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. To Mind a Quarterly Review of philosophy.

Jean-François Monteil, ancien maître de conférences de linguistique générale à l’Université Michel de Montaigne de Bordeaux

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Jean-francois.monteil@neuf.fr

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I suggest to those who are interested in the content of the present paper to contact me by e-mail. Here is my e-mail adress: jean-francois.monteil@neuf.fr In my opinion the celebrated sentence of Alfred Tarski, triggers a revolution in logic.

 

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.

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.

In any of the three situations, the law of excluded middle is preserved. This law can be represented thus: (l) p w not-p. Les faits p et non-p sont nécessairement et par définition (l) contradictoires. Ils sont incompatibles et ne peuvent pas être tous les deux exclus de la réalité.