The Law of Contradiction in the Light of Recent Investigations of Bertrand Russell

Chwistek, Leon

doi:10.1007/978-3-319-52869-4_13

The Law of Contradiction in the Light of Recent Investigations of Bertrand Russell

Leon Chwistek⁵

Chapter
First Online: 27 July 2017

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Part of the book series: Vienna Circle Institute Yearbook ((VCIY,volume 21))

Abstract

In this article I endeavour to discuss a question on a topic that after simplification can be expressed as the following great problem: is contradiction an essential feature of the human mind? Philosophers have discussed that problem since ancient times. And since ancient times one has discussed it together with the problem of Existence. Even before the Law of Contradiction was formulated, and even before one found criteria to distinguish contradiction from non-contradiction, the Eleatics had based the criterion of existence on the concept of contradiction. We know that Parmenides rejected the existence of the world of phenomena because he saw everywhere in it a contradiction. Indeed Greek philosophy is full of observations of contradiction, starting with the sophisms without value and ending with the deep and currently interesting paradoxes. Although all philosophers are interested in the observation of contradictions, not all of them follow the way of the Eleatics. The need to deal with contradictions, their removal and depreciation was not so universal in Greece as it is today and views asserting contradictions were not rare. Already the great Heracleitos had the opinion that one and the same is and is not and Protagoras stated categorically that every opinion is true and therefore accepted that two contradictory propositions can be true.

