Table of contents
Introduction 3
Part I Creating the scene 16
Chapter 1: Kant and Hegel 16
§1.1: A case study: Kant’s category of ‘limitation’ and his ‘infinite’ judgment 16
§1.2: Kant and Hegel on the antinomic structure of reason or knowledge 22
Chapter 2: Basic concepts of Hegel’s philosophy 28
§2.1: The ‘in itself’, ‘for itself’, and ‘in and for itself’ 28
§2.2: Contradiction in the Logic 31
Chapter 3: Reinforcing the scene 35
§3.1 The Third Man Argument in Plato’s Parmenides, which, in a significant way, anticipates Kant’s problem of the ‘synthesis of the empirical’ and Hegel’s dialectical speculation 35
§3.2: Tarski and ‘the semantic conception of truth’: its systematic vagueness, a metaphor for ‘semantically closed’ and ‘semantically open’, and on a ‘transcendental pre-understanding of language’ 38
Part II: Dialectical speculation and the logical paradoxes of self-reference 44
Chapter 1: The logical paradoxes of self-reference 44
Introduction 44
1.1: The Liar paradox (representing the semantic paradoxes of self-reference) 46
1.2: The Russell-Cantor paradox (representing the set-theoretic paradoxes of self-reference) 46
Chapter 2: Dialectical speculation 51 2.1 The Liar paradox considered from a dialectical-speculative point of view 51 2.2 Considering the Russell-Cantor paradox from a dialectical-speculative point of view 58
Chapter 3: Concluding 63
Bibliography 69
What are paradoxes of self-reference? It seems that there are powerful reasons in favor of the view that they aren’t philosophically interesting at all. The semantic paradoxes of self-reference, – for example, – culminate in an endless series of semantic terms such as ‘true’, ‘false’, ‘true’, false’, ‘true’ …, as in the simple Liar paradox: ‘This sentence is false’. As such, there is little excitement about them – as a flip-flop toy, they are easily outdone by pinball. At the same time, they are widely considered to be violating the law of non-contradiction (from now on, abbreviated to LNC): the principle saying that for any A, it is impossible for both A and not A1 (or ¬A2, as is written in contemporary formal logic) to be true3, – or for A and not A to be true ‘in the same sense – or relation – at the same time’4. Aristotle is the first Western philosopher whom we know of to have given a defence of the LNC, – and to assure the world that it is “the most certain of all principles”5. In Book Γ of his Metaphysics, he states:
It is impossible that the same thing can at the same time both belong and not belong to the same object and in the same respect, and all other specifications that might be made, let them be added to meet local objections6.
The way in which Aristotle articulates the LNC suggests that he thinks that for any object – for example a horse – it is impossible to have both A and not A as properties ‘in the same
1 I prefer not to write not-A, since, as Kant has pointed out, a judgment such as ‘Socrates is non-mortal’, means that Socrates belongs to the set of all things that are non-mortal, so that one does not actually negate the
property in question, but make an affirmation. In making such an ‘infinite’ judgment, our understanding does
not make much progress in determining the meaning of the name ‘Socrates’, as Socrates is only taken to be one among the infinite things that are non-mortal; our knowledge of this name is not really increased by expressing it. Accordingly, Kant also distinguishes between the concept of ‘negation’, which occurs in a ‘negative
judgment’, like ‘Socrates is-not mortal’, in which our knowledge of the subject is increased or its meaning is
determined, and the concept of ‘limitation’, which leaves the meaning of ‘Socrates’ undetermined. In our discussion of Kant’s categories, we will be focusing on this category of limitation. In a ‘negative’ judgment, the negation affects the copula, while in an ‘infinite’ judgment, the negation affects the predicate. I hope that it is now clear why I don’t use ‘not-A’, namely, since this could be taken to express Kant’s concept of limitation, rather than that of negation, – so that I will be working throughout with ‘not A’.
2 I assume that the negation of A in standard formal logic, ¬A, is intended to mean that which Kant calls ‘negation’, not that which he calls ‘limitation’. That this is so, I also have experienced in my elementary logic and semantics courses. This ‘experience’ does not count as a proof – and why am I entitled to say that I’ve had a certain ‘experience’? – but I figure it is better than saying nothing at all about it.
3 See Berto, F., ‘Dialetheism’ (2013). In: Stanford Encyclopedia of Philosophy, URL: http://plato.stanford.edu/entries/dialetheism/: see the introduction.
4 See Aristotle, Metaphysics, translated from ancient Greek to English by Hugh Tredennick (1933), appearing in the series of Loeb Classical Library (Cambridge, MA: Harvard University Press, 2013. Original title: unknown. Written around 350 BCE): Book Γ, 1005b, 17-24.
5 See ibid., 1005b, 24. 6 See ibid., 1005b, 19-20.
respect at the same time’. We should keep in mind that Aristotle was probably the first full-fedged biologist, and that his biological research informed his interpretation of the LNC. Nevertheless, one could say that, with this, Aristotle already proclaims in many – if not all – cases, for any sentence A – or for any proposition or statement A –, it is impossible for both A and not A to be true in the same sense at the same time. In the case of a horse – for example, – Aristotle would state that a horse cannot both have four legs and not have four legs in the same respect at the same time. In this we may read that, for Aristotle, the sentence ‘the horse has four legs’ and its negation or opposite, ‘the horse does not have four legs’, are mutually exclusive, – so that we are entitled to say that Aristotle stated that for any sentence A and its negation or opposite, not A, it is impossible for both to be true in the same sense at the same time.
It seems that, since Aristotle’s defence of the LNC, very little efforts have been made to defend it once again; we do find one in a 1764 philosophical writing by Thomas Reid, in which it is said “[t]hat no proposition can be both true and false at the same time”7, – Reid states it here as a ‘principle of common sense’. It seems that, since Aristotle’s defence of it, the LNC has been considered to be a self-evident truth – belonging to ‘common sense’, – not standing in any substantial need of additional defence. But, as paradoxes of self-reference are considered to be violating the LNC, and many of them have been known at least since antiquity (Eubulides of Miletus (4th century BCE) already brought up the Liar as well as a looped Liar, which we will hereby introduce: ‘The next sentence is true. The previous sentence is false’), it is strange (and this is an understatement) that the LNC has enjoyed this lofty status all the way through to around the turn of the 20th century AD. We will have to push this interesting as well as important question aside, and await plausible replies on it by
historians of ideas.
As we just already hinted at, the awareness that one had to find a so-called ‘non-trivial’ way of dealing with the challenge that paradoxes of self-reference pose to the LNC reached a climax at the beginning of the 20th century AD. New paradoxes of self-reference – set-theoretic ones – had entered the scene. The situation became in particular acute when people like Bertrand Russell and A. N. Whitehead, proposed – inspired by the work of Gottlob Frege, – that some or all of mathematics could be reduced to formal logic8. From this period
7 Cf. Reid, T., An Inquiry into the Human Mind on the Principles of Common Sense (1764), Chapter VI, statement 2. In: Thomas Reid: Selected Philosophical Writings, ed. by Giovanni B. Grandi (Upton Pyne, Exeter, UK: Imprint Academic, 2012).
