[Hol-info] The two characteristics of an antinomy: self-reference and negation

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Dear Members of the Research Community,

A frequent and ongoing impediment in mathematical research is an only partial 
understanding of the nature of antinomies, in which _self-reference_ is 
(correctly) identified as a characteristic of the antinomy, but in which it is 
also omitted that - as is well known in philosophy - _negation_ is the second 
necessary characteristic of an antinomy.

For example, in his article "The Foundation of a Generic Theorem Prover" (1988) 
introducing Isabelle's meta-logic, Larry Paulson writes: "A logic for the 
formalization of mathematics must presuppose the very minimum. Philosophers 
have debated whether a logic must be _predicative_ - free of 'vicious circles'" 
[Paulson, 1988, p. 33, emphasis as in the original].
	http://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-130.pdf

Or, for example, when I met Peter (Peter Andrews, the creator of the 
mathematical higher-order logic Q0) in 2010 and mentioned that in my logic R0 
there is a universal type (omega) containing all mathematical objects, 
including itself, he made a joke: "That's when logicians get nervous!"

Obviously, the origin of this view is Bertrand Russell's article "Mathematical 
Logic as based on the Theory of Types" (1908), in which type theory was 
introduced in order to avoid the inconsistency arising from Russell's paradox 
(found independently by Russell and Zermelo) that Russell discovered in 1901, 
communicated in a letter to Frege in 1902, and first published in "The 
Principles of Mathematics" in 1903:

I. "In all the above contradictions (which are merely selections from an 
indefinite number) there is a common characteristic, which we may describe as 
self-reference or reflexiveness." [Russell, 1908, p. 224; see also Russell, 
1967b, p. 154]

II. "IV. / _The Hierarchy of Types_. / [...] The division of objects into types 
is necessitated by the reflexive fallacies which otherwise arise. These 
fallacies, as we saw, are to be avoided by what may be called the 
'vicious-circle principle;' i.e., 'no totality can contain members defined in 
terms of itself.'" [Russell, 1908, pp. 236 f., emphasis as in the original; see 
also Russell, 1967b, p. 163]

However, the classical antinomy has two characteristics, or, actually, three:

1. self-reference,
2. negation,
3. and the negation has to be applied to the self-reference (and not to 
something else).

It can easily be observed that all three characteristics hold for Russell's 
paradox:
    The set of all sets that do NOT contain THEMSELVES.

In order to make this philosophical insight plausible for mathematicians, I 
have compiled three quotes from (some of) the most important philosophical 
works on this topic (citations in German as written in the original works, 
followed by my own translations):

1. "Strikte Antinomien sind von bloßen Widersprüchen unterschieden. Eine 
strikte Antinomie weist immer zwei sich gegenseitig negierende und zugleich 
wechselseitig implizierende Seiten auf, während bei einem einfachen Widerspruch 
bloß eine Konjunktion kontradiktorisch entgegengesetzter Bestimmungen bzw. 
Aussagen vorliegt. [...] / Zwei Aspekte sind für Antinomien konstitutiv, das 
_Me[r]kmal der Selbstbeziehung_ und das _Merkmal der Negation_, wobei der 
interne Zusammenhang dieser beiden Merkmale so aussieht, daß dasjenige, was 
negiert wird, nicht irgend eine beliebige Eigenschaft ist, sondern die 
Selbstbeziehung. Die Negation, die Negation von Selbstbeziehung ist, ist aber 
genau das, was bei Hegel sich auf sich beziehende Negation heißt. Die Negation 
von Selbstbeziehung ist darin antinomisch, daß sie sich als ein Modus von 
Selbstbeziehung erweist: Die Negation ist als Negation der Selbstbeziehung 
selbstbezügliche Negation." [Iber, 1990, S. 482, Hervorhebungen im Original]

"Strict antinomies are distinct from mere contradictions. A strict antinomy 
always has two sides that simultaneously negate and reciprocally imply each 
other, whereas, in a simple contradiction, there is just a conjunction of two 
contradictorily opposing properties or propositions. [...] / Two aspects are 
constitutive for antinomies, the _property of self-reference_ and the _property 
of negation_, whereas the internal relationship of these two properties is such 
that what is negated is not some arbitrary property, but the self-reference. 
However, the negation, which is the negation of self-reference, is exactly what 
Hegel calls the self-referencing negation. The negation of self-reference is 
antinomical insofar, as it proves to be a mode of self-reference: The negation 
as a negation of self-reference is a self-referencing negation." [Iber, 1990, 
p. 482, emphases as in the original; translation by the author]

2. "[...] Negation und Selbstbeziehung [sind] 'die beiden Charakteristika 
strikter Antinomien', wobei das Negierte die Selbstbeziehung selbst sein muss 
und nicht irgendeine Eigenschaft (wie bei dem Begriff 'nicht farbig', der eben 
[selbst] nicht farbig ist)." [S. Schick, 2010, S. 343]

"[...] Negation and self-reference [are] 'the two characteristics of strict 
antinomies', while that which is negated has to be the self-reference itself 
and not an arbitrary property (as with the notion 'not colored', which [itself] 
is not colored)." [S. Schick, 2010, p. 343; translation by the author]

