BEFOREBOOLEAN SYSTEM

Ä. Äèìèòðîâ
6 þíè 2010 ã.


Notation:

Ï - intersection
e - belongs to
no-x - x' ( absolute complement )

BEFOREBOOLEAN SYSTEM - Definition


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:

  1. The system is closed.
  2. The system is commutative.
  3. The system is associative for both operations.
  4. The system is compatible.
  5. The elements 1 and 0 are such, that:
    a v 0 = a;
    a ^ 0 = 0;
    a v 1 = 1;
    a ^ 1 = a
  6. a^no-a=0;
    a v no-a-1
  7. The system respons to the laws of De Morgan
  8. The system in the general case is not distributive.
    Only when: a + b + c < = 1 it is distributive.
  9. 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.

DEGREE OF VERACITY


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.

GENERALIZED SUBSETS(GS) AND FUNCTIONS OF VERACITY(FV)

  1. 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.
  2. 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.

DETERMINATIONS

  1. FV of an UNION of two GS M and N is:

    Fv( xeM U xeN)=min[Fv(xeM) + Fv(xeN), 1]

  2. FV of a SECTION of two GS M and N is:

    Fv(xeM Ï xeN)=max[Fv(xeM)+Fv(xeN) - 1, 0]

  3. FV of an ADDITION of GS M is:

    Fv(xe no-M)=1 - Fv(xeM)

BEFOREBOOLEAN OF A SET

It is easy to check up that the set of all the GS of A is a beforeboolean system.
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.

FLT


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.