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Notes

1.
Wasserberg: On intuition in Bergson, Przegląd Filozoficzny, 1912.
2.
Arist. Met. $\Gamma$ 3.
3.
“Zasada sprzeczności w świetle nowszych badań Bertranda Russella”, Rozprawy Akademii Umiejętności Wydzial Historyczno-Filozoficzny, Seria II. Tom XXX, Kraków 1912, 270–334.
4.
Translated by Rose Rand. The manuscript of this translation is in the Rose Rand Papers of the Archive of Scientific Philosophy, University of Pittsburgh. It was located by Nika Pona, who provided translations of some of the material. Footnotes and a section missed by Rand have been translated by Adam Trybus. The resulting English version has been edited by Bernard Linsky.
5.
Wasserberg: On intuition in Bergson, Przegląd Filozoficzny, 1912.
6.
Arist. Met. $\Gamma$ 3.
7.
Arist. Met. $\Gamma$ 5.
8.
Überweg-Heinze: Grundriss der Geschichte der Philosophie I, 324.
9.
[Rand has written above this, as an alternative: “mood”]
10.
Compare quotes e.g. Łukasiewiez on pages 58–59.
11.
Überweg-Heinze: Grundriss IV, p. 57.
12.
Als ob das, was conträr ist, nicht ebenso sehr als contradictorisch bestimmt werden müsste. Hegel Wissenschaft der Logik. Werke V 55, Berlin 1841.
13.
Myślini, Poznań 1844 I, pp. 416–417.
14.
Rand has: “consequently” and “consistently” as alternatives.
15.
“even our Trentowski repeated after the master, that this law of ancient logic isn’t worth much” O zasadzie sprzezności u Arystot. (About the Law of Contradiction in Aristotle). Kraków 1910.
16.
loc.cit. p. 418.
17.
loc.cit. p. 150.
18.
The Laws of Thought 1854.
19.
Begriffschrift, Halle 1879. Grundlagen der Arithmetik, Breslau 1884. Grundgesetze der Arithmetik, Jena 1893, 1903.
20.
Algebra der Logik 1890, 1891, 1895.
21.
Russell: The Principles of Mathematics, Cambridge 1903 p. 4.
22.
R.M.M. 1905 p. 803.
23.
Principles.
24.
Bemerkungen zu den Paradoxieen von Russell u. Burali-Forti. (Abh. der Friesschen Schule p. 305).
25.
On difficulties in the theory of transfinite Numbers and Order types. Proceedings of the London Mathem. Society 1906 Vol IV, cited by Poincaré R.M.M. 1906 page 17.
26.
Zermelo, Grundlagen der Mengenlehre, Mathemat. Annalen 65.
27.
In a public discussion in Göttingen July 1909.
28.
Les mathématiques et la logique §IX R. M. M. 1906.
29.
Neuer Beweis für die Wohlordnung, Mathem. Annalen 65 p. 117.
30.
Amer. Journ. XXX, Russell.
31.
Einheiten u. Relationen, Leipzig 1902.
32.
Lipps doesn’t add this, but it must be put down to his disregard of precision in definitions.
33.
Meinong: Über die Stellung der Gegenstandstheorie im System der Wissenschaftslehre, Leipzig 1907 p. 17.
34.
loc.cit. p. 40.
35.
Cf. Husserl: Logische Untersuchungen II p. 313.
36.
Amer. Journ. of Mathematics vol. XXX p. 222, 1908.
37.
La logique de l’infini, lipiec 1909.
38.
R. d. Mét. Et de Mor. 1910.
39.
Cambridge 1910.
40.
[Rand has as an alternative: ‘treatises’.]
41.
[Rand has as an alternative: ‘judgment.’]
42.
[Rand has as an alternative: ‘fundamental’.]
43.
[Rand considers ‘theses’ as an alternative. The Polish is ‘sądami’, the plural of ‘sąd’. The word ‘sąd’ can be translated as both ‘proposition’ and ‘judgment’, which are very different for Russell. Here Rand’s choice will be followed, although the Polish for these terms and for “propositional function” [funkcyą propozycyonalną] (or whatever grammatical case is appropriate) will be indicated in brackets and sporadically in what follows them as a reminder.]
44.
For propositional functions in Russell stand symbols $\phi \hat{x}$, $\psi \hat{x}$, etc., their arbitrary values are then ϕx, ψx, etc.
45.
Compare also: Le réalisme analytique, Bulletin de la Societé française de Philosophie, Mars 1911, p. 53.
46.
If a function $\phi \hat{x}$ is given, then symbol (x)ϕx = all values of the function $\phi \hat{a}$ are true.
47.
If p, q are propositions (sądy), then ∼ p, ∼ q are their negations.
