Consistent generalization of classical two-valued
Boolean theories into real-valued
Dragan Radojević
Mihajlo Pupin Institute, Volgina 15, 1100 Belgrade, Serbia
[email protected]
Classical mathematical theories in the Boolean frame – the theory of sets,
mathematical logic, theory of relations, probability, etc., are based on a black-white
outlook. The black-white or two-valued outlook is inadequate for treating
impreciseness inherent to many real problems (from cognition, natural languages to
quant physics phenomena). This was a motive for the mathematical treatment of
gradations and/or for developing real-valued theories: fuzzy sets [1], fuzzy logic [2],
[3], fuzzy relations, etc. Another very important motive for treating gradations is
reducing the complexity immanent to the mathematical analysis of real problems by
classical approaches. For example: two elements of the analyzed universe of
discourses can be discerned by the chosen property in the black-white or two-valued
approaches, only if one has and the other hasn’t this property. As a consequence, the
number of necessary properties (complexity of problem) increases with the increasing
number of elements which should be discerned between them. The complexity of this
problem can be drastically reduced by introducing gradation since it is possible to
discern elements by the same property on the basis of its realized intensities.
Consistent Boolean generalization of two-valued into a real-valued theory means
preservation of all of its algebraic – value indifferent characteristics: Boolean axioms
and theorems. Actually two-valued theories in Boolean frame are based on the
celebrated two-valued realization of Boolean algebra (BA) and their real-valued
consistent generalization should be based on a real-valued realization of BA.
The conventional real-valued theories: fuzzy sets, fuzzy logic, fuzzy relations, fuzzy
probability, etc., are not in Boolean frame [4] and/or they are not consistent Boolean
generalization of classical theories.
Interpolative Boolean algebra (IBA) [5], [6] is a real-valued realization of atomic or
finite BA. The real-valued realization of atomic or finite BA is adequate for any real
problem since gradation offers superior expressiveness in comparison to the blackwhite outlook.
TECHNIQUES OF IBA REALIZATION
Technically, IBA is based on generalized Boolean polynomials (GBP-s), [6]. GBP
 uniquely corresponds to the analyzed element  BA of atomic BA. GBP is a
mapped disjunctive canonical form of analyzed BA element, on a value level as its
polynomial “figure”, which can process values from a real unit interval [0, 1] so as to
preserve all algebraic properties of this element by corresponding arithmetic
consequents. For example: if one element of BA is included in another
  ,  ,  BA (algebraic property) then, in all possible value realizations, the
value of the first element is less or equal to the value of the second   
(arithmetic property).
Disjunctive canonical form and structure (content) of BA element
Any element  BA   of analyzed atomic BA generated by the finite set of free
(primary) variables   a1 ,...,an  can be represented in a disjunctive canonical form
as the union (join) of relevant atoms   S   def
Ca j ,  S       :
ai
ai S
a j \S
S  .

S     S  1
Which atoms are relevant for the analyzed element is determined by its structure
(content) – defined by the following set function:
1,   S   ;    S       S  

  S   def 
;  S      ; , 0  BA     .

0 ,   S   ;    S     0 
Structure is a value indifferent – algebraic characteristic and/or it is invariant on
the type of value realization (two-valued, many-valued and/or real-valued).
The structure or content   of any element  BA   of the analyzed BA is
homomorphism
  S     S     S  ,
  S     S     S  ,
 S    ,
C  S     S  ;

,  BA    ,   S  ,  S   0,1 .
and/or it homomorphically maps the analyzed BA -  :      0, 1    BA     .
Structure functionality principle is a fundamental algebraic principle defined in the
following way: the structure (content) of any combined element of analyzed BA can
be determined directly on the basis of structures of its components.
All Boolean axioms and theorems defined for elements of BA satisfy the structures
of these elements too. Since the values of structure components of analyzed element
of BA are coincident with the values of this element in two-valued realizations:


1, ai  S
  a1S ,...,anS     S  ,  aiS  def 
; ai  , S      
0, ai  S


From above it follows that truth functionality principle is the realization of structure
functionality principle on a value level in and only in the case of two-valued
realization. This principle is arithmetic and as a non-algebraic characteristic it
couldn’t be a basis for consistent generalization.
Generalized Boolean polynomials
A generalized Boolean polynomial (GBP) uniquely corresponds to any element of
analyzed atomic BA. GBP is the superposition of relevant atomic GBP-s. Which
atoms are relevant for the analyzed element of BA is defined by its structure (content).
Atomic GBP is defined by the following expression:
  S   a1 ,...,an  
ai
1 a j .
 
ai S
a j \S

Where:  is generalized product (from min function to the standard product  ).
Atomic GBPs have the following properties: (a) their sum is identically equal to 1
  S   a1 ,...,an   1 ; and (b) they are nonnegative   S   a1 ,...,an   0.