8 See Tennant, N., ‘Logicism and Neologicism’ (2013). In: Stanford Encyclopedia of Philosophy, URL: http://plato.stanford.edu/entries/logicism/, see the introduction.
onwards, paradoxes of self-reference have foisted a serious problem on logicians and philosophers. The most famous example of a set-theoretic paradox of self-reference is the Russell-Cantor paradox, which runs as follows. Consider the set of all sets that do not contain themselves as a member, R. Does R contain itself or not? If R is a member of itself, then by definition it is not a member of itself; but if R is not a member of itself, then by definition, it should contain itself, – for it is the set containing all sets that do not contain themselves as a member. So that it contains itself insofar as it does not contain itself, and does not contain
itself, insofar as it does contain itself9.
One radical (re)action to the issue in question, in particular addressed at the semantic paradoxes of self-reference (it seems), came from logician Alfred Tarski (1901-1983). He argued that semantic paradoxes of self-reference need to be avoided:
In my judgment, it would be quite wrong and dangerous from the standpoint of scientific progress to depreciate the importance of this [that is, the Liar paradox] and other antinomies, and to treat them as jokes or sophistries. It is a fact that we are here in the presence of an absurdity, that we have been compelled to assert a false sentence (since … [the Liar], as an equivalence between two contradictory sentences, is
necessarily false)10.
So, according to Tarski: (1) semantic paradoxes of self-reference, and contradictions in general, are false – not just false, but necessarily false; (2) as such, they are a threat to science or, more broadly we might say, to ‘intellectual life’11. Tarski went on to identify the premises upon which semantic paradoxes of self-reference (such as the Liar) operate, in order to reject at least one of them. The problem, according to Tarski, is that the language in question is semantically closed: it contains semantic terms such as the term “true” referring to sentences of itself; furthermore, all sentences which determine the adequate usage of this
term can be asserted in the language12.
Tarski’s conclusion is that languages may not contain ‘is true’ as a regular predicate, and may not contain a theory of truth of themselves13. By introducing the distinction between object language and meta-language, Tarski makes sure that the Liar, for example, cannot be
9 See Irvine, A. D., ‘Russell’s Paradox’ (2014). In: Stanford Encyclopedia of Philosophy, URL: http://plato.stanford.edu/entries/russell-paradox/, see the introduction.
10 Cf. Tarski, A., ‘The Semantic Conception of Truth and the Foundations of Semantics’. In: Philosophy and
Phenomenological Research 4 (3), pp. 341-376 (1944): section 7 (the insertions are mine).
11 One should take the phrase ‘intellectual life’ with a grain of salt: we do not wish to use a concept informed by all kinds of biologistic or vitalistic considerations which we don’t want to invoke. Having said that, it is hopefully clear in what direction we are pointing here.
12 See ibid., section 8. 13 See ibid., section 8.
asserted any longer, and he also arranges that a theory of truth in a particular object language L (determining what counts as a true sentence in L) is to be given in a meta-language M, containing (copies of) all sentences of L14. So, in addition to proclaiming that paradoxes of self-reference are necessarily false, a threat to science, and that they – as such – need to be avoided, Tarski also did the job of showing how they can be avoided. Although Tarski only addresses the semantic paradoxes of self-reference in ‘The Semantic Conception of Truth and the Foundations of Semantics’, it seems proper to assume that, for Tarski, the set-theoretic paradoxes of self-reference have to avoided somehow as well, that is, “from the standpoint of scientific progress”, as contradictions in general would be necessarily false.
Should philosophy be happy with this maneuver?
In a different philosophical tradition than the one in which, for example, Russell was active – in what would nowadays be called ‘continental philosophy’, – there has been a thorough debate on how contradiction and self-reference figure in the work of G. W. F. Hegel (1770-1831). The central theme of his philosophy, I contend, is thought thinking itself – so that it is a thought that is referential and, thereby, somehow, contradictory or self-undermining, I contend. Actually, this is not only my position, it is the position of Graham Priest (1948) as well15.
In this thesis, we will focus on this – at least, what I take to be, – basic feature of Hegel’s philosophy. From now on, we will refer to this as ‘Hegel’s paradox of self-reference’ – abbreviated to Hpsr, – and we consider its meaning to be purely dialectical-speculative. According to some, Hegel was a dialetheist16. Dialetheism is the view that there are true contradictions. This is not quite the same as saying that all contradictions are true17 - whereas in this statement Hegel certainly seems to suggest just that. In dialetheism, the LNC is not altogether rejected by the prevailing versions of dialetheism; rather, it is considered not to hold unrestrictedly. That is to say, a dialetheist accepts the LNC, but incorporates certain sentences in his theory that are inconsistent with it, that is, true sentences whose negations are true as well18. Dialetheism’s revival in the 20th century could be seen as a repudiation of the Tarskian program. Its analytic philosophical adherents have tried to embed it in an overall formal theory, using a paraconsistent logic. This logic assumes that the logical consequence
14 See ibid., section 9.
15 See Priest, G., ‘Dialectic and Dialetheic’. In: Science & Society, 53-4 (Winter, 1989/1990), pp. 388-415: p. 388 ff.
16 See Berto, F., ‘Dialetheism’, section 2.
17 Usually, the view that all contradictions are true is characterized as trivialism, – but it can hardly be called that with respect to the motivational force that underlies it, which will hopefully become clear in this thesis. 18 See ibid., introduction.
relation ‘isn’t explosive’, or – in more ordinary language, – that it isn’t the case that anything
follows from a contradiction19.
Obviously, we will be focusing on instances of both contradiction and self-reference. As holds for virtually every subject matter, some distinction is called for.
Re-introducing our main counterpart in this thesis, Graham Priest20 – one of the main figures in contemporary dialetheism, – points out that one has to differentiate between ‘logical’ and ‘dialectical’ contradictions (from now on, we leave out the punctuation marks). As Priest points out, in a logical contradiction, of the form A and not A or ¬A, one wouldn’t find an essential connection between the contradictory conjuncts, that is, its contradictory conjuncts do not somehow depend upon one another for their very existence, – we may also say that they do not presuppose and “reciprocally condition” one another21. In the case of a dialectical contradiction, however, this is precisely what seems to be going on, so that there is an intimate relation between its contradictory conjuncts. Priest speaks respectively of an external or accidental conjunction and an internal or essential conjunction22. So it would be a mistake to try to grasp a dialectical contradiction by using the formal logical operator of mere external or extensional conjunction23.
It seems that Priest accordingly distinguishes between logical paradoxes of self-reference and dialectical paradoxes of self-reference, – in any case, this is what we will do here, as could already be seen, having characterized Hpsr as purely dialectical-speculative, where we, as is obvious, have added ‘speculative’ to ‘dialectical’. That we call Hpsr purely dialectical-speculative, implies that we think that there are also paradoxes of self-reference that are not purely dialectical-speculative – this is actually what we take both the semantic and set-theoretic paradoxes of self-reference to be.
