie 4: 20-40.

Cogburn, Jon. 2005. “The Logic of Logical Revision: Formalizing Dummett’s Argument.”

Australasian Journal of Philosophy 83.1: 15-32.

Cogburn, Jon. 2011a. “Moore’s Paradox as an Argument Against Anti-Realism.” In The

Realism-Antirealism Debate in the Age of Alternative Logic. Edited by Shahid Rahman,

Giuseppe Primiero, and Mathieu Marion. Heidelberg: Springer.

Cogburn, Jon. 2011b. “Fitch style proper natural deduction formulation of T,S4, and S5.”

http://www.newappsblog.com/2011/07/fitch-style-proper-natural-deduction-formulation

of-ts4-and-s5.html (accessed 3/18/2014).

Cogburn, Jon. 2012. “(modal logical) Meditations on the Meillassoux versus Harman.”

http://drjon.typepad.com/jon_cogburns_blog/2012/01/more-modal-logic-and-

meillassoux.html (accessed 4/10/2014).

Cogburn, Jon and Mark Silcox. 2005. “Computing Machinery and Emergence.” Minds and

Machines 15.1: 73-89.

Cogburn, Jon and Mark Silcox. 2011. “The Emergence of Emergence: Computability and

Ontology.”American Philosophical Quarterly 48.1: 63-74.

Davies, Martin. 1981. Meaning, Quantication, Necessity: Themes in Philosophical Logic.

London: Routledge and Kegan Paul.

Delanda, Manuel. 2000. A Thousand Years of Non-Linear History. Cambridge: The MIT Press.

Delanda, Manuel. 2003. Intensive Science and Virtual Philosophy. London: Bloomsbury.

Divers, John. 2002. Possible Worlds. London: Routledge.

Dummett, Michael. 1978. ‘The Reality of the Past.’ Reprinted in his Truth and Other Enigmas.

33

London: Duckworth.

Field, Hartry. 2008. Saving Truth From Paradox. Oxford: Oxford University Press.

Förster, E. 2012. The Twenty-Five Years of Philosophy: A Systematic Reconstruction.

Cambridge: Harvard University Press.

Forrest, P. & Armstrong, D.M. 1984. An argument against David Lewis’ theory of possible

worlds. Australasian Journal of Philosophy 62:2: 164-168.

Gabriel, Markus. 2013a. Transcendental Ontology. New York: Continuum.

Gabriel, Markus. 2013b. Warum es die Welt nicht gibt. Berlin: Ullstein Verlag GmbH.

Garcia, Tristan. 2014. Form and Object. Translated by Mark Ohm and Jon Cogburn. Edinburgh:

Edinburgh University Press.

Hamilton-Grant, Iain. 2008. Philosophies of Nature after Schelling. London: Bloomsbury

Academic.

Harman, Graham. 2011a. Quentin Meillassoux: Philosophy in the Making. Edinburgh:

Edinburgh University Press.

Harman, Graham. 2011b. The Quadruple Object. Zero Books.

Heidegger, Martin. 2010. Being and Time. Translated by Joan Stambaugh and Dennis Schmidt.

Albany: State University of New York Press.

Hiddleston, Eric. 2005. “A Causal Theory of Counterfactuals.” Noûs 39:4, 632-657.

Incurvati, Luca and Julian Murzi. Forthcoming. “Maximally Consistent Sets of Instances of

Naïve Comprehension.” Mind.

Jacobi, F.H. 2009. The Main Philosophical Writings and the Novel Allwill. Translated by George

D. Giovanni. Montreal: McGill-Queens.

Kaplan, David. 1995, “A Problem in Possible World Semantics.” In W. Sinnott-Armstrong, D.

34

Raffman, and N. Asher (eds.) Modality, Morality and Belief: Essays in Honor of Ruth

Bqarcan Marcus. Cambridge: Cambridge University Press, 41-52.

Livingston, Paul. 2012. The Politics of Logic: Badiou, Wittgenstein, and the Consequences of

Formalism. London: Routledge.

Meillassoux, Quentin. 2008a. After Finitude. Translated by Ray Brassier. New York:

Continuum.

Meillassoux, Quentin. 2008b. Time without Becoming. Middlesex: Middlesex University.

Meillassoux, Quentin. 2012. “Iteration, Reiteration, Repetition: A Speculative Analysis of the

Meaningless Sign.” http://oursecretblog.com/txt/QMpaperApr12.pdf (accessed 3/19/24).

Miller, Alexander. 2008. The Philosophy of Language. Montreal: McGill-Queen’s University

Press.

Oksanen, Mika. 1999. "The Russell-Kaplan Paradox and Other Modal Paradoxes: A New

Solution." Nordic Journal of Philosophical Logic 4:1, 73-93

Priest, Graham. 2002. Beyond the Limits of Thought. Oxford: Oxford University Press.

Priest, Graham. 2005. Towards Non-Being: The Logic and Metaphysics of Intentionality.

Oxford: Oxford University Press.

Roland, Jeffrey. 2009. “A Euthyphronic Problem for Kitcher’s Epistemology of Science,”

Southern Journal of Philosophy 47, 205–23.

Sartre, Jean Paul. 2007. Nausea. Translated by Lloyd Alexander. New York: New Directions.

Schelling, Friedrich. 1993. System of Transcendental Idealism. Translated by Peter Heath.

Richmond: University of Virginia Press.

Schelling, Friedrich. 1989. Ideas for a Philosophy of Nature. Translated by Errol Harris, Peter

Heath, and Robert Stern. Cambridge: Cambridge University Press.

35

Simpson, Alex. 1994. The Proof Theory and Semantics of Intuitionist Modal Logic. Ph.D.

Thesis. http://homepages.inf.ed.ac.uk/als/Research/thesis.pdf (accessed 3/18/2014).

Wright, Crispin. 1994. Truth and Objectivity. Cambridge: Harvard University Press.