FR MNC II Contingency and necessity. The a priori conditions of experience. Interpretation of the formula M L(p) v M Mp v M M~p. 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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M L(p) v M Mp v M M~p

represents

the a priori conditions of experience.

x

x

– 1  M L(p)

         and

   ~M L(p), L M(p), ~L M(p)

x

M L(p) represents the possibility of certainty L(p) or Lp w L~p  and therefore the possibility of knowledge.

 ~M L(p) represents the im-possibility of knowledge.

 ~M L(p) the im-possibility of  knowledge is equivalent to L M(p) the necessity of the bilateral possible, in other words, to the necessity of a state of ignorance.

   L M(p) represents  the necessity of a bilateral possible directly, represents the necessity of a state of ignorance about p and not-p.

If p and not-p are both possible, that means, I repeat, that neither p nor not-p is certain.

Mp & M~p is equivalent  to  ~L~p & ~Lp.

L M(p) symbolizes more explicitly than its equivalent ~M L(p) the necessity of a state of ignorance

~L M(p) excludes the necessity of a state of ignorance.

As ~L M(p) is equivalent to  M L(p), ~L M(p) tells us the sense conveyed  by the initial  M L(p).

M L(p) excludes the fact that we might be condemned to non-knowledge concerning the facts p and not-p.

Suppose one does not know whether Lady Jane left at five o’clock or did not leave at five 0′clock. The state of ignorance concerning the two contingent facts p and not-p in question is itself a contingent fact. A priori, the conceivable fact that Lady Jane left at five and the conceivable fact that she did not are both possible. Only experience can tell us which of the facts is the case. Obviously, we may also be in a state of ignorance concerning the two conceivable contingent facts in question. but nothing in them condemns us to be in a state of ignorance about them. A priori, it may happen that Lady Jane left,  it may happen that she didn’t, and it may happen that you cannot say. A priori, the world where we live and act does not exclude any of those three possibilities.

  On the contrary, the world where we live and act  does not allow us to think that in certain circumstances the sum of three angles of a triangle is not 180 degrees and that it may happen that we should be in a state of ignorance about the fact. We know as necessary that the sum of the three angles is 180 degrees. This sum can be nothing else than 180 degrees whatever the circumstance may be. If the fact p in question is envisaged and duly symbolized by L Lp or by ~M M~p, to be read im-possibility of the possibility of not-p, the possibility of not-p is excluded and with it the possibility of ignorance.

  For M~p, which excludes Lp is equivalent to L~p w M(p).

 Lp, L~p, M(p) being mutually contrary facts, M~p i.e ~Lp implies that one of two things, either L~p or M(p) is the case.  L~p excludes Lp and so does also Mp & M~p, the bilateral possible. Therefore L Lp (or ~M M~p)  a priori excludes two things : L~p, of course, but also M(p) (or Mp & M~p), the bilateral possible corresponding to a state of non-knowledge.

More about that necessary non-knowlege which  M L(p) wards off.  M(p) is the abridgement of  Mp & M~p symbolizing the bilateral possible or state of ignorance. If L precedes M(p), that means that  the state of ignorance in question is inescapable, necessary when certains facts are considered. Let p be the conceivable fact that the world had a beginning and not-p the conceivable fact that the world had no beginning. Both Lp and L~p are excluded and the state of ignorance: ~L~p & ~Lp, Mp & M~p and finally M(p) is necessary and results from the nature of the facts envisaged.

  Quite different is the case of Lady Jane having left or not at five o’clock. If one is not in a position to say  Lady Jane left at five or Lady Jane did not leave at five, we are also in a state of ignorance Mp & M~p but the non-knowledge now is a contingent fact. Nothing in the matter of the judgement now considered, I think of the subject Lady Jane and of the predicate left at five condemns us to an inescapable and definitive state of non-knowledge about p and not-p.

 For this reason, to describe such a state of ignorance,  one must add to the symbol M(p) of the bilateral possible, the symbol M L(p) signifying ~L M(p). The representation  of the contingent fact consisting in our non-knowledge about the comings and goings of the lady will be something like this:

  M(p) . M L(p)

We have to do with a state of non-knowledge but a state of non-knowledge which is not necessary but contingent.

The present case contrasts with the case of the world having  a beginning or not.

 In the case of the world having a beginning or not, our state of non-knowledge is a necessary fact to be symbolized by ~M L(p) im-possibility to reach certainty about p and not-p or by L M(p) necessity of the state of ignorance.

