Re: Probability Theory Needs Fuzzy Logic

James N Rose (integrity@prodigy.net)
Sun, 22 Nov 1998 19:39:00 +0100 (MET)

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To: BISC Group
From: James Rose <integrity@ceptualinstitute.com ; integrity@prodigy.net>=

Re: Probability Theory Needs an Infusion of Fuzzy Logic to Enhance its
Ability to Deal with Real World Problems [Zadeh 19981111]

In all deference to Prof. Zadeh and everyone else who perceives a need to=
reconcile conventional Probability Theory with Zadeh (nee,
"fuzzy") Logic, the issue really goes to the entire architecture of mathe=
matics. It calls into question the relationships among and
accessibility between, functions and content, in whatever possible config=
urations are defined, discerned, discovered, or proposed.
Especially as we're required to establish viable connections between ind=
uctively reachable, but not necesarily immediately
available, information domains.

Succinctly - we can't meld PT with Zadeh Logic (ZL) unless we discuss the=
m in terms of Godel's theorems, and, find a reasonable way
to surmount the Incompleteness conclusions.

For any bounded set (defined more by the finite content than by any expli=
cit boundary ... such as the temporal moment "now", which
is forever changing and inclusive of more "explicit information" as it ad=
vances), any particular fixing of a temporal locus along a
timeline can formalize the corresponding "content" to an amount that is n=
ormalizable as "one", the principal upper-bound of standard
probability and Boolean Logic.

Essentially, it randomly fixes and defines the Godel boundary of a consid=
ered system. And yet, we recognize that there is
transboundary information, potential, factors, and relationships, which c=
an't be willy nilly left unconsidered merely because we
don't have the mathematical language or functions to fully acknowledge or=
access what is potentially there. And, we're prompted to
establish just such valid working relationships with all of that open pot=
ential, if only for the fact that ZL says it's viable
useful/usable information.

That notion is inherent in all ZL functions, because the "net probability=
" can and more typically does, surpass the value "one". It
explicitly uses the probability field outside the Boolean/Godelian accept=
ed 'unit' boundary(ies). So, it becomes unavoidably
necessary to examine and specify these factors and relationships which ar=
e seen as common to both systems.

I did this analysis in 1991, and included it in "Understanding the Integr=
al Universe" (1992), which relevant text is online at
<http://www.ceptualinstitute.com/uiu_plus/UIUcomplete1199.htm> Section 5=
=2E "Charting a Path through the Maze". My conclusion is
that all of these relationships and functions can be canopied under the n=
ominal nomenclature of Stochastic Logic. It seems that
Probability Theory does not need "an infusion of Fuzzy Logic to enhance i=
ts ability to deal with real world problems", as Prof Zadeh
has phrased it, because Zadeh Logic is already the more general form of p=
robabilities-relations, of which standard PT is a
restricted sub-set. Comprehensively, Stochastic Logic also includes th=
e companion logic system of Quantum Mechanics, and, even
the post-quantum system of Bohm/Hiley (in which case, information coding =
and transduction takes center stage).

The fact of this has been masked, ever since Huntington (1905) reductioni=
stically pared down Boole's extensive 1853 treatise which
was originally titled "An Investigation of the Laws of Thought on which a=
re founded the Mathematical Theories of Logic and
Probabilitites." Much of Boole's original discourse is very reminiscent =
of the soft-bounded relationships and flexible causal
interdependencies of Zadeh Logic and soft computing.

In his attempt to break away from Aristotelian Logic and that of Clarke a=
nd Spinoza, Boole in fact enunciated the notion of
'conditional existence' (my labeling). Where existence - and non-existen=
ce - occur in relation to other events, dynamics and things
- being present or not, in diverse combinations. So, while he tried to o=
ppose the logic systems of 2.5 millenia, he in fact placed
them in context of a larger group of relations and boundaries.

And this is what Zadeh Logic does also. Let me explain how.

I'll use the notations "a", and "<a>" (for bar-a), and (dot) for the dot=
function.