48.
Logical sum of propositions p, q is written p ∨ q.
49.
A. Padoa: La logique déductive, Revue de Mét. Et de Morale, Novembre 1911 p. 871.
50.
Sign of assertion is ⊢.
51.
For elementary functions Russell writes symbols $\phi !\hat{x}$, $f!(\hat{x}\hat{y}\hat{z})$; other first order functions look like e.g. $(x)f!(x\hat{y})$ and the like, but also can be written as symbol $F!\hat{y}$.
52.
These definitions don’t overlap completely with the definitions Russell gives in Principia that are not completely precise. I constructed it from completely clear descriptions of the ∗ 12.
53.
Second order matrix $f!(\phi !\hat{x})$. Second order function $(\phi )f!(\phi !\hat{x},\hat{x})$. Second order predicative function $(x)f!(\phi !\hat{x},x)$.
54.
Cf. Principia I p. 56. Compare on page 172: Russell introduces different meaning of notion of predicative function, identifying it with notion of matrix. In order to avoid misunderstandings I will use only the notion of matrix.
55.
Über das sogenannte Erkenntnisproblem, Göttingen 1908.
56.
p implies q Russell writes as p ⊃ q. The definition of that symbol is then p ⊃ q = ∼ p ∨ q.
57.
The product of propositions p, q is written as p. q. According to that we said p .  q =   ∼ ( ∼ p ∨ ∼ q).
58.
In symbols for p is equivalent to q we write p ≡ q. We have then p ≡ q = p ⊃ q .  q ⊃ p.
59.
I use these phrases only as abbreviations for phrases that single out relevant individuals.
60.
[Rand has: In the propositional calculus plays a significant role the concept of such propositions as: …]
61.
(Ex)ϕx [This in the original, but beginning two pages later we always have the more standard quantifier expression $(\exists x)\phi x$.]
62.
(Ex).  ϕx =   ∼ [(x). ∼ ϕx]
63.
The symbolical definition of identity looks like: x = y. =  : (ϕ): ϕ! x ⊃ ϕ! y.
64.
(xy). ψ(xy) = (x): (y).  ψ(xy).
65.
($\exists x,y)\psi (xy) = (\exists x)\!:\! (\exists y).\:\psi (xy)$. In order to simplify the symbolism Russell writes, following Peano: ϕx ⊃_x ψx instead of (x).  ϕx ⊃ ψx, ϕx ≡_x ψx instead of (x).  ϕx ≡ ψx, ϕ(xy) ⊃_xy ψ(xy) instead of (xy).  ϕ(xy) ⊃ ψ(xy). In the above I omitted many of Russell’s definitions, namely those transferring the theory of elementary propositions to propositions of 1st degree.
66.
American Journal of Mathematics vol. XXX.
67.
If we have the assertion ⊢ p, and ⊢ p ⊃ q, then we can say ⊢ q. One can formulate this axiom, like many others, without words.
68.
Number ∗ 1 ⋅ 11.
69.
p ∨ p . ⊃.  p, number ∗ 1 ⋅ 2.
70.
q . ⊃ .  p  ∨ q, number ∗ 1 ⋅ 3.
71.
p ∨ q . ⊃ .  q ∨ p number ∗ 1 ⋅ 4.
72.
p ∨.  (q ∨ r) . ⊃ q . ∨.  (p ∨ r), number ∗ 1 ⋅ 5.
73.
q ⊃ r: p ∨ q ⊃ p ∨ r number ∗ 1 ⋅ 6. [This should be: q ⊃ r. ⊃  : p ∨ q . ⊃ .  p ∨ r ].
74.
Number ∗ 1 ⋅ 7.
75.
Number ∗ 1 ⋅ 71.
76.
Number ∗ 1 ⋅ 72.
77.
Number ∗ 1 ⋅ 172. In other words, function ϕp ∨ψp has the same range of variable as functions ϕp and ψp taken separately.
78.
$\phi x\:. \supset (\exists z).\:\phi z$, number ∗ 9 ⋅ 1.
79.
$\phi x \vee \phi y\:. \supset (\exists z).\:\phi z$, number ∗ 9 ⋅ 11.
80.
Number ∗ 9 ⋅ 12.
81.
Number ∗ 9 ⋅ 13.
82.
One should remember that this statement isn’t at all equivalent to the statement ϕy ⊃ (z)ϕz, as one could think. The latter statement is just false.
83.
If ϕa is a proposition or if a and x are of the same type, then ϕx is also a proposition, number ∗ 9 ⋅ 13.
84.
If a proposition p looks like ϕa, then there is a function ϕx, number ∗ 9 ⋅ 14.
85.
∗ 11 ⋅ 07
86.
[In English:] the axiom of reducibility.
87.
$\vdash \::\! (\exists f)\!:\phi x \equiv _{x}.f!x$, number ∗ 12 ⋅ 1
88.
$\vdash \::\! (\exists f)\!:\phi (xy) \equiv _{xy}.f!(xy)$
89.
Russell, L’importance philosophique de la logistique R.M.M. 1911 p. 287.
90.
It will be interesting to see how Russell introduces the Law of Contradiction. I give the proof in extenso. The principle of Sum is