Once again – bis repetita placent –  in the case of Lady Jane having left or not, our state of ignorance is a contingent fact, which could have not occurred and therefore is to be represented by M(p). M L(p)

M(p) represents our ignorance about what the lady did at five 0′ clock. By adding M L(p), possibility of certainty, we want to indicate that this ignorance is a contingent fact, not  necessary, not resulting from the nature of the facts p and not-p considered.  Being equivalent to ~L M(p),  M L(p) excludes the necessity of the bilateral possible, that is to say, the necessity of the state of ignorance.  We don’t know, we can’t say but like other contingent facts  this state of non-knowledge could have not occurred.

M(p) can be necessary, for example, in the case of the world having a beginning or not

M(p) can be contingent, for example, in the case of Lady having left or not.

Let us write about  M(p):

(l)   M(p)   ≡    L  M(p)  w   M(p) . M L(p)

Let us read the equivalence thus:

 A state of ignorance  M(p) is

 either necessary L  M(p)

e.g  Maybe the world had a beginning, maybe the world had no beginning

or contingent  M(p) . M L(p)

 e.g  Maybe Lady Jane left at five o’clock, maybe she didn’t

x

x

 

– 2  M  Mp

         and

 ~M Mp, L L~p, ~L L~p

 

 M Mp represents the possibility of  Mp possibility of p.

x

 ~M Mp is equivalent to L L~p

The im-possibility of Mp the possibility of p is equivalent to L L~p  the necessity of the fact not-p.

L L~p symbolizes the necessity of the fact not-p more explicitly than its  equivalent ~M Mp.

 

~ L L~p  excludes the necessity of the fact not-p.

As ~L L~p is the strict equivalent of  M Mp, ~L L~p tells us the sense conveyed  by the initial

M Mp

M Mp excludes the necessity of the fact not-p.

x

M Mp represents the possibility of the possibility of p. The expression  possibility of the possibility of p is not readily understood and sounds rather st range.  Still, it has a sense and  when one wants to describe what contingency consists in we have to use it.

Suppose I say Lady Jane did not leave at five o’clock. My statement makes known as certain a fact not-p. The fact not-p envisaged is certain indeed but not at all necessary. The fact is certain and its representation must contain L~p but the certainty of it does not result from the nature of the subject and from that of the predicate and so is contingent.

Quite different is the fact that the boiling-point of water is not at 0 degree. The fact not-p now is not only certain but also necessary, on account of the well known fact that the boiling-point of water is at 100 degrees. The boiling-point is precisely the criterion employed to define by convention the temperature of 100 degrees.

The necessary fact in question is the content of an assertive sentence negative in form: The boiling-point of water is not at 0 degree. It must be symbolized by ~M Mp or by the still more explicit L  L~p.

~M Mp  Im-possible is the possibility of p, if p is described as the fact that the boiling-point of water is at 0 degree.

Therefore,

 L  L~p  Necessary is not-p, the fact that the boiling-point is not at 0 degree.

A fact not-p which is certain and must be symbolized by L~p is

either necessary  L  L~p

e.g  The boiling-point of water is not at 0 degree

or contingent       L~p . M Mp

e.g   Lady Jane did not leave at five o’clock

Let us write L~p the following formula of equivalence :

(l)   L~p   ≡    L  L~p  w   L~p . M Mp

 

 

 

– 3  M  M~p

         and

 ~M M~p, L  Lp, ~L  Lp

 

 M M~p represents the possibility of  M~p possibility of not-p.

 ~M M~p is therefore equivalent to L Lp

The im-possibility of M~p the possibility of not-p is equivalent to L  Lp  the necessity of the fact p.

L  Lp symbolizes the necessity of the fact p more explicitly than its  equivalent ~M M~p.

~ L Lp  excludes the necessity of the fact p.

As ~L Lp is the strict equivalent of  M M~p, ~L Lp tells us the sense conveyed  by the initial

M M~p

M M~p excludes the necessity of the fact p.

M M~p represents the possibility of the possibility of not-p. The expression  possibility of the possibility of not-p is not readily understood.  Still, it has a sense and a use.

Suppose I say Lady Jane left at five o’clock. My statement makes known as certain a fact p which obviously is not necessary. This fact is certain and and its representation must contain Lp. But it is no less obvious that this certain fact Lp is not at all necessary. The certainty of it does not result from the subject and  the predicate constituting the matter of the judgement  apprehending it.

Quite different is the fact that the sum of the angles of a triangle is 180 degrees. The fact is not only certain but also necessary. The necessary fact in question is the content of an assertive sentence affirmative in form: The sum of the angles of a triangle is 180 degrees. It must be symbolized by ~M M~p or by the still more explicit L  Lp.

~M M~p  Im-possible is the possibility of not-p, if not-p is described as the fact that the sum of the angles of a triangle is not 180 degrees.

Therefore,

 L   Lp  Necessary is p, the fact that the sum of the angles of a triangle is 180 degrees.

