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fuzzy logic (was: scalar polarity)

On Wed, 22 Nov 1995, Scott Brickner wrote:

> Ever hear of "fuzzy logic"?  The whole point of multi-valued logics is
> to permit degrees of truth.  The entire field says that not only is the
> statement useful, it is critical.  Statements about the real world are
> *never* completely true nor completely false.  Requiring them to be so
> leads to sorites paradoxes and Goedel's theorem, ultimately undermining
> all of mathematics.

The reason that statements about the real world are never completely true
or completely false is that human language is about abstraction.  So,
statements about reality are only as good as the abstractions.  Also,
statements can be self-contradictory (paradoxes).  In the *real* world,
paradoxes are resolved by correcting abstraction problems or by isolating
false elements.

> While everyone might agree that "At six feet and three inches, John is
> a tall man" is true, most would agree that "At seven feet and four
> inches, Joe is a tall man" is *more* true.  Truth values near 1/2 (the
> range is generally [0,1]) represent "I don't know" or "I'm not sure"
> answers, with leanings toward one side or another.

The dictionary definition of "tall" doesn't rely soley on measurement.  It
includes the idea of comparison:  Having greater than ordinary height.
You can't learn anything about tall by making a measurement, unless you
then compare it to something.

Someone is tall if they are perceived as having greater height than what
the perceiver judges as average, normal or ordinary.  A person will either
be perceived as having greater height than normal (thus, they are "tall")
or they will be perceived as *not* having greater height than normal
(thus, they are "not tall").  Of course, different people may define
"average, normal or ordinary" in different ways.  That's called
subjectivity.  The whole fuzzy-logic concept is just a way of trying (and
failing) to simulate subjectivity in a mechanistic fashion.

> Fuzzy logic disagrees completely.  They have a formal solution to
> Goedel's paradoxes which cannot be solved otherwise.  Take Bertrand
> Russell's barber, for example, the man who shaves all and only those
> who do not shave themselves.  Who shaves the barber?  Formally, the
> question is whether the barber is a member of the set of those whom
> the barber shaves.

The logical answer is that the statement contradicts itself (it fails a
linguistic "checksum" analysis) and that it must contain incorrect
information.  The statement contains two premises:

1) The barber is shaved.
2) The barber shaves all and only those who do not shave themselves.

Since these sets have a point for which both cannot be valid (which is the
barber), both of them cannot be true.  In the real world, the barber
either shaves himself (#2 is false) or the barber doesn't get shaved (#1
is false).

> For this to be true, t(A) = t(not(A)) must be
> true.  Fuzzy logic permits one to solve for t(A) as follows:
>     t(A) = t(not(A))
>     t(A) = 1 - t(A)   (by the fuzzy definition of "not")
>     t(A) = 1/2
> So the question is maximally fuzzy, admitting to neither a true
> nor a false answer.  Traditional logic must throw up its hands, while
> fuzzy logic gives a definite answer.

If "1/2" is a definite answer, so is "the statement is a half-truth".
And I didn't even need to do any math!  :)  The only reason that
this sort of calculation gives the result "1/2" is because the original
statement is binary, and we already knew that there was a flaw in it...
that is, there is an exception to either the first premise (making that
part untrue) or the second premise (making that part untrue).  The "fuzzy
logic" analysis is just another way of saying the same thing with a
number instead of with words.

> Another important argument favoring fuzzy logic is that it shows the
> solution to sorites paradoxes.  How many grains are necessary for a
> "heap" of sand?

Again, the misuse of words (as with "tall") is the problem here.  A heap
is simply *not* defined as "a collection of X grains."  It is, in fact,
defined as "a group of things haphazardly gathered or in disorder; a
pile." So, a heap has to look like a group, has to look gathered, and has
to look disordered.

> If I remove one grain from a heap, is it still a
> heap?

If it still looks like a group, looks gathered, and looks disordered,
yes.  If it doesn't, no.

> If I keep removing grains, precisely when does it stop being a
> heap?

When it stops looking like a group, stops looking gathered or stops
looking disordered.

> As long as the statement "this is a heap" must have either a
> completely true or completely false answer, the transition from heap to
> not-heap is meaninglessly arbitrary.

Not arbitrary, but subjective.  And *yes*, heap is a manifold subjective

> These and the engineering successes of fuzzy logic suggest that
> including it in lojban is useful.

I'm not convinced.  Perhaps fuzzy logic is of some use when you don't want
to make a definite decision, but you still want to call it a definite
decision.  Or perhaps it is helpful when your information is contradictory
and you don't have the ability or desire to root out the logical problem,
yet you want to pretend that you have some sort of definite conclusion.

Peter Schuerman                                    plschuerman@ucdavis.edu
                        Co-editor, SPECTRA Online
          for back issues: http://www.well.com/user/phandaal/