q ⊃ r ⊃  : p ∨ q . ⊃ .  p ∨ r (1)

Then substitute ∼ p for p, thus

q ⊃ r ⊃  :   ∼ p ∨ q . ⊃ .   ∼ p ∨ r

Using the definition of implication I have then p ⊃ q. =. ∼ p ∨ q (2) then

q ⊃ r ⊃  : p ⊃ q . ⊃ .  p ⊃ r (3)

the so-called Principle of Syllogism. If here I substitute p ∨ p for q and p for r then:

p ∨ p . ⊃ p ⊃  : p . ⊃ p ∨ p . ⊃ .  p ⊃.  p

Thus, the premise of that conclusion is the Principle of Tautology p ∨ p . ⊃ .  p; I can then assert the conclusion:

p ⊃ p ∨ p . ⊃ .  p ⊃ p.

The premise of that proposition is an application of the Principle of Addition q ⊃ p ∨ q, and so is true. Then we can assert a proposition p ⊃ p, (4), that Russell calls the Principle of Identity, although it differs from the principle that is usually called the Principle of Sameness or Identity. Let’s consider now the Principle of Permutation:

p ∨ q . ⊃ .  q ∨ p

and substitute ∼ p for p and ∼ q for q, and so ∼ p ∨ ∼ q . ⊃ .   ∼ q ∨ ∼ p. From the definition of implication we will have then p  ⊃   ∼ q . ⊃ .  q  ⊃   ∼ p (5), what is called the Principle of Transposition. From another direction if we substitute p for q into the definition of implication we will have

p ⊃ p . =.   ∼ p ∨ p

from which follows: ∼ p ∨ p (6).
Now consider the Principle of Permutation and substitute ∼ p for p and p for q; then

∼ p ∨ p . ⊃ .  p ∨ ∼ p

Here the premise is true, therefore the conclusion is also true, the Principle of Excluded Middle p ∨ ∼ p (7). From the Principle of Identity and the definition of the product we have

p .  q ⊃ .   ∼ ( ∼ p ∨ ∼ q)

From transposition: p .  q ⊃ .   ∼ ( ∼ p ∨ ∼ q):   ⊃  :   ∼ p ∨ ∼ q . ⊃. ∼ (p q). Thus, substituting ∼ p for q we will get:

∼ p ∨  ∼ ( ∼ p) . ⊃ .   ∼ (p . ∼ p)