Another move made by Priest that I applaud, is his statement that dialectical contradictions are instances of the Hegelian notion of ‘identity in difference’, (which Priest
19 See ibid., introduction; section 1. See also Priest, G., Tanaka, K., and Weber, Z., ‘Paraconsistent Logic’ (2013). In: Stanford Encyclopedia of Philosophy, URL: http://plato.stanford.edu/entries/logic-paraconsistent/:
see the introduction.
20 So our chief counterpart will not be Tarski. We claim that his ambitions were logical and semantic in character, as well as pragmatic with respect to wanting to contribute to the progress of science – accordingly, we consider Tarski’s work, although, obviously, closely related to philosophy, not to be philosophical in character. This also means that we do not criticize Tarski himself for his development of a ‘semantic conception of truth’ – for his own standards we consider to have been mathematical logical and scientific in nature, not philosophical. Priest’s work is obviously philosophical in character, so that we are entitled to criticize his work. 21 See Priest, G., ‘Dialectic and Dialetheic’. In: Science & Society, 53-4 (Winter, 1989/1990), pp. 388-415: pp. 396-7. The term ‘reciprocally conditioning’ we derive from a fragment of a text quoted by Priest, namely one by Gustav A. Wetter, which seems to be his work, Der dialektische Materialismus (München: Pustet, 1958). 22 See ibid., pp. 396-7
labels as: (a = b) & (a ≠ b)24), – we will discuss this notion later on. In this respect, Priest also speaks of an ‘identity of opposites’, in which the opposites would go over into each other – Hegel speaks also of a “transmitting” (“Ubersetzen”) of these opposites to one another in his section on contradiction in the Logic25.
Finally, we are inclined to believe that the boundary between semantic and set-theoretic paradoxes of self-reference is a shifting one, so that, at some point, the distinction between them becomes arbitrary26. In this thesis, we will stick to this distinction.
But, although we definitely share certain views with Priest, we contend that the semantic and set-theoretic paradoxes of self-reference ought – from a Hegelian point of view, – be given a different treatment than Priest gives them.
According to Hegel, dialectics arises, or, we may say, ‘dialectical reason’ (Hegel also speaks about ‘the dialectical’ as the ‘negative-rational’: “… [das] dialektische oder negativ-vernünftige …”27) is active, when reflective understanding (“der reflektierende Verstand”28) is compelled to go beyond her separating determinations and obtain them29. Hegel claims that on the vantage point of this obtaining, the conflict of the separating determinations emerges30. It is reason (“Vernunft”) that carries out this obtaining of reflection31. The next step would be gaining insight into this conflict of separating determinations, which is only achieved by a raising beyond these determinations – it is this that would be “the great negative step to truthful concepts of reason”32. These ‘truthful concepts of reason’ are arrived at by speculation (the by speculative reason grasping the unity of the separating determinations in their opposition, that is, the affirmative or positive (Hegel speaks about the speculative, on its turn, as the positive-rational: “ … [das] spekulative oder
positiv-24 See ibid., pp. 410-412. We will go into this further in discussing Hegel’s notion of contradiction and our reading of the paradoxes of self-reference bears on this discussion.
25 See Hegel, G. W. F., Wissenschaft der Logik, Bd. II, p. 51 26 See Priest, G., Beyond the Limits of Thought, p. 142
27 See Hegel, G. W. F., Enzyklopädie der philosophischen Wissenschaften, §79 28 See Hegel, G. W. F., Wissenschaft der Logik (Erster Teil), p. 26
29 See ibid., p. 26; see Hegel, G. W. F., Enzyklopädie der philosophischen Wissenschaften, §81
30 See Hegel, G. W. F., Wissenschaft der Logik (Erster Teil), p. 26; see Hegel, G. W. F., Enzyklopädie der
philosophischen Wissenschaften, §81
31 See Hegel, G. W. F., Wissenschaft der Logik (Erster Teil), p. 26; see Hegel, G. W. F., Enzyklopädie der
philosophischen Wissenschaften, §81
32 See Hegel, G. W. F., Wissenschaft der Logik (Erster Teil), p. 26; see Hegel, G. W. F., Enzyklopädie der
vernünftige”33) contained in their resolution and transition34. As to what this positivity may precisely amount to, we hope to find this out later on.
Now, I contend that something different but analogous to Plato’s example of the activity of the geometer and arithmetician, on the one hand, and that of the philosopher, on the other, is happening within the contemporary debate on paradoxes of self-reference: too much ‘thought’ is involved, too little ‘understanding’ is reached35. There is a danger in considering semantic and set-theoretic paradoxes of self-reference as mathematical objects ,– since, as such, they may be reified to such an extent that their reflexive character gets lost. I suspect that this is going on in the contemporary revival of dialetheism, in which, today, Priest plays a decisive role.
The clue to grasping the intended meaning of ‘understanding’, is to be sought in the ‘fact’ that Hegel’s philosophical system is not just formal, but also substantive in character – this is what Theodor W. Adorno (1903-1969) makes very clear in the introduction to his Negative Dialectics:
Through Hegel, philosophy had regained the right and the capacity to think substantively instead of being
put off with the analysis of cognitive forms that were empty and, in an emphatic sense, null and void.
Where present philosophy deals with anything at all, it lapses either into the randomness of a weltanschauung or into that formalism, that “matter of indifference”, against which Hegel had arisen … The fundament and result of Hegel’s substantive philosophizing was the primacy of the subject, or – in the famous phrase from the Introduction to his Logic – the “identity of identity and non-identity.”36
I contend that Hegel’s dialectical speculation is a promising way to preserve the semantic and set-theoretic paradoxes of self-reference, since, first of all, the central theme of Hegel’s own philosophy, I contend, is thought thinking itself – so that it is a thought that is self-referential and, thereby, somehow, contradictory or self-undermining, what I call Hpsr. The semantic and set-theoretic paradoxes of self-reference, I contend, are not purely dialectical-speculative, that is, they are to be formalized as well as be given a dialectical-speculative treatment – they are something ‘in between’ the logical and dialectical-speculative.
33 See ibid., §79
34 See Hegel, G. W. F., Wissenschaft der Logik (Erster Teil), p. 26; see Hegel, G. W. F., Enzyklopädie der
philosophischen Wissenschaften, §82
35 See Plato, Republic (Books 6-10), translated from ancient Greek to English by Chris Emlyn-Jones and William Preddy, appearing in the series of Loeb Classical Library (Cambridge, MA: Harvard University Press, 2013. Original title: Politeia, written around 360 BCE): 510-511
36 See Adorno, T. W., Negative Dialectics, trans. E. B. Ashton (New York: The Continuum Publishing Company, 1973. Original title: Negative Dialektik. Frankfurt am Main: Suhrkamp Verlag, 1966), p. 7. Italics are mine.