 A fact p which is certain and must be symbolized by Lp is

either necessary  L   Lp

e.g   the sum of the angles of a triangle is 180 degrees

or contingent       Lp . M M~p

e.g   Lady Jane left at five o’clock

Let us write about Lp the following formula of equivalence:

(l)   Lp   ≡    L  Lp  w   Lp . M M~p

 

x

Interpretation of the formula

M L(p) v M Mp v M M~p

x

 M L(p) v M Mp v M M~p

≡

(M L(p)  &  M Mp  & M M~p)  w ( ~M L(p) w~M Mp w ~M M~p)

x

In the developped form of

ML(p) v MMp v MM~p

x

1)    ML(p) & MMp & MM~p

     represents contingency or non-necessity

2) ~ML(p) w ~MMp w ~MM~p

represents necessity or  non-contingency

x

 

 In  ~M L(p) , ~M Mp , ~M M~p  is used the modal prefix M corresponding to the idea of possibility. The prefix M is preceded by a sign of negation.

We saw above  that

~M L(p) , ~M Mp , ~M M~p

can be replaced by expressions in which is used the modal prefix L :

L  M(p), L L~p, L Lp.

So

~M L(p) w ~M Mp w ~M M~p

representing necessity will be advantageously replaced by

 L M(p) w L  L~p   w L  Lp

This exclusive disjunction  represents more explicitly the idea of necessity. Moreover it shows that as foreseen two of the three necessities : necessity of non-knowledge about p and not-p, necessity of p, necessity of not-p cannot coexist in reality.

M L(p) & M Mp & M M~p

represents contingency, in other words, non-necessity. Therefore it may be convenient sometimes to replace  M L(p) & M Mp & M M~p by its strict equivalent:

~L M(p) &  ~L  L~p  &  ~L  Lp

 

Note this:

L conveys the sense of necessary only when employed before another L in the expressions

L  L~p , L  Lp

or before M(p)

in L M(p)

symbolizing an inescapable state of ignorance.

x

x

We are describing  now the mould in which are cast our scary symbols

M L(p) , M Mp , M M~p,  ~M L(p) , ~M Mp , ~M M~p

x

The mould in question is:

a v b v c

≡

(a & b & c)   W  (~a  w ~b  w ~c)

x

– First remark:

a v b v c

is equivalent to

(a v b)  & ( a v c)   &  (b v c)

– Second remark:

In virtue of  De Morgan’s laws,

(a v b) is equivalent to ~(~a & ~b)

( a v c) is equivalent to ~(~a & ~c)

(b v c)  is equivalent to ~(~b & ~c)

– Third remark:

(a v b) is equivalent to ~(~a & ~b)

~(~a & ~b) is the fact that the conjunction of the facts not-a and not-b is excluded from reality.

~(~a & ~b)

 exclusion of the conjunction  ~a & ~b

is equivalent to the following exclusive disjunction:

(a & b)  w  (a & ~b)  w  (~a w b)

If  ~a & ~b is excluded,

one of three things,

either  a & b   or  a & ~b  or  ~a w b

x

Any of those three conjunctions excludes the conjunction ~a & ~b

Obviously  (a & ~b)  w  (~a w b) can be replaced by the simpler and more elegant:

~a w ~b

(a & b)  w  (a & ~b)  w  (~a w b) is equivalent to

(a & b) w (~a w ~b)

As through the sign ‘w’ called operator of the exclusive disjunction, ~a  and ~b are described as contradictory facts , one  is  the exclusion of the other.

~a excludes ~b and therefore means b

~b excludes ~a and therefore means a

(a & b) w (~a w ~b)

 means that

a v b

represents the fact that either one has a & b, the conjunction  of the two facts a and b or the exclusion of only one of them.

In the same way

( a v c)   ≡   (a & c) w (~a w ~c)

(b v c)    ≡   (b & c) w (~b w ~c)

x

Therefrom it follows that

a v b v c,

that is to say,

a v b &  a v c  &  b v c

is equivalent to

(a & b & c)   W  (~a  w ~b  w ~c)

The facts a b c may be all three coexistent . If they are not all three coexistent, only one of them is excluded from reality.

x

The expression ML(p)  v  MMp v   MM~p

is of the form  a v b v c  which we have just interpreted.

Therefore

ML(p)  v  MMp v   MM~p

≡

(ML(p)  &  MMp  & MM~p)  w ( ~ML(p) w~MMp w ~MM~p)

x

In the developped form of

ML(p) v MMp v MM~p

 

1)    ML(p) & MMp & MM~p

     represents contingency or non-necessity

2) ~ML(p) w ~MMp w ~MM~p

represents necessity or  non-contingency