3. "Die Reflexionslogik behandelt entsprechend die Kategorien der Identität und 
des Unterschieds, der Verschiedenheit, des Gegensatzes und schließlich des 
Widerspruchs. [...] Die Beziehungen, die die Reflexion als formallogisches 
Denken _setzt_, sind zunächst solche abstrakter Identität bzw. ebenso 
abstrakter Unterscheidung. Die Reflexion auf die Kategorien von Identität und 
Unterschied zeigt indessen, dass diese für sich isoliert gar nicht bestimmbar 
sind, d.h. ihr Anderes _logisch_ in sich enthalten: '... der Unterschied an 
sich ist der sich auf sich beziehende Unterschied; so ist er die Negativität 
seiner selbst, der Unterschied nicht von einem Andern, sondern seiner von sich 
selbst. [...] - Dies ist als die _wesentliche Natur der Reflexion_ und als 
_bestimmter Urgrund aller Tätigkeit und Selbstbewegung_ zu betrachten. - 
[...]'" [Wagenknecht, 2013b, S. 74 f., Hervorhebungen im Original, ursprünglich 
1997 veröffentlicht] (Das Zitat im Zitat stammt aus dem zweiten Kapitel der 
Wesenslogik in Hegels _Wissenschaft der Logik_ [Hegel, 1986 ff., Bd. 6, S. 46 
f., ursprünglich 1813 veröffentlicht]. Diesen Teil, die Reflexionslogik, 
bezeichnet Hegel selbst als "(de[n] schwerste[n]) Teil" [Hegel, 1986 ff., Bd. 
8, S. 236 (Enz. § 114), ursprünglich 1817 veröffentlicht, zitiert nach der 
letzten (dritten) Ausgabe seiner _Enzyklopädie_ von 1830] seines Hauptwerks 
_Wissenschaft der Logik_, des mit Abstand wichtigsten Werks Hegels überhaupt.)

"Accordingly, the logic of reflection treats the categories of identity and of 
difference, of diversity, of opposition, and, finally, of the contradiction. 
[...] The relations _set_ by the reflection as formal logic are, first of all, 
those of abstract identity or those of likewise abstract distinction. The 
reflection on the categories of identity and difference indeed shows that those 
cannot be determined in an isolated manner, i.e., _logically_ contain their 
other: '... the difference in itself is self-related difference; as such, it is 
the negativity of itself, the difference not of an other, but of itself from 
itself. [...] - This is to be considered as the _essential nature of 
reflection_ and as the _specific, original ground of all activity and 
self-movement_. - [...]'" [Wagenknecht, 2013b, pp. 74 f., emphases as in the 
original, originally published in 1997; translation by the author] (The quote 
within the quote is taken from the second chapter of the logic of essence in 
Hegel's _Science of Logic_ [Hegel, 1986 ff., vol. 6, pp. 46 f., originally 
published in 1813; translation by the author]. Hegel himself calls this part, 
the logic of reflection, "(the most difficult) part" [Hegel, 1986 ff., vol. 8, 
p. 236 (Enc. § 114), originally published in 1817, quoted from the last (third) 
edition of his _Encyclopedia_ from 1830; translation by the author] of his main 
work _Science of Logic_, which is by far Hegel's most important work.)

Finally, it should be noted that the antinomy is distinct from the mere 
contradiction in formal logic and mathematics, as from the strict antinomy, a 
contradiction is obtained without premises (without non-logical axioms, 
assumptions, or hypotheses), i.e., from the logic (the formal language, the 
logistic system) only, as in the case of Russell's paradox:

4. "Tatsächlich aber ist das, was Kant als Antinomie bezeichnet, weder eine 
echte Antinomie noch eine Antinomie im Sinne Hegels. Wenn Antinomien logische 
Widersprüche sind, 'die _prämissenfrei_ abgeleitet werden', dann liegen bei 
Kant keine eigentlichen Antinomien vor, denn die entgegengesetzten Behauptungen 
implizieren einander nicht, sondern beruhen auf verschiedenen Voraussetzungen." 
[S. Schick, 2010, S. 339 f., Hervorhebung im Original]

"But, indeed, that which Kant calls an antinomy is neither a true antinomy nor 
an antinomy as understood by Hegel. If antinomies are logical contradictions, 
'which are derived _without premises_', then in Kant's case there are no actual 
antinomies, since the opposing propositions do not imply each other, but depend 
on different assumptions." [S. Schick, 2010, pp. 339 f., emphasis as in the 
original; translation by the author]

Using this terminology, it becomes clear that the chapters "The Contradiction" 
("Der Widerspruch") both in book two of Hegel's _Science of Logic_, _The 
Doctrine of Essence_ (1813), [Hegel, 1986 ff., vol. 6, pp. 64 ff.] and in 
Russell's "The Principles of Mathematics" (1903) [Russell, 1903, pp. 101 ff.] 
should actually be entitled "The Antinomy" ("Die Antinomie").
	http://www.deutschestextarchiv.de/book/view/hegel_logik0102_1813?p=77
	https://archive.org/stream/principlesmathe00russgoog#page/n132/mode/1up

In this chapter in _The Doctrine of Essence_ (_Die Lehre vom Wesen_), commonly 
called the logic of essence (Wesenslogik), Hegel wrote:

5. "Dies ist also derselbe Widerspruch [...], nämlich [...] Negation, als 
Beziehung auf sich." [Hegel, 1986 ff., Bd. 6, S. 66, ursprünglich 1813 
veröffentlicht]

"This is therefore the same contradiction [...], namely [...] negation as 
self-reference." [Hegel, 1986 ff., vol. 6, p. 66, originally published in 1813; 
translation by the author]

	http://www.deutschestextarchiv.de/book/view/hegel_logik0102_1813?p=79

For philosophical references, please see: http://doi.org/10.4444/100.110
For mathematical references, please see: http://doi.org/10.4444/100.111

Kind regards,

Ken Kubota

____________________________________________________

Ken Kubota
http://doi.org/10.4444/100