The starting point of this thesis, is that we contend that reading the paradoxes of self-reference (whether they are semantic or set-theoretic) and understanding them, has, potentially, significant philosophical gains, that is: they prepare us somehow for engaging in Hspr, the thinking of thinking itself. In addition, we contend that Priest does not do justice to this philosophical potential of the paradoxes of self-reference. However, things are not as clean-cut as is suggested by this: Priest seems not only to distinguish between what he calls dialectical contradictions and logical contradictions; he also seems to give the paradoxes of self-reference a dialectical-speculative treatment, in any case, at least one of the set-theoretic ones, – what he calls ‘the Fifth Antinomy’. So, the following is a legitimate question at my address: why are you challenging Priest’s treatment of the paradoxes of self-reference, if it seems dialectical-speculative in character, and, accordingly, not to be reifying at all?
First of all, I contend that Priest does not sufficiently acknowledge the nature of the relationship between consciousness and the Absolute. This is what makes his interpretation of dialectics and speculation, from a Hegelian point of view, problematic from the start. There is something at stake in this relationship, namely, the development of the Absolute: somehow, this development is dependent on what consciousness does or, more appropriately, on what it thinks or in what kind of thinking it is engaged. “Knowing-oneself-purely-in-being-absolutely-other” (“Das reine Selbsterkennen im absoluten Anderssein”)37: this is, I contend, how Hegel characterizes the relation between consciousness and the Absolute in the Preface to the Phenomenology. Hegel’s idea is, I suggest, that one can know oneself in relating oneself to the Absolute in a certain way, which consists in the knowledge that one is absolutely other than the Absolute, – the idea is, then, that the development of the Absolute is somehow dependent upon this (self-)knowledge. We will elaborate on this phrase more in what follows, that is, on what the character of this relationship between consciousness and the Absolute is; moreover, we will elaborate in what follows also on what it means that the Absolute is in development, – which at this point may sound rather counter-intuitive.
Second, I contend that, according to Hegel, there is, as Plato points out in the fragment of the Republic we just referred to, a difference between ‘thought’ and ‘understanding’ or – if we roughly translate this into Hegel’s vocabulary, between ‘understanding’ (“Verstand”) and reason (“Vernunft”), the ‘powers’ of which, I contend, from a Hegelian point of view, Priest uses inappropriately, – I can also say ‘eclectically’ – in order to characterize the paradoxes of
self-reference.
37 See Hegel, G. W. F., Phänomenologie des Geistes, p. 29. This translation is mine; Hegel’s own statement is put in italics by me as well.
This move by Priest in the attempt to understand the paradoxes of self-reference seems promising from a dialectical-speculative point of view. However, not long after Priest seems to provide the Fifth Antinomy with a dialectical-speculative treatment, he reduces the set-theoretic paradoxes of self-reference – Russell-Cantor’s, Burali-Forti’s, Mirimanoff’s, and the Fifth Antinomy, – to a formal logical structure, that which he calls ‘Russell’s schema’, a schema based on a formula constructed by Russell himself, that would generate a whole
family of set-theoretic paradoxes of self-reference38.
Very importantly, Priest does not say: ‘But this is not how Hegel’s notion of the qualitative true infinite is to be presented’. On the contrary, Priest is referring here to his earlier given, what seems to be, dialectical-speculative exposition of the Fifth Antinomy, in which this paradox of self-reference was explicated as an instance of the qualitative true infinite39, and now, he says that Russell’s schema is “providing a precise specification of the notion of a true infinity”40. I don’t know whether Priest is contradicting himself, or whether what just looked like a candidate for dialectical-speculative treatment of the paradoxes of self-reference, was intended to culminate in its formalization as in Russell’s schema. It is important to keep in mind, that, even if Priest is contradicting himself, and indeed wants to give the set-theoretic paradoxes of self-reference or the paradoxes of self-reference in general a dialectical-speculative treatment, he is losing touch with Hegel in that he does not take the relationship between consciousness and the Absolute, that is at the center of both the Phenomenology of Spirit and the earlier written The Difference Between Fichte’s and Schelling’s System of Philosophy – to which we will refer throughout as the Differenzschrift, – as a starting point for gaining understanding the paradoxes of self-reference. Although Priest does not need this in particular – he is perfectly justified in developing his own point of view, – the reflexivity and dynamic that are involved in this relationship are, I conted, very important for gaining what we call ‘understanding’ of the paradoxes of self-reference.
As Priest is not the only one to have pointed out that the LNC doesn’t hold unrestrictedly, I certainly am not the only one who is suspicious about the formal logical treatment of the putatively non-dialectical, non-speculative, formal logical paradoxes of self-reference. For example, Arend Kulenkampff (born 1936 – (?), (former?) professor in Philosophy at Goethe University of Frankfurt) rejects Tarski’s evasion of the antinomies such as the Liar by using the distinction between object language and metalanguage, and he does so too by appealing
38 See Priest, G., Beyond the Limits of Thought, p. 131 39 See ibid., p. 131
to Hegel. I would like to recognize, at this point, the significance of Kulenkampff’s work. His little work on contradiction is the closest of all I have read to what I’m going to do in this thesis, and I will use this work as my guide –, with my own additions to it, to be sure.
In the Introduction to the Phenomenology, Hegel says that the way to truth is a way of doubt (“Zweifel”) and of despair (“Verzweiflung”)41, – in this work, Hegel gives a presentation of how natural or naïve consciousness can elevate itself to the point of view of what Hegel calls ‘science’ (“Wissenschaft”42, or “das Wissen im allgemeinen”43). This involves a development of shapes of consciousness, that has a dialectical structure which consists in moments of self-contradiction, in that each shape of consciousness is implicitly committed to the point of view of Hegel’s system, that of ‘absolute idealism’, in which consciousness, according to Hegel, elevates itself to the perspective of the Absolute, that is: it makes itself into its own object44. Hegel speaks about the series of shapes through which consciousness must go through in order to, finally, reach ‘Absolute Knowledge’ (“das absolute Wissen”45), as the experience of itself46. The gradual development of Spirit is what makes Spirit historical, as shapes of consciousness will become conscious of their limited truth, and will accordingly be subject to “determinate negation”, the ‘negative of the negative’, by means of which the contradiction between a shape of consciousness and the Absolute is ‘sublated’ (‘aufgehoben’) into a new shape of consciousness – this contradictoriness is what the dialectic primarily consists in. This ‘sublation’ or ‘Aufhebung’, Hegel explains as being both an abolishment and preservation of the concept, natural or spiritual phenomenon, or shape of consciousness in question: basically, through this process, a new category, a new natural phenomenon, a new spiritual phenomenon, or a new shape of consciousness emerges47.
I indeed distinguish between the Absolute and Absolute Spirit or Absolute Knowledge – for this is precisely that which consciousness helps to develop by engaging in the thinking of thinking. But before consciousness reaches this point, it has to go through these different series of shapes of consciousness – such as ‘sense-certainty’, ‘perception’, ‘understanding’, which Hegel distinguishes from self-consciousness, in consciousness becomes its own object. But, although this ‘self-objectification’ may seem already the point of ‘science’, at which
41 See Hegel, G. W. F., Phänomenologie des Geistes, p. 72 42 See ibid., for example pp. 29-31; p. 47
43 See ibid., p. 29
44 See Hegel, G. W. F., Differenz des Fichte’schen und Schelling’schen Systems der Philosophie, p. 40 45 See Hegel, G. W. F., Phänomenologie des Geistes, p. 81
46 See ibid., p. 72
consciousness elevates itself into the perspective of the Absolute, this can hardly be the case: for ‘reason’ (“Vernunft”) then has yet to be developed, and it is dialectical-speculative reason – of course, – that carries out dialectical speculation.