The basic Boolean identities are :

(1) a (dot) <a> =3D 0
(2) a (plus) <a> =3D 1

and are an entr=E9e to replacing the Aristotelian, A=3DA, identity functi=
on which is inherent in equation (2) ... the binary and equal
probabilities that "a" either exists or it doesn't.

If we translate (1,2) in terms of probabilities, the relationships become=
clearer.

(1) The probability that "a" and "<a>" can co-exist simultaneously, is, z=
ero.
(2) The probability "a"'s existence, plus, the probability of "<a>"s exis=
tence, sums to, one.

Under Boolean Logic, the probabilities of (2) are equal by assumption and=
convention.
Under ZL, they need not be. Nor are the number of parameters limited to =
two. Nor is the function product fixed as, or limited to,
"one".

In the Quantum Mechanical application of function (1), the function produ=
ct is allowed - in fact expected - to be "not-zero", where
the formal enunciation of QM fixes the value at "one"; and the EPR-Bell-A=
spect formulations inherently specify "not-zero, not-one".
Bohm/Hiley, too, allow different product values - which is why parameter =
specifications of their general relations perforce 'reduce
to' standard QM equation forms.

Now, in essense, we are describing the most general form of mathematical =
architecture possible, allowing and expecting "open
unboundedness" as the most accessible apriori form. This, in contra-posi=
tion to the strict interpretation of Godel Incompleteness.
The full house can be sub-partitioned in many alternative ways, (includin=
g the Godel Cut) but it is still one-house, where all
aspects and qualities are presumed compatible and accessible, regardless =
of cut, and regardless of local access restrictions. Each
of the partitionings illuminates important relationships unique to the ch=
osen partitions and definitions, so it is not a case of
"which one is better or more accurate". Each is valid under the chosen o=
r applied constraints.

Any model will always display a 'variance from observation' since any and=
all events must be subject to the full spectrum of
relations and dynamics of all possible partitionings rather than merely a=
ny local focus-group of parameters used by a particular
partitioning method (PT, ZL, etc.).

This basically sums up why I concur with Prof Zadeh's essential message, =
that some form of melding and compatibility among the two
immdediate approaches (partitionings) of concern - PT and ZL - needs to =
be secured, to the benefit of both perspectives. And
specifically, Zadeh Logic is the more expansive of the two approaches, be=
ing able to embrace PT, even as they have their individual
information-processing capabilities, which the other method may not be ab=
le to handle or produce.

=2E..There are 2 other threads which grow out of the above thoughts.

The above has a bearing on the Fractal approach to chaos & complexity. I=
t can be considered a specific partitioning also, which is
modular within a larger frame of reference. In this case, fractal expone=
nts ought to be perceived as dimension-states, albeit
non-integer ones. In general, all exponents are therefore referential to=
the dimensionality of a function or partial function.
With this notion, the landscape of Complexity broadens, and expands to in=
clude the notion that non-zero probabilities of
information/energy exchange .... exponents greater than zero ... are requ=
isite for the enactment of complex organizations which are
behaviorally interconnected. There may be specific recursive exponent r=
elationships ... Mandelbrot sets, etc...., but the primary
dynamic is implicit ... the greater than zero probability of information =
transferred from one configuration set to some next one ...
which in turn illuminates emergent patterns that viewing the simple gener=
al base/exponent definition might not show too readily.

=2E..The last thing I'll talk about here is a specific example (DB 6, if =
you will) of another probability problem experienced
everyday, but having no methodology of solution at the present. It is t=
he "substitution problem" :

If a basketball team is defined as 5 players actively participating in ba=
ll handling on the court, in competition with another team,
what single formula can process the full team roster of non-active as wel=
l as active players? The "team" is the "complexity", as
defined above. It "exists" as long as certain parameters of interaction/=
participation are met. But at any given time the players
on the bench have a participation probability of zero. And, there are co=
nditions which enable the complexity called "team" to exist
in potentia when no games are going on.