Here the conclusion is true, when the premise is true, as it follows from (7).
91.
(ϕ). ∼ {ϕ! A .   ∼ ϕ! A}
92.
One needs to add that although Russell’s proof is made for elementary propositions, it can be transferred without difficulties to propositions of other types.
93.
If ϕx = x is the winner from Jena, then the incomplete object the winner from Jena is written as: ( ι x)(ϕ x) Every proposition f[( ι x)(ϕ x)] with that symbol Russell replaces with proposition $(\exists c):\phi x\:. \equiv _{x}.\:x = c: fc$.
94.
One can like Russell use an expression “plurality” in the intuitive meaning, keeping “class” for the precise notion.
95.
Principia p. 75.
96.
Let $f(\phi \hat{x})$ be the function whose variable depends on type $\phi \hat{x}$ in a way that it always has sense. The derivative function from that function is $f(\hat{x}(\phi x)) = (\exists \psi !).\:\psi !z \equiv _{z}\phi z.\:f(\psi !z)$.
97.
[The Rand manuscript is missing section 20. The translation of section 19 ends on page 186 of the notes, and the next sheet, page 187, starts with section 21.]
98.
$\hat{z}(\phi z) =\hat{ z}(\psi z):\!. =.\!: (\exists \chi,\theta )\!:\phi x \equiv _{x}\theta !x:\chi !\hat{z} =\theta !\hat{z}$.
99.
$a\:\epsilon \:\hat{z}(\phi z) = (\exists \psi )\phi x \equiv _{x}\psi !x\:.\:a\:\epsilon \:\psi \hat{x}$, thus $a\:\epsilon \:\psi !\hat{x} =\psi !a$.
100.
α ⊂ β: = : x ε α ⊃_x x ε β.
101.
$f(R) = f(\hat{x}\hat{y}.\;\phi !(xy))$.
102.
In symbols: f(R‘$y) =\:: (\exists b)\!: xRy\:. \equiv _{x}x = b: fb$.
103.
[Rand has: reverse].
104.
$\overrightarrow{R}$‘$y =\hat{ x}(xRy)$, the class of referents y. $\overleftarrow{R}$‘$y =\hat{ x}(xRy)$ the class of relata x. It is clear that $\overrightarrow{R}$‘$y =\hat{ x}(xRy)$ and $\overleftarrow{R}$‘$y =\hat{ x}(xRy)$ are relations.
105.
$D =\hat{\alpha }\hat{ R}[\alpha =\hat{ x}\{(\exists y)xRy\}]$ domain D $=\hat{\beta }\hat{ R}[\beta =\hat{ y}\{(\exists x)xRy\}]$ converse domain.
106.
For that class Russell writes the symbol R “β.
107.
A many-one relation is written in Russell’s symbolism as Cls → 1, its definition is ${\ast}71\! \cdot \! 02\;\;Cls\! \rightarrow \! 1 =\hat{ R}(\overleftarrow{R}$“D‘R ⊂ 1) and can be written: $Cls\! \rightarrow \! 1 =\hat{ R}[\hat{x}\{(\exists y).\:y\:\epsilon \:D$‘$R: x =\hat{ z}(yRz)\} \subset 1\:]$ where α ⊂ 1 means that class a has 1 and only 1 element.
108.
$R\:\epsilon \:1 \rightarrow Cls. \supset.\hat{y}[E!R$‘y] = D ‘R.
109.
1 → 1 = (1 → Cls) ∩ (Cls → 1) where ∩ is a sign of product of two classes or a class of common elements of the two classes.
110.
∗ 101 ⋅ 1 ⊢. Nc‘$\alpha =.\;\hat{\beta }(\beta$ sm $\alpha ) =\hat{\beta } (\alpha$ sm β). Nc‘α = cardinal numbers of class α, sm = similis.
111.
Principia Mathematica, p. 65.
112.
R.M.M. 1909.
113.
Loc.cit. p. 482.
114.
One must admit that Russell is a supporter of this postulate. (Principia I p. 64). I must confess that I can’t understand why Russell didn’t pay attention to the consequences for his system that follow from this postulate.
115.
[There is no section 2 in the original, and Rand follows that numbering.]
116.
Acta math. 32.
117.
That object is a class with letter L and number n as elements. Letters are considered as individuals.
118.
One can’t say that all individuals have proper names.
119.
Über eine vermeintliche Antinomie der Mengenlehre, Acta Mathematica 32, 1909.
120.
Réflexions sur les deux notes précédentes, ibid.
121.
I give here the main points of Russell’s proof: Vol.II p. 32.
⊢ R ε Cl → 1 .  D‘R ⊂ α . D ‘R ⊂ Cl‘$\alpha \:.\! \supset \!.\:\exists !\:Cl$‘α − D‘R.
Proof: hypothesis and $\varpi =\hat{ x}(x\:\epsilon \:D$‘$R\:.\:x \sim \epsilon \:\breve{ R}$‘x). ⊃  : .
$$\displaystyle\begin{array}{rcl} & x\:\epsilon \:D\text{`}R\:. \supset _{x}\:: x\:\epsilon \:\varpi. \equiv x \sim \!\epsilon \:\breve{ R}\text{`}x: & {}\\ & \supset _{x}:\;\sim \! (x\:\epsilon \:\varpi. \equiv x\:\epsilon \:\breve{R}\text{`}x) & {}\\ & \supset _{x}:\varpi \neq \breve{R}\text{`}x & {}\\ & \supset.\!:\varpi \sim \epsilon D \text{`} R(1) & {}\\ & \mbox{ hypothesis and }(1). \supset.\varpi \subset D\text{`}R\:. \supset.\varpi \subset \alpha (2)& {}\\ \end{array}$$
From these statements and the definition of identity follows the desired statement. Russell doesn’t consider the case when ϖ doesn’t have any element.
122.
R.M.M. p. 296.
123.
The above assumption makes necessary the introduction of a new symbol. Let ϕxTψx be a proposition: ϕx is of the same type as ψx. Then our definition will be: $f[\hat{z}(\phi z)] = (\exists \psi ).\:\psi zT\phi z\:.\:\psi z \equiv \phi z\:.\:f(\psi \hat{x})$. Here we must remember that variable $\phi \hat{z}$ must have a definite type.
124.
Acta math. 32. p. 199.
125.
Acta math. 32, p. 199.
126.
Acta math, Vol. 32.
127.
Cf. Zermelo, loc.cit., p. 188.
128.
A. J. XXX p. 242.
129.
Ibid., p. 327.
130.
Compare for example Jerusalem: Der kritische Idealismus u. seine Logik. Wien und Leipzig 1905.
131.
[The Polish is: (Wolno na miejscu układu położyć dwie litery i powstaje układ). These strings thus have the same number of letters.]
132.
R.M.M. 1905 p. 815.
133.
This project has been a group effort, both by persons who understand Polish and others without Polish. Nika Pona located the Polish text on line, as well as Rand’s translation in the Archives of Scientific Philosophy. Brigitta Arden of the ASP provided a copy of the manuscript and information about Rose Rand and her papers. My thanks to Piotr Rudnicki for downloading the original and discussing the translation of logical terminology from 1912. I am especially indebted to Adam Trybus, a Polish logician in his own right, for help with translating material missing from Rand’s draft, and for assessing and improving Rand’s translation. Emma Kennedy, Joshua St Pierre and Christopher Johnson transcribed Rand’s manuscript.
134.
See Feferman and Feferman (2004) and Linsky (2009) for discussions of the appointment and of Russell’s letter of reference for the position.
135.
See Linsky (2004).
136.
See Linsky (2011, 54–57) for a discussion of the significance of this reference.
137.
See Linsky (2011) for an extended discussion of this argument and the controversy about the error Russell makes in it. Chwistek’s argument is similar to Russell’s later proof and is subject to the same difficulties as Russell’s argument.
138.
See Jadacki (1986).
139.
See Hamacher-Hermes (2003) for an account of Rose Rand’s life, career in philosophy, and for a complete bibliography of Rand’s works. The chronology here comes from Hamacher-Hermes. There are extensive references to Rand’s involvement with the Vienna Circle in Stadler (2001) and two in Dawson (1997) which connect her with Gödel.
140.
See Stadler (2001), p. 234 and Dawson (1997) 73,125.
141.
Rand (1937).
142.
Rand (1962) was originally published in 1939.

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Institute of Philosophy, University of Warsaw, Warsaw, Poland
Leon Chwistek

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Institute of Philosophy, University of Warsaw, Warsaw, Poland
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Institute Vienna Circle, University of Vienna, Vienna, Austria
Friedrich Stadler
University of Information, Technology and Management, Rzeszów, Poland
Jan Woleński

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Chwistek, L. (2017). The Law of Contradiction in the Light of Recent Investigations of Bertrand Russell. In: Brożek, A., Stadler, F., Woleński, J. (eds) The Significance of the Lvov-Warsaw School in the European Culture. Vienna Circle Institute Yearbook, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-319-52869-4_13

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DOI: https://doi.org/10.1007/978-3-319-52869-4_13
Published: 27 July 2017
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-52868-7
Online ISBN: 978-3-319-52869-4
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)

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