To think substantively – according to Hegel’s view – consciousness has to get in the mode of ‘knowing-oneself-purely-in-being-absolutely-other’48: only this can engender knowledge in the speculative sense49. The speculative sentence generates “a going beyond the concept, without the concept being gone beyond”50. Very significant in this regard, is Hegel’s view that the judgment is unable to express speculative truths51, which is why it must be ‘destroyed’ (“zerstört”)52 – but it is not so much a destruction as it is a transformation, in which the difference between its different moments, subject and predicate, is sublated, and grasped in its unity. According to Kulenkampff, philosophy must objectify the form of judgments that state that something is ‘such-and-such’, that is, objectify the form ‘to be so-and-so’ to something, which is on its turn ‘so-so-and-so’53. As Gilbert Ryle (1900-1976) puts it, this amounts to a “breach of logical syntax”54. Kulenkampff states that a systematic “breach of logical syntax” precisely presents a model of dialectical-speculative thinking. Ryle also brings up the question, by means of what we become conscious, that we contravene logical syntax. His answer: “The deductive derivation of absurdities and contradictions shows it, and nothing else can”55.
Kulenkampff concludes that Hegel’s philosophical system is precisely that which Tarski deemed to be so dangerous for ‘scientific progress’: its language would be ‘semantically closed’. That is, its language would contain semantic terms such as the term “true” referring to sentences of itself; furthermore, all sentences which determine the adequate usage of this term could be asserted in the language56. As we have already mentioned, Hegel states that speculative logic contains already the logic of common sense57: one can attain this logic of common sense by leaving out the dialectical and rational from speculative logic58. With bare abstractions or formal thoughts – with which formal logic or the logic of common sense is
48 See ibid., p. 29.
49 See ibid., p. 29 ff.; pp. 59-63. See Hegel, G. W. F., Wissenschaft der Logik (Erster Teil), p. 76. See also Kulenkampff, A., Antinomie und Dialektik. Zur Funktion des Widerspruchs in der Philosophie, pp. 1-6 50 See ibid., p. 1
51 See Hegel, G. W. F., Wissenschaft der Logik (Erster Teil), p. 76 52 See Hegel, G. W. F., Phänomenologie des Geistes, p. 59
53 See Kulenkampff, A., Antinomie und Dialektik. Zur Funktion des Widerspruchs in der Philosophie, p. 76 54 See ibid., p. 76
55 See ibid., p. 76. Italics are mine.
56 See Kulenkampff, A., Antinomie und Dialektik. Zur Funktion des Widerspruchs in der Philosophie, pp. 58-9 57 See Hegel, G. W. F., Enzyklopädie der philosophischen Wissenschaften, §82
involved –, Hegel states, philosophy would have nothing to do59. I conclude that, therefore, – from the point of view of Hegel –, there is something more profound to be learned about paradoxes of self-reference than a formal structure that supposedly expresses its import altogether.
So our methodical starting point in this thesis is the ‘relation of reflection to the Absolute’, Hegel’s insight that the Absolute is “essentially result, so that it is only at the end (“zu Ende”) that which it is in truth”60, and the seeming need for consciousness entering into a relation with the Absolute in order to make the Absolute into what it is in truth. The logical paradoxes of self-reference should become more intelligible than is their reduction to a formal logical structure. As in Hegel’s philosophy a thinking is central that thinks itself, so that this thinking is self-referential and thereby undermining itself, yet in a positive fashion, namely, as sublation, it will provide, I hope, for a refreshing perspective on the contradiction and self-reference that is present in the logical paradoxes of self-reference. As we already said, we consider the semantic and set-theoretic paradoxes of self-reference to be essentially different from Hpsr, namely, to be something in between the abstract, formal logical and the dialectical-speculative. In the contemporary debate on the logical paradoxes of self-reference, virtually only the former side has been given attention. Now is the time to provide an alternative, that is, to pull the logical paradoxes of self-reference out of the formal logical rust, and to present them in their reflexivity and content, – as well, we might say.
The seeming need for consciousness entering into a relation with the Absolute in order to make the Absolute into what it is in truth, Hegel does not express literally, as it is spirit (“Geist”) that is only “for us” in and for itself, and that consciousness must relate with Spirit in order to make it also in and for itself for itself, taking its self-production as its object61. But just before this, he says that it is “the absolute stated as spirit” which is “the grandest conception of the new times and its religion”62. What I take to be substantiating this, is that right after Hegel talks about this relation of consciousness to spirit, he states that it is “knowing-oneself-purely-in-being-absolutely-other” that is the foundation for science and knowledge in general63. I take this to mean that what are absolutely different from one another are consciousness on the one hand and the Absolute on the other, so that one can
59 See ibid., §82
60 See Hegel, G. W. F., Phänomenologie des Geistes, p. 24 61 See ibid., p. 28
62 See ibid., p. 28 63 See ibid., p. 29
know oneself in being absolutely other than the Absolute, and can relate oneself in this to the Absolute, such that this has impact on the development of the Absolute.
We need to make one crucial comment concerning terminology. The word ‘paradox’ – at least in my mother’s tongue Dutch – has the connotation of apparent contradiction, rather than actual contradiction. But in the literature on paradoxes or paradoxes of self-reference I read, the meaning is clearly that of actual contradiction. In any case, we assume here that the proper meaning for ‘paradox’ is indeed actual contradiction.
So the thesis is divided as follows. We begin with creating the appropriate setting for the paradoxes of self-reference to be given a dialectical-speculative treatment, so that the first part consists of the Kant-Hegel debate on contradiction and self-reference, and the exposition of the for us very important analysis of the Kant-Hegel dialectic by Arend Kulenkampff. In addition to this, we will provide some basic concepts and insights that we need developed in Hegel’s philosophy, such as that of the ‘in itself’, ‘for itself’, and the ‘in and for itself’ – focusing on the Phenomenology of Spirit, and, of course, the account Hegel gives of contradiction in the Science of Logic. Throughout, we will use passages from the Encyclopaedia of the Philosophical Sciences as well. Another important text that we use in order to show what his philosophy is about is The Difference of Fichte’s and Schelling’s System of Philosophy. We have read all of these works in the original German versions: we refer to pages in these versions, but have provided translations of the original passages both by ourselves and using translations we have encountered in reading secondary literature. As for the second part of our thesis, here we present the logical paradoxes of self-reference in full detail and will consider them from a dialectical-speculative point of view. We have chosen to limit ourselves to the Liar paradox on the semantic side, and to the Russell-Cantor paradox on the set-theoretic side. Although we cannot extrapolate the results of our discussion to the other paradoxes of self-reference just like that, we believe that our discussion has relevance for them as well. Another thing that needs to be mentioned, is that our treatment of the Liar paradox differs from that of the Russell-Cantor paradox. For both paradoxes of self-reference these dialectical-speculative treatments are appropriate, we believe. The other, however, is more profound than the other. We’ve chosen to engage in the more profound one with respect to the Russell-Cantor paradox. This is a shortcoming, but the less profound dialectical-speculative treatment still provides us with useful philosophical results. That we chose to do give them separate treatments, has much to do with the fact that these profound dialectical-speculative considerations are rather time-consuming and we were dealing with a deadline.