In other words, if the team is bounded by the definition of "5 players ac=
tively engaged against another 'team'", then it is
unavoidable that transboundary trans-Godel Limit information/energy must =
be mathematically accessible in order to specify the
alternative conditions under which a "team" remains a viable entity and c=
omplexity. We can't limit the probabilities to only 5
specific players and whether they are active in a game.

My regards to all. I thank you for reading and considering these ideas.

James Rose
Ceptual Institute
integrity@ceptualinstitute.com
integrity@prodigy.net

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To: BISC Group
From: James Rose <integrity@ceptualinstitute.com ; integrity@prodigy.net>

Re:  Probability Theory Needs an Infusion of Fuzzy Logic to Enhance its
Ability to Deal with Real World Problems  [Zadeh 19981111]
 

In all deference to Prof. Zadeh and everyone else who perceives a need to reconcile conventional Probability Theory with Zadeh (nee, "fuzzy") Logic, the issue really goes to the entire architecture of mathematics.  It calls into question the relationships among and accessibility between, functions and content, in whatever possible configurations are defined, discerned, discovered, or proposed.  Especially as we're required to establish  viable connections between inductively reachable, but not necesarily immediately available, information domains.

Succinctly - we can't meld PT with Zadeh Logic (ZL) unless we discuss them in terms of Godel's theorems, and, find a reasonable way to surmount the Incompleteness conclusions.

For any bounded set (defined more by the finite content than by any explicit boundary ... such as the temporal moment "now", which is forever changing and inclusive of more "explicit information" as it advances), any particular fixing of a temporal locus along a timeline can formalize the corresponding "content" to an amount that is normalizable as "one", the principal upper-bound of standard probability and Boolean Logic.

Essentially, it randomly fixes and defines the Godel boundary of a considered system.  And yet, we recognize that there is transboundary information, potential, factors, and relationships, which can't be willy nilly left unconsidered merely because we don't have the mathematical language or functions to fully acknowledge or access what is potentially there.   And, we're prompted to establish just such valid working relationships with all of that open potential, if only for the fact that ZL says it's viable useful/usable information.

That notion is inherent in all ZL functions, because the "net probability" can and more typically does, surpass the value "one".  It explicitly uses the probability field outside the Boolean/Godelian accepted 'unit' boundary(ies).  So, it becomes unavoidably necessary to examine and specify these factors and relationships which are seen as common to both systems.

I did this analysis in 1991, and included it in "Understanding the Integral Universe" (1992), which relevant text is online at <http://www.ceptualinstitute.com/uiu_plus/UIUcomplete1199.htm>  Section 5. "Charting a Path through the Maze".   My conclusion is that all of these relationships and functions can be canopied under the nominal nomenclature of Stochastic Logic.  It seems that
Probability Theory does not need "an infusion of Fuzzy Logic to enhance its ability to deal with real world problems", as Prof Zadeh has phrased it, because Zadeh Logic is already the more general form of probabilities-relations, of which standard PT is a restricted sub-set.   Comprehensively,  Stochastic Logic also includes the companion logic system of Quantum Mechanics, and, even the post-quantum system of Bohm/Hiley (in which case, information coding and transduction takes center stage).

The fact of this has been masked, ever since Huntington (1905) reductionistically pared down Boole's extensive 1853 treatise which was originally titled "An Investigation of the Laws of Thought on which are founded the Mathematical Theories of Logic and Probabilitites."  Much of Boole's original discourse is very reminiscent of the soft-bounded relationships and flexible causal interdependencies of Zadeh Logic and soft computing.

In his attempt to break away from Aristotelian Logic and that of Clarke and Spinoza, Boole in fact enunciated the notion of 'conditional existence' (my labeling).  Where existence - and non-existence - occur in relation to other events, dynamics and things - being present or not, in diverse combinations.  So, while he tried to oppose the logic systems of 2.5 millenia, he in fact placed them in context of a larger group of relations and boundaries.

And this is what Zadeh Logic does also.  Let me explain how.

I'll use the notations  "a", and "<a>" (for bar-a), and (dot) for the dot function.