S  
GBP  of analyzed element  BA   is the superposition of relevant atomic
GBP-s, defined by its structure (as a polynomial figure of disjunctive canonical form):
  a1 ,...,an  
  S   a1 ,...,an  
  S    S   a1 ,...,an 

S     S  1

S  
In a classical two-valued realization, as a special case, only one atom is equal to
one and all others are equal to zero. All elements of BA in which a realized atom (with
value 1) is included are equal to 1 and all other are equal to 0. As a consequence in a
classical case for a given realized atom any element of BA has a value equal to 1 or 0
(excluded middle) and can’t be 1 and 0 (contradiction).
In a general value-irrelevant case, excluded middle means that any atom of BA is
included in the analyzed element of BA or in its complement and contradiction means
that any atom can’t be included in the analyzed element and in its complement.
ILLUSTRATION OF APPLICATIONS
Many-valued logic and fuzzy logic in the narrow sense and IBA
Real valued logic based on IBA is in the Boolean frame contrary to conventional
fuzzy logics and many-valued logic. Example: all phenomena in quant physics can be
explained by real-valued logic based on IBA and or algebra of consistent quantum
logic is Boolean, contrary to conventional quantum logics.
Non-classical logical measures and integrals and IBA
Any logical measure can be represented as a linear convex combination of 0-1
measures and/or structures of corresponding logical functions. As a consequence a
generalized integral is equal to a linear convex combination of corresponding GBP-s
in which a generalized product figures: all operators from ordinary product to min
function. For examples: (a) Choquet integral is a linear convex combination of
monotone GBP-s with a generalized operator defined as min function; (b)
Generalized Choquet integral is a linear convex combination of not only monotone
GBP-s, with a generalized operator defined as a min function too.
Decision theory and preference modeling and IBA
Any DM’s partial logical consistent demand in a general case can be represented as
a logical function and as a consequence by a corresponding GBP. A global demand in
a general case is a linear convex combination of partial demands with coefficients
which reflect the importance of partial demands. So, a global demand is a linear
convex combination of corresponding GBP-s.
A preference structure is a set of atomic GBP of BA generated by two real-valued
relations: great preferences    and their transpose        , [7].
T
Relations to probability theory and statistics and IBA
Classical additive probability is based on the classical theory of sets and as a
consequence it is in Boolean frame. Real probability [8] is a consistent generalization
of classical additive probability based on real sets consistent generalizations of
classical sets based on IBA, [6].
Classical non-additive probability is based on classical relations and as a
consequence it is in Boolean frame too. Real non-additive probability is a consistent
generalization of classical case based on real relations, consistent generalizations of
classical relations [8].
CONCLUSION
In the analysis of real problems, in a general case, by using gradation – real-valued
realization, one can do much more (efficiency) by much less (complexity) than by a
classical black-white outlook, on which are based classical two-valued theories in
Boolean frame. Consistent Boolean generalization of classical two-valued theories
into real-valued theories means preservation of all value irrelevant – algebraic
characteristics (all axioms and theorems of BA). Since classical theories are based on
celebrated two-valued realization of BA, consistent generalized theories have to be
based on real-valued realization of BA. Interpolative Boolean algebra (IBA) [6] is a
real-valued realization of BA. In IBA any Boolean function (element of a
corresponding BA) is uniquely mapped into a generalized Boolean polynomial (GBP),
able to process values from a real unit interval [0, 1], so that it preserves on a value
level algebraic properties by corresponding arithmetic properties. Any element of BA
and/or GBP has a structure (content) as an algebraic characteristic. Structure
functionality principle as an algebraic one, has its value interpretation known as a
truth functionality principle for and only for two-valued realizations. As a
consequence all generalizations based on truth functionality principle are nonconsistent generalizations since they produce theories which can’t be in Boolean
frame. Superior characteristics of theories based on IBA are mentioned in this paper.
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