Part I: Creating the scene
Chapter 1: Kant and Hegel
§1.1 A case study: Kant’s category of ‘limitation’ and his ‘infinite’ judgment
In his Critique of Pure Reason, Immanuel Kant (1724-1804) provides a list of what he calls the ‘categories of pure understanding’ – there would be twelve of these, and Kant groups them into four kinds64. With these four kinds of categories there would correspond a certain condition, bringing about a specific dialectical argument. In accordance with the kind of condition that generates them, Kant calls these the ‘antinomies of pure reason’. Although you may be acquainted with Kant’s theory of knowledge, in order to grasp why Kant thinks that reason is engaged in these arguments because of them, I will very briefly expound what Kant intended the ‘categories of pure understanding’ to accomplish.
In the part of the Critique of Pure Reason called “Analytic of Concepts”, Kant engages in both what he calls a ‘metaphysical deduction’ and what he calls a ‘transcendental deduction’65. In principle, these deductions are rather hypothetical in nature, as they doesn’t state that, given such and such, this or that actually is the case. But since Kant is talking about something we all can agree on – that we (perhaps more accurately, that I) have ‘experience’ – it would be odd to state that Kant intends the deductions to stay in their
hypothetical shell.
In the ‘metaphysical deduction’, Kant intends to show that, in order to have experiences, we must have concepts that are a priori related to objects of experience, and that means that we must have a faculty of judgment as well, because we wouldn’t be able to use these
64 See Kant, I., Critique of Pure Reason, edited and translated by Paul Guyer and Allen W. Wood (Cambridge, UK: Cambridge University Press, 1998. Original title: Kritik der reinen Vernunft. A-edition first appeared in 1781 at Hartknoch (Riga), edition first appeared in 1787 at Hartknoch (Riga). For some passages we will use the B-edition, for others the A-edition.): B 95-109
65 We will definitely not discuss the reasoning pattern of both the metaphysical and transcendental deduction in a technical way, as to ascertain its (in)validity. A sketch of its general meaning and set-up will do here. That the involved strategy is both innovative and obscure, is for example proposed by Rohlf, M., ‘Immanuel Kant’. In:
concepts otherwise66. In fact, it seems that Kant deduces the a priori concepts of the understanding out of his deduction of the logical functions of the understanding in judgments – for Kant calls the understanding the ability to judge67. As these judgments are of four kinds: quantity, quality, relation, and modality68, the pure concepts of the understanding are of the same four kinds69. What Kant intends to show is that the necessity and strict generality of the rules that we take to hold in objective reality – the law of gravitation, for example, – cannot be derived from experience insofar as it is solely based on sense perception. Kant states that it is only by means of a priori concepts – and ipso facto, by the logical functions of judgment – that objects of experience can be thought at all70: so that our understanding, with the logical functions of judgment and pure concepts it contains, is the a priori condition of the
possibility of all experience.
According to Kant, the sensory data provided by the stimulation of the five senses provide us with sensory representations that are characterized by the crude and unprocessed manifolds they contain. Insofar as these sensory representations are related to objects external to the subject, Kant calls these intuitions71. In the ‘synthesis of apprehension in intuition’, the impressions of sense intuition are given a separate (spatio?-)temporal location in the mind – for every representation is an absolute unity, so that it seems that Kant thought that one can apprehend only one item at a time – and synthesized afterwards, such that they can be taken up by the imagination (“Einbildungskraft”)72. In the operation that Kant calls the ‘synthesis of reproduction in the imagination’, these sensory representations are added together, and their manifold is gone through, recorded, and associated, so to apprehend it in a unit of knowledge73. But for knowledge to arise this is not enough. Knowledge can only arise if a certain unity has been brought about in the synthesis of the manifold of intuition. Bringing unity to the synthesis of the manifold, is something that is done by the understanding by imposing concepts on it. This is what Kant calls the ‘synthesis of recognition in a concept’74. As we pointed out, only a priori concepts can impose necessity and strict generality upon the
manifold of intuition.
66 See Kant, I., Critique of Pure Reason, B 93 67 See ibid., B 94
68 See ibid., B 95 69 See ibid., B 106
70 See ibid., B 93-102; 102-109
71 These empirical intuitions arise in accordance with the pure intuitions, that is, the forms of intuition in which all empirical intuitions arise: space and time. Interestingly, Kant calls space also “outer sensibility”, and time “inner sensibility”, – but I’m afraid we’re transgressing the limits of all possible relevance here.
72 See Kant, I., Critique of Pure Reason, A 100 73 See ibid., A 100-103
These three syntheses seem to be all part and parcel of the ‘original synthetic unity of apperception’: the representation ‘I think’, Kant states, that must be able to accompany all the representations I have on account of sense intuition, because these representations could otherwise never be thought, or would be nothing for me75 – as such, Kant states, this representation is an act of spontaneity, that is, of thinking, and does not belong to sensibility in any way76. Now, it is because I add one representation to the other that I am conscious of their synthesis, so that this connection of synthesis of different representations in one consciousness is what makes the identity of consciousness in all these representations conceivable in the first place77, – so that Kant says: “that these representations are all my representations means that I can unite them in a self-consciousness or at least can unite them therein”78. The unity of the representation ‘I think’, Kant calls the transcendental unity of self-consciousness, for the possibility of all a priori knowledge is only possible on account of this unity79. Accordingly, for Kant, this ‘synthetic unity of apperception’ is “the highest principle for all use of the understanding”80, and the whole of logic and transcendental philosophy are fundamentally attached to it81.
We hope that this suffices as far as Kant’s ‘synthesis of the empirical’ or his notion of ‘original synthetic unity of apperception’ is concerned.
Now, let’s turn to the central topic of this section. The a priori concepts that the understanding contains are – as Kant claims – of four kinds: quantity, quality, relation, and modality. Let us discuss one of these a priori concepts of the understanding or ‘categories’ and the logical function of judgment corresponding with it: the category of ‘limitation’, and the ‘infinite’ judgment.