The basic Boolean identities are :
 
     (1)    a  (dot)  <a> = 0
     (2)    a  (plus) <a> = 1

and are an entrée to replacing the Aristotelian, A=A, identity function which is inherent in equation (2) ...  the binary and equal probabilities that "a" either exists or it doesn't.

If we translate (1,2) in terms of probabilities, the relationships become clearer.

(1) The probability that "a" and "<a>" can co-exist simultaneously, is, zero.
(2) The probability "a"'s existence, plus, the probability of "<a>"s existence, sums to, one.

Under Boolean Logic, the probabilities of (2) are equal by assumption and convention.
Under ZL, they need not be.  Nor are the number of parameters limited to two.  Nor is the function product fixed as, or limited to, "one".

In the Quantum Mechanical application of function (1), the function product is allowed - in fact expected - to be "not-zero", where the formal enunciation of QM fixes the value at "one"; and the EPR-Bell-Aspect formulations inherently specify "not-zero, not-one".  Bohm/Hiley, too, allow different product values - which is why parameter specifications of their general relations perforce 'reduce to' standard QM equation forms.

Now, in essense, we are describing the most general form of mathematical architecture possible, allowing and expecting "open unboundedness" as the most accessible apriori form.  This, in contra-position to the strict interpretation of Godel Incompleteness. The full house can be sub-partitioned in many alternative ways, (including the Godel Cut) but it is still one-house, where all aspects and qualities are presumed compatible and accessible, regardless of cut, and regardless of local access restrictions.   Each of the partitionings illuminates important relationships unique to the chosen partitions and definitions, so it is not a case of  "which one is better or more accurate".  Each is valid under the chosen or applied constraints.

Any model will always display a 'variance from observation' since any and all events must be subject to the full spectrum of relations and dynamics of all possible partitionings rather than merely any local focus-group of parameters used by a particular partitioning method (PT, ZL, etc.).
 
This basically sums up why I concur with Prof Zadeh's essential message, that some form of melding and compatibility among the two immdediate approaches (partitionings) of concern -  PT and ZL - needs to be secured, to the benefit of both perspectives.  And specifically, Zadeh Logic is the more expansive of the two approaches, being able to embrace PT, even as they have their individual information-processing capabilities, which the other method may not be able to handle or produce.
 

...There are 2 other threads which grow out of the above thoughts.

The above has a bearing on the Fractal approach to chaos & complexity.  It can be considered a specific partitioning also, which is modular within a larger frame of reference.  In this case, fractal exponents ought to be perceived as dimension-states, albeit non-integer ones.  In general, all exponents are therefore referential to the dimensionality of a function or partial function.  With this notion, the landscape of Complexity broadens, and expands to include the notion that non-zero probabilities of information/energy exchange .... exponents greater than zero ... are requisite for the enactment of complex organizations which are behaviorally interconnected.   There may be specific recursive exponent relationships ... Mandelbrot sets, etc...., but the primary dynamic is implicit ... the greater than zero probability of information transferred from one configuration set to some next one ... which in turn illuminates emergent patterns that viewing the simple general base/exponent definition might not show too readily.

...The last thing I'll talk about here is a specific example (DB 6, if you will) of another probability problem experienced everyday, but having no methodology of solution at the present.   It is the "substitution problem" :

If a basketball team is defined as 5 players actively participating in ball handling on the court, in competition with another team, what single formula can process the full team roster of non-active as well as active players?   The "team" is the "complexity", as defined above.  It "exists" as long as certain parameters of interaction/participation are met.   But at any given time the players on the bench have a participation probability of zero.  And, there are conditions which enable the complexity called "team" to exist in potentia when no games are going on.

In other words, if the team is bounded by the definition of "5 players actively engaged against another 'team'", then it is unavoidable that transboundary trans-Godel Limit information/energy must be mathematically accessible in order to specify the alternative conditions under which a "team" remains a viable entity and complexity.   We can't limit the probabilities to only 5 specific players and whether they are active in a game.

My regards to all.  I thank you for reading and considering these ideas.

James Rose
Ceptual Institute
integrity@ceptualinstitute.com
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