Ä. Äèìèòðîâ

6 þíè 2010 ã.

Notation:

Ï - intersection

e - belongs to

no-x - x' ( absolute complement )

Let us define in the interval [0,1] the following equations:

a v b= min[a+b,1];

a ^ b=max[a+b-1,0];

no-a=1-a

It is easy to check up the next properties:

- The system is closed.
- The system is commutative.
- The system is associative for both operations.
- The system is compatible.
- The elements 1 and 0 are such, that:

a v 0 = a;

a ^ 0 = 0;

a v 1 = 1;

a ^ 1 = a - a^no-a=0;

a v no-a-1 - The system respons to the laws of De Morgan
- The system in the general case is not distributive.

Only when: a + b + c < = 1 it is distributive. - The system is not idempotent.

We give a name for this system a beforeboolean system(BS).

It is also easy to check up,that the subset 1 and 0 is closed, distributive and idempotent i.e. it is a Boolean algebra.

In the Aristotel’s logic the contentions have degrees of veracity only 1 and 0:

T[I] = 1; T[0] = 0.

Let us look at the contention: x: T[x]=0.

It is equal to: no-x: T[x]=1 and no-x: T[no-x]=0.

In the Boolean algebra of degrees of veracity these equations have no solution.

But in BS it has a solution: x = no-x = 0,5.

- For an arbitrary element x of the set A and an arbitrary generalized subset M there is only one degree of veracity of the contention,that x belongs to M.
- Function of veracity is every single-valued function with area of determination A(all the elements) and area of direction [0,1].FV is associated with GS one-to-one.

- FV of an UNION of two GS M and N is:
Fv( xeM U xeN)=min[Fv(xeM) + Fv(xeN), 1]

- FV of a SECTION of two GS M and N is:
Fv(xeM Ï xeN)=max[Fv(xeM)+Fv(xeN) - 1, 0]

- FV of an ADDITION of GS M is:
Fv(xe no-M)=1 - Fv(xeM)

So,we give a name to the set of all the GS of A as a beforeboolean of A.

In it a zero element is the null set and A is the single element.

Let us define the subsets:

1^n,2^n…a^n < c ^n and 1^n,2^n…b^n < c^n

on the set A( 1^n,2^n…c^n), where a,b,c,n are positive integers.Let us define the beforeboolean of A with the set of FV.

Does the beforeboolean contain a Boolean as a subset?

If it does,then the equation

(1) a^n+b^n=c^n

may have a solution as FV with degrees of veracity 1 and 0 only.

If it does not the equation (1) has not such a solution.

As we saw, the beforeboolean in the general case is not distributive and does not contain a Boolean except of zero and single elements.

To be the beforeboolean distributive it is necessary and enough(a1,a2,a3-positive integers):

Fv(xea1^n)+Fv(xea2^n)+Fv(xea3^n) < =1.

It is easy to check up that this is possible at:

a1^n+a2^n+a3^n<=c^n.

Let the case is:

(2) a1^n+a2^n+a3^n=c^n.

Then it is easy to calculate:

(3) Fv(xea1^n Ï (xea2^n U xea3^n))=Fv(xea1^n)+Fv(xea2^n)+Fv(xea3^n)-1

(4) Fv((xea1^n Ï xea2^n) U (xea1^n Ï xea3^n)) = 0

If:(3)=(4)

Fv(xea1^n)+Fv(xea2^n)+Fv(xea3^n)=1

The deduction is that (1) may have a solution only when (2) has a solution in general case.