So, according to Kant, a judgment may be ‘affirmative’, ‘negative’, or ‘infinite’, for example: ‘Socrates is mortal’ is an affirmative judgment. So, the predicate, ‘being mortal’, is ascribed to the subject, ‘Socrates’, and one increases one’s knowledge therewith of the person named ‘Socrates’ – or, we may say that the meaning of ‘Socrates’ gets determined in this. Now, the negative judgment, ‘Socrates isn’t mortal’, this denies that this property of ‘being mortal’ is something that should be described to the subject, ‘Socrates’, and so, we may say that the meaning of ‘Socrates’ gets determined as well in this, or our knowledge of the person
75 See ibid., B 132 76 See ibid., B 132 77 See ibid., B 133 78 See ibid., B 134 79 See ibid., p. 132
80 See ibid., B 134, footnote; B 136 81 See ibid., B 134, footnote
named ‘Socrates’ is increased by denying this property to be one of the properties that this person has. Now, there is another type of judgment, according to Kant, which does not increase our knowledge at all, which, if we stick to our example, is ‘Socrates is non-mortal’: in this, the person named ‘Socrates’ is only considered to be one among the infinite things
that are non-mortal.
In another work by Kant, Logik – which I only know of as it is mentioned in my own Dutch edition of the first Critique – Kant explains this also as follows: “In negative judgments, the negation always affects the copula, in infinite judgments the negation doesn’t affect the copula, but the predicate”82. This is why Kant also says that the infinite judgment is, qua logical form, really an affirmation,83 and, in what Kant calls ‘general logic’ (what for Kant counts as ‘formal logic’) – a logic that abstracts from all content of the predicate,84 also when it is negative, which enquires only whether the predicate be ascribed to the subject or is rather opposed to it, – these two are not even distinguished, according to Kant85. But a transcendental logic considers also what may be the content of a logical affirmation that is thus made by means of a merely negative predicate, and what is thereby achieved in the way
of addition to our total knowledge86.
As does my Dutch edition, the negative judgment in question may be written as ‘Socrates is-not mortal’, the negation affecting the copula, while, as we have seen, the infinite judgment in question is one in which the negation affects the predicate, ‘Socrates is non-mortal’87. Now, as we already said, Kant claims that this infinite judgment locates Socrates in the unlimited sphere of non-mortal beings. The mortal constitutes one part of the whole extension of possible beings, and the non-mortal the other – so, nothing more is said in this infinite judgment than that Socrates is one of the infinite number of things which remain over when we take away all that is mortal. Now, the infinite sphere of all that is possible is thereby only so far limited that the mortal is excluded from it, and that Socrates is located in the remaining part of its extension. Even allowing for such exclusion, this extension, Kant states, still remains infinite, and several more parts of it may be taken away without the the meaning of the name ‘Socrates’ being thereby in the least determined in an affirmative manner, or our knowledge of this name being in the least increased thereby. The infinite judgment, though
82 See Kant, I., Kritiek van de zuivere rede, translated from German to Dutch by Jabik Veenbaas and Willem Visser (Boom: Amsterdam, 2004), p. 161
83 See Kant, I., Critique of Pure Reason, B 97 84 See ibid., B 97
85 See ibid., B 97 86 See ibid., B 97
infinite in respect of its logical extension, is, in respect of the content of the knowledge it conveys, limitative only, and, therefore, Kant states, they cannot be passed over in a transcendental table of all moments of thought in judgments, because the function of the understanding thereby expressed may be important in the field of its (that is, the understanding’s) pure a priori knowledge.
It seems that from this, we should proceed to the category of limitation, in the table of a priori concepts of the understanding, that is classified under the part called ‘Quality’, and the other concepts belonging to this, according to Kant, are ‘reality’ and ‘negation’. It is clear from what we just said on the corresponding logical functions of judgment, what these concepts amount to. ‘Reality’ may be taken to mean just as a property, ‘mortal’, while negation needs to be put in the context of the negative judgment, since in a negative jugment the negation affects the copula, not the predicate, so that we should conceive of it as ‘being-not mortal’, while limitation is just this property negated, that is, ‘non-mortal’, as the negation in an infinite judgment does not affect the copula, but the predicate.
Although Hegel criticized Kant’s transcendental logic, he quite agreed with its motivation, namely, “developing a science of pure understanding and rational cogition (“Vernunfterkenntnis”), by means of which we may cogitate objects entirely a priori”88. The problem for Hegel is that the table of categories is rather something that belongs to philosophic psychology than to philosophy – Kant sticks to the realm of the mental or psychological.89 In Hegel’s speculative logic, this mental or psychological dimension would disappear90, by focusing on the content of concepts themselves, not in the least, those concepts that conceptualize themselves, which takes place when conceptual thinking becomes its own object91. But Hegel agrees with the need for the project that Kant sets out in the first Critique, namely that of developing a science of pure understanding and rational cognition (“Vernunfterkenntnis”), “by means of which we may cogitate objects entirely a priori”92, – so, these are thoughts that do not thrive on intuition, but are acts of pure thought. Transcendental logic must “ … determine the origin, the scope, and the objective validity of such knowledge… ”93. As we already said, Hegel agrees that such a project must be carried out, but Kant would fail to maintain the appropriate level of generality, which has to do with this focus on, what Hegel calls, subjective understanding, so that Kant is “… treat[ing] concepts
88 See Kant, I., Critique of Pure Reason, B 81-2. Italics are mine.
89 See Wartenberg, T. E., ‘Hegel’s idealism: The logic of conceptuality’, pp. 114-7 90 See ibid., 115-7
91 See Hegel, G. W. F., Wissenschaft der Logik (Erster Teil), p. 23; p. 43 92 See Kant, I., Critique of Pure Reason, B 81. Italics are mine.
as mental entities rather than as entities within “logical space” ”94. For Hegel, the absolute character of the concept consists in its operating in both nature and spirit95.
The rather obvious question we are left with is: what would Hegel think or say about the ‘infinity’ involved in Kant’s infinite judgment and in the corresponding pure concept of ‘limitation’?
If Kant says that with ‘Socrates is non-mortal’ nothing more is said than that Socrates is one of the infinite number of things which remain over when we take away all that is mortal, it seems that he is pointing toward a quantitative infinite – after all, it concerns a quantity. But how does Kant come to say that on account of this judgment Socrates is taken to be among those infinite things that are non-mortal? Why does this concern an infinity? How do you know that there are infinitely many things, the one half of which is mortal and the other half of which is non-mortal? Kant just says that there are infinitely many things, the halves of which are infinite too. Hegel says in the Logic that Kant’s notion of infinity is a “progress to infinity”, that is the quantitative false infinite. The argument Hegel gives is that for Kant, in what he calls ‘the measurement of a quantum’, the successive synthesis of the unity can never be completed96. But such a progression, according to Hegel, never attains to infinity, it is only the ‘perpetual generation’ of it. So, when Kant is saying that ‘Socrates is non-mortal’ is an ‘infinite’ judgment, it may be that the infinity of things to which Socrates would be stated to belong in this judgment is only a false infinity, so that it is not really infinite. But if that is so, Kant would have to reconsider his statement that this ‘infinite’ judgment. And as Kant states that this extension of falsely infinite non-mortal things still remains infinite, several more parts of it may be taken away without the meaning of the name ‘Socrates’ being thereby in the least determined in an affirmative manner, or our knowledge of this name thereby being in the least increased – when Socrates for example also turns out to be non-Greek, non-female, non-alcoholic, etc., – it seems that Kant has even more reason to revise his position, for if one takes these things away from what is only a false infinity the result can hardly be that the resulting amount is still truly infinite: if repeated enough, maybe we can arrive at a sufficiently delimited group of things, such that our knowledge of Socrates is increased after all. So, Hegel may agree with Kant that the ‘infinite’ judgment is affirmative, but for a very different reason. And, of course, this all applies to the category of ‘limitation’ too, for this concept only occurs in an ‘infinite’ judgment.
