La philosophie mathématique de Bertrand Russell

Couverture
Vrin, 1993 - 509 pages
Par le terme de philosophie mathematique , Bertrand Russell designe une philosophie qui s'efforce d'expliquer les principes logiques sur lesquels reposent les mathematiques. Portant sur l'ensemble de l'oeuvre logico-mathematique de Russell, depuis les Principles of Mathematics jusqu'aux Principia Mathematica en passant par On Denoting , cette etude reconstitue la genese de la logique russellienne a partir d'une reflexion grammaticale et scrute l'analyse philosophique des concepts et propositions mathematiques qui gouverne leur reduction logiciste. L'auteur, adoptant une approche historique, souligne, a travers evolutions et ruptures, la coherence d'une authentique pensee.
 

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The book of Bertrand Russell is, no doubt, one to take if you are condemned to " the desert island". To my mind, the climax of it is the chapter entitled The law of excluded middle. In light of Tarski and most probably thanks to the use of the logical hexagon of the Frenchman Robert Blanché in modal logic, a lot of problems raised by Russell in his book and particularly in the twentieh chapter can be solved.
Tarski said: the proposition "Snow is white" is true, if and only if snow is white. One may conclude that instead of saying the proposition p is true, one must say that the fact p is certain and symbolize the certainty of the fact p by Lp.
If we are in a position to assert: ‘It snowed on Manhattan Island on the first of January in the year 1 Anno Domini’, the fact p in question must be symbolized by Lp, to be read It is a certain fact that it snowed on Manhattan Island on the first of January in the year 1 Anno Domini.
If we are in a position to assert: ‘It did not snow on Manhattan Island on the first of January in the year 1 Anno Domini’, the fact not-p in question must be symbolized by L~p, to be read : It is a certain fact that it did not snow on Manhattan Island on the first of January in the year 1 Anno Domini.
If we are in a state of ignorance concerning the two contradictory facts p and not-p, in other words, if we are unable to assert ‘It snowed on Manhattan Island on the first of January in the year 1 Anno Domini’ as well as ‘It did not snow on Manhattan Island on the first of January in the year 1 Anno Domini’, we experience a fact, the fact that neither p nor not-p is certain. This third fact can be symbolized by ~L~p & ~Lp, both the certainty of the fact not-p and the certainty of the fact p are excluded.
~L~p, the non-certainty of the fact not-p is equivalent to the possibility of the fact p to be symbolized by Mp, ~Lp, the non-certainty of the fact p is equivalent to the possibibity of the fact not-p to be symbolized by M~p.
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There exist three situations corresponding to the case envisaged by Bertrand Russell in the chapter 20 of his An inquiry into meaning and truth and entitled the law of excluded middle.
One of three things, either Lp the certainty of the fact p or L~p the certainty of the fact not-p or Mp & M~p the possibility of both p and not-p to the extent that both are non-certain.
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In any of the three situations, the law of excluded middle is preserved. This law can be represented thus: (l) p w not-p.
The facts p and not-p are necessarily, by definition ( this is the meaning of the symbol (l) here used) contradictory. They are incompatible and they cannot be both excluded of reality.
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In order to get a precious diagram, type on Google the title of the article reproduced above :
"FR MNC Tarski said: the proposition "Snow is white" is true, if and only if snow is white. One may conclude that instead of saying the proposition p is true, one must say that the fact p is certain and symbolize the certainty of the fact p by Lp."
The author of these lines thinks that the solution of the Russellian problem renders possible a consistent formula of strict implication
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KNOLmnc 1 Modal logic. The three ingredients of strict implication. Calcutta
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FR MNC (I) Modal logic. The final touch in defining the strict implication of a fact q by a fact p. If the bilateral possible Mp & M~p is to be symbolized by L ( p w ~p) , the formula of strict implication is L (p ≡ Lq).
FR MNC (II) Modal logic. The final touch about strict implication L (p ≡ Lq). The two facts containing it.
FR MNC (III) Modal logic. The three ingredients of L (p ≡ Lq). Unconditional certainty and conditional certainty.
FR MNC (IV) Modal logic. The three facts to be considered: L (p ≡ Lq), the strict implication of q by p and the two facts containing it, namely, L ((p & Lq) w (~p & M(q)) on the one hand and L ((p & Lq) w (~p & L~q) on the other.
 

Table des matières

Table des abréviations
7
LA RÉVOLUTION INITIALE
19
ANALYSE DE LA PROPOSITION
30
LA QUESTION DU SYMBOLISME
54
DÉFINITION DE LA CLASSE
75
LE STATUT DES RELATIONS
102
DÉVELOPPEMENTS ET LIMITES
123
LES PRÉMISSES PHILOSOPHIQUES
156
RÉSOLUTION DE LA CONTRADICTION
271
LES DESCRIPTIONS DÉFINIES
306
ONTOLOGIE DE LA LOGIQUE
318
LÉCONOMIE DES PRINCIP1A MATHEMAT1CA
332
Théorie des nombres
356
DISCURSIVITÉ DU SENS ET DE LA VÉRITÉ
362
LES LIMITES DU LOGICISME RUSSELLIEN
381
LA RÉDUCTION DE LARITHMÉTIQUE
399

LÉVOLUTION MAJEURE
175
LA CRITIQUE DE MEINONG
189
LA CRITIQUE DE FREGE
201
Le sens
215
Intensionnalitéextensionnalité
225
Discours artistiquediscours scientifique
238
LA CONNAISSANCE DISCURSIVE
244
LOPUS MAGNUM
253
LULTIME AMENDEMENT
429
LAUTOFONDATION DELA LOGIQUE
442
CONCLUSION
453
Table des principaux symboles logiques
459
Bibliographie
469
Index des noms
492
Table des matières
505
Droits d'auteur

Expressions et termes fréquents

À propos de l'auteur (1993)

Denis Vernant est professeur de philosophie a l Universite Pierre-Mendes-France de Grenoble.

Informations bibliographiques