If the symbol (l) represents a priori necessity, (l) p w ~p means that the fact p and the fact not-p are a priori necessarily contradictory. On the one hand, they are necessarily in-compatible in reality, on the other hand they cannot be both excluded from reality.
Hence the fact p ≡ ~~p. That means that the fact p is the fact excluding the fact not-p as the fact not-p is the fact excluding p. The author of this remark refers the potential reader to An inquiry into meaning and truth, chapter 20 by Bertrand Russell and to what is devoted to the said chapter entitled The law of excluded middle in the following papers:
To my mind, the twentieth chapter entitled The law of excluded middle, constitutes a sort of climax in the celebrated An inquiry into meaning and truth. In light of Tarski and 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. I emphasize here that the third fact I mention must be given as much importance as the facts Lp and L~p we are led to consider when we are in a state of knowledge. The third fact is the fact we have to envisage when we are in a state of ignorance. It corresponds to what is called the bilateral possible. ~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 possibility 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. 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.
The author of these lines thinks that the solution of the Russellian problem renders possible a consistent formula of strict implication