94 See Wartenberg, T. E., ‘Hegel’s idealism: The logic of conceptuality’, pp. 115-6 95 See ibid., p. 116
§1.2 Kant and Hegel on the antinomic structure of reason or knowledge
The a priori concepts that the understanding contains are again, – as Kant claims, – of four kinds: quantity, quality, relation, and modality. With each kind of category, there corresponds a kind of condition that can be taken iteratively to bring about a particular series97 - these conditions we encounter in basic concepts such as before, part of, cause, depends on98, which correspond to these four kinds of categories. Graham Priest explicates the condition in question as a ‘generator’: iterating applications of the generator as far as possible – asking for the condition of an arbitrary phenomenon x, asking for the condition of this condition, and so on – we can obtain the limit, the totality of all such conditions, which Kant calls ‘the unconditioned’99. The limit may also be considered to be “the sequence generated in this way, or the result of applying the operator as often as possible”100. With the application of each kind of generator – or condition, – reason carries out dialectical arguments, which Kant discusses as the ‘antinomies of pure reason’ in the part of the Critique called the ‘Transcendental Logic’ and in the part of this logic Kant calls the ‘Transcendental Dialectic’. The dialectical arguments in question – according to Kant – come in pairs, consisting of a thesis and an antithesis, and these point out that the pertinent limit is characterized by contradictory properties101. Although the form of the antinomies tends to differ – and the arguments for the statements they contain aren’t as solid as can be, – the idea is that these limits all generate contradictions. The point that the argument for the antithesis makes, is that ‘one can always apply the generator again’. If we may adduce one of the quotes illustrative for the title of Priest’s major book, Beyond the Limits of Thought, Kant also says that “… the limit, if conceived of as obtained after a finite number of steps, is ‘too small for the concept’ which is generating it”102. The argument for the thesis is always an argument to the effect that if the limit is generated by an infinite number of applications of the generator, we end up in all kinds of trouble, like if we consider Kant’s point that the world would not have been able
97 Cf. Priest, G., Beyond the Limits of Thought, p. 87
98 Cf. Berto, F., ‘Dialetheism’, Stanford Encyclopedia of Philosophy, URL: http://plato.stanford.edu/entries/dialetheism/, section 2
99 See Priest, G., Beyond the Limits of Thought, p. 87 100 Cf. ibid., p. 87
101 See ibid., p. 87 102 Cf. ibid., p. 87
to come to be if an infinity of moments in time had to be bridged. Again, we may quote Priest: “As Kant puts it, the limit, so conceived, is ‘too large for the concept’ ”103.
As such, reason is engaged in the following antinomies: (1) that the world has a beginning in time and is limited in space – that the world has no beginning in time and isn’t limited in space, that it is infinite both in time and space104; (2) that every composed substance in the world consists of simple parts, and that everything that exists is either a simple or something compounded of simples – that no compounded thing in the world consists of simple parts, and there exists nothing simple in the world105; (3) that causality according to laws of nature is not the only causality by means of which all the phenomena in the world can be derived, to explain all phenomena, that one must also state that there is a causality according to freedom – that there is no freedom, that everything in the world happens solely by virtue of laws of nature106; (4) that there belongs something to the world that, either as a part of it, or as its cause, is an absolutely necessary being – that there is no absolutely necessary being, either in the world, or outside of the world, as its cause107. Let us discuss the third antinomy, which I both quote and paraphrase from the edition and translation by Paul Guyer and Allen W. Wood. Although the rather extensive remarks by Kant on both the thesis and antithesis are very interesting, it is sufficient for our purposes just to refer to the proofs by reductio that Kant gives for both the thesis and antithesis of the antinomy in question, – so that we leave these remarks out.
The argument for the thesis of the third antinomy, – that is, for the position that causality in accordance with laws of nature is not the only causality from which the appearances of the world are to be derived, that another causality, namely, that of freedom, must be assumed in order to explain these appearances, – is provided by Kant with the following proof by reductio. Assume that there is no other causality than that in accordance with laws of nature. If it is the case that there is another causality than this one, then everything which takes place presupposes a preceding state upon which it follows inevitably, that is, according to a rule. Assuming that this preceding state itself is something that takes place, as it has come to be in a time in which it did not exist previously, – since, if it has always existed, then its effect must also always have existed, and would not have only arisen at the moment we in fact see it has. The idea is, therefore, that the causality of the cause through which something takes
103 Cf. ibid., p. 87
104 See Kant, I., Critique of Pure Reason, B 454-461 105 See ibid., B 462-471
106 See ibid., B 472-9 107 See ibid., B 480-489
place is itself something which has taken place, that on its turn presupposes, in accordance with laws of nature, a preceding state and its causality, and this in similar manner a still earlier state, and so on. Now, if, then, everything indeed takes place in accordance with the laws of nature, Kant proposes, “there will always be only a relative and never a first beginning, and consequently no completeness of the series on the side of the causes that arise the one from the other”108. However, nothing would take place without a cause sufficiently
determined a priori.
This is why the proposition that no causality is possible save in accordance with laws of nature, when taken in unlimited universality, is self-contradictory, so that this cannot, be regarded as the sole kind of causality. Kant states that, on this account, we must assume a causality through which something takes place, but the cause of which is not itself determined, that is, in accordance with necessary laws, by another cause antecedent to it: Kant calls this an “absolute spontaneity of the cause”109, through which a series of appearances, which proceeds in accordance with laws of nature, begins of itself. Kant calls this transcendental freedom, a freedom, “without which, even in the [ordinary] course of nature, the series of appearances on the side of the causes can never be complete”110. In the Remark on the thesis, Kant states that the transcendental idea of freedom “does not by any means constitute the whole content of the psychological concept of that name, which is mainly empirical”111.
The argument for the antithesis of the third antinomy, – that is, for the position that there is no transcendental freedom, that everything in the world takes place solely in accordance with laws of nature, – is provided by Kant with the following proof by reductio. Kant says, suppose that there is freedom in the transcendental sense, that is, as a special kind of causality in accordance with which the events in the world can have come about, what he calls “a power of absolutely beginning a state, and therefore also of absolutely beginning a series of consequences of that state”112; from this it would follow “that not only will a series have its absolute beginning in this spontaneity, but the very determination of this spontaneity to originate the series, that is to say, the causality itself, will have an absolute beginning; there will be no antecedent through which this act, in taking place, is determined in accordance with fixed laws”113. However, every beginning of action, Kant states, would presuppose a
108 See ibid., B 472
109 See ibid., B 474. Italics are Kant’s. 110 See ibid., B 474. Italics are mine. 111 See ibid., B 476
112 See ibid., B 473 113 See ibid., B 473
