FR MNC I Contingency and necessity. The a priori conditions of experience. Something about Bertrand Russell’s An inquiry into meaning and truth.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

Adresse électronique :

Jean-francois.monteil@neuf.fr

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My treatise on modal logic for grammarians treats of contingency in a somewhat detailed way. I refer the reader thereto immediately. Precious diagrams are to be found there.

KNOLmnc 1 Traité de logique modale

KNOLmnc 1 Crucial importance of the bilateral possible M(p), the third contrary fact represented in the logical hexagon of Robert Blanché applied to modal logic.

 

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Above a diagram for modal logic. It represents the two triads to be considered first:

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-the  triad of contrary facts:

Lp,      certainty of the fact p

L~p,    certainty of the fact not-p

M(p) that is to say Mp & M~p, the bilateral possible symbolizing a state of ignorance

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-the triad of subcontrary facts:        

L(p) that is to say Lp w L~p excluding the bilateral possible M(p) and so symbolizing a state of knowledge

Mp, possibility of the fact p, that is to say, ~L~p, exclusion of the certainty of the fact not-p

M~p,  possibility of the fact not- p, that is to say, ~Lp, exclusion of the certainty of the fact p

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In  An inquiry into meaning and truth Chapter 20, Bertrand Russell envisages as a conceivable fact p the fact that it snowed on Manhattan Island on the first of January in the year 1 Anno Domini and as a conceivable fact not-p the 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 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.

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~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.
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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.

Mp & M~p is equivalent to  ~L~p & ~Lp

Due to its importance, the bilateral possible Mp & M~p will be represented by the abridgement M(p)

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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~p

or

(l) p  ≡ ~~p

The facts p and not-p are necessarily, a priori contradictory. They are incompatible and they cannot be both excluded from reality.

The prefix  (l) symbolizes here the a priori necessity of the fact  p w~p

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The two  principles  of modal logic

-1  The law of excluded middle

(l)  p w ~p   The conceivable facts p and not- p are a priori contradictory. They are necessarily in-compatible and cannot be both excluded.

Here is found the law of excluded middle. It is formulated in terms of fact and not in terms of proposition, as is usually done.

-2 The law of the opposition knowledge versus ignorance

(l)  Lp w L~p w (Mp & M~p)

 The bilateral possible Mp & M~p corresponding  to a state of ignorance will often be replaced by M(p).

Likewise  Lp w L~p corresponding to a state of knowledge and to the ability to ascribe certainty either to p or to not-p will usually be replaced by the abridgement L(p).

The law  knowledge versus ignorance can conveniently be represented thus:

(l)  L(p) w M(p)

Diagram for modal logic

The document called diagram for modal logic which I supply once again shews the reader the fortunate consequences in modal logic of Robert Blanché’s logical hexagon being used in its simplified form. The latter is a single triangle. The hexagon can be simplified and appear in the form of a single triangle.

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 The sides of it correspond to the three contrary facts:

-Lp  certainty of p

-L~p  certainty of not-p

– M(p) that is to say Mp & M~p

the most important bilateral possible

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The bilateral possible M(p) is the third contrary fact Y introduced by Robert Blanché and duly represented in his hexagon when applied to modal logic.

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The  vertices of the triangle correspond to the three subcontrary facts:

– L(p) or Lp w L~p

the state of  knowledge contradicting the bilateral possible or state of ignorance

– Mp possibility of p

– M~p possibility of not-p

 

 L(p) is the third subcontrary fact U introduced by Robert Blanché and duly represented in his hexagon when applied to modal logic. There is a relation of contradictoriness between the subcontrary fact U and the contrary fact Y.  L(p) is the abridgement of Lp w L~p. So L(p) signifies One of two things, either p is certain or not-p is certain. In both cases,  the bilateral possible M(p) that is to say, Mp & M~p is excluded for Mp & M~p is nothing else than ~L~p & ~Lp.

Lp certainty of p and L~p certainty of not-p are contrary facts. This relation of contrariness between Lp and L~p shall be here symbolized by (l) ~(Lp & L~p) to be read

(l) ~ Necessarily excluded is Lp & L~p the conjunction of the certainty of p and the certainty of not-p

Lp and L~p are contrary facts: firstly they are in-compatible,  secondly they can be both excluded.

The fact that they can be both excluded explains why we must say that Lp and L~p are contrary facts and not contradictory facts.

Contradictory facts are the facts p and not-p and that necessarily.

They are in-compatible but moreover they cannot be both excluded.

The latter feature distinguishes the relation of contradictoriness existing between p and ~p from the relation of contrariness existing between Lp and L~p.

To stress the difference between the contradictoriness of p and ~p and the contrariness of Lp and L~p, let us write:

(l) p w ~p   One of two things, either the fact p or the fact not-p

(l) Lp w  L~p w (~L~p & ~Lp) One of three things, either certainty of p or certainty of not-p or exclusion of both.

The exclusion of either certainty represents the bilateral possible  Mp & M~p. The facts  p and not-p are said to be both possible if neither p nor not-p can be can be said  to be certain.

I am of opinion that

(l) p w ~p   One of two things, either the fact p or the fact not-p

is not a bad formulation of the law of excluded middle.

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• KNOLmnc 1 To defend his views about modal logic and strict implication, Jean-François Monteil utilizes the chapter of Bertrand Russell’s An inquiry into meaning and truth entitled The law of excluded middle.

• 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.

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The diagram below shows that the triad of contrary facts A, E, Y becomes the triad Lp, L~p, M(p) in modal logic and the triad of subcontrary facts U, I, O becomes the triad L(p), Mp, M~p.

 

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Bis repetita placent. Once again, let us say that the bilateral possible  Mp & M~p or M(p) is the fact that both p and not-p are possible because neither not-p nor p is certain.

If the prefix L is employed instead of the prefix M, the bilateral possible Mp & M~p can be represented as ~L~p & ~Lp. For Mp, the fact that p is possible is nothing else than ~L~p  the fact that not-p is not certain. In the same manner  M~p the fact that not-p is possible is nothing else than the fact that p is not certain.

In so far as Lp w L~p or L(p) is the fact that either p or not-p is certain, necessarily it excludes the bilateral possible  Mp & M~p or M(p). So there is a relation of contradictoriness between the third subcontrary U introduced by Robert Blanché and the third contrary Y introduced by the same in modal logic.

– Mp possibility of p,

Mp is the traditional subcontrary fact I , already represented in the square of opposition when applied to modal logic.

There is a relation of contradictoriness between  the subcontrary fact I that is Mp in modal logic and the contrary fact E that is L~p in modal logic.

. Clearly, Mp, possibility of p, is equivalent of  ~L~p, the exclusion of the certainty of not-p.

– M~p,  possibility of not-p

M~p is the traditional subcontrary fact O , already represented in the square of opposition when applied to modal logic. There is a relation of contradictoriness between the subcontrary fact O that is M~p in modal logic and the contrary fact A that is Lp in modal logic. Clearly, M~p, possibility of not-p, is equivalent to  ~Lp,  exclusion of the certainty of p .

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The document below shows how the logical triangle can be used for studying the process of quantification in a system of six propositions of natural language.

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carré  hexagone  triangle

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So, the logical hexagon that Robert Blanché described in his Structures intellectuelles ( Vrin, 1966) can be simplified and appear in the form of a single triangle the sides and the vertices of which can be used to symbolize  a triad of contrary facts on the one hand and a triad of subcontrary facts on the other. The hexagon in its simplified form is very useful both in the study of quantification and in that of modal logic. Particularly, it shows the existence and importance of the notion of bilateral possible. Grasping the existence and importance of the bilateral possible is the condition sine qua non of the problem of strict implication  L   p ≡ Lq  being at last  solved.

Mp & M~p

or

M(p)