SUO: Re: Plea for Relements
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[Reposting after 16 hrs]
John,
I think that we have probably reached one of those
"look at what I'm indicating, don't bite my indicator"
types of situations. The true function of truth values
is to serve as a functional range of indicator values --
the codomain M of a predicate p : X -> M. They are used
to make different levels of indication in the functional
domain X, and X is where the things of actual value rest.
As such, truth values are much like the gold stars in
school or silver certificates, with no intrinsic value
in themselves, but just place holders for indicating
something of value in the relevant domain X. For all
practical purposes, step functions f : X -> B = {0, 1)
are enough to construct anything else that anybody might
need, and the generalizations that are most often useful
are things like probability densities that we need in
order to define information. So I really don't care
about the number of values in M. What is significant
here is the elevation from 2-adic relations of set
membership to a 3-adic relation of L c O x S x J,
where O is a domain of objects, S is a domain of
signs, and J is a domain of "judges", say, just
to be soppy, and <o, s, j> is in L iff judge j
judges object o to fall under the sign s.
John F. Sowa wrote:
>
> Jon,
>
> There is a difference:
>
> JS: There is a very big difference between fuzzy logic and
> fuzzy sets. I approve of fuzziness in set membership,
> but not fuzziness in truth value.
> >
> JA> No argument there. The same thing applies to many-valued logics.
>
> Peirce also noted the need for a third truth value, Unknown, between T and F.
> And I approve of Cyc's 5 truth values: Certainly T; T by default; Unknown;
> F by default; Certainly F.
Peirce's attempts at 3-value logic are some of the first things
that I studied when I used to spend my summers buried in his
microfilmed manuscripts. The situation is not that simple,
and none of these systems persisted very long in his work.
What does persist is the importance of having adequate
arity in the relations that one uses to model things,
over and above the cardinality of the domains used.
If Cyc's Likert scale has any meaning, it will reside in the axioms
that are used to combine and pass along these values, and whether
this can be done in a consistent fashion. This is the big if.
Experience with the mathematical properties of such systems
has made me (5) Extremely Skeptical.
> But those values can be introduced without destroying
> the relationship between the logic and the model theory.
> None of the many formulas for computing fuzzy truth values
> are compatible with any reasonable version of model theory.
>
> JA> The important thing is not the multitude M of functional values that
> > we use in making decisions or in expressing propositions f : X -> M,
> > which need for analytic purposes, and can without loss of generality,
> > be constructed from distinctions of the form f : X -> B = {0, 1}, but
> > the arity, dimensionality, or valence of the relations that we use to
> > model the realities behind the phenomena. And here I think it is way
> > past time to put the interpreter back into the picture of logic and
> > semiotics, as they were forced to even in physics, long long ago.
>
> Yes. The 5 truth values of Cyc can be interpreted as the
> result of a triadic relation that relates (1) a proposition,
> (2) a model, and (3) some interpreter:
>
> - T, F, and Unknown represent the interpreter's
> knowledge about how the proposition is related
> to the model.
>
> - T or F by default represent the interpreter's
> presupposition in terms of some default model,
> which the statisticians call the "null hypothesis".
>
> My quarrel with the fuzzy logicians is that they ignore
> the complex relationships between the logic, the model
> the interpreter, and the world.
Again, I am not making a case for fuzzy logic or fuzzy sets.
I am simply pointing to a logically more fundamental 3-adic
relation between objects, setlike concepts, and interpreters,
that can be recognized to generate the degrees of membership
as a derivative, secondary, or even optional appurtenance.
> I have brought such issues to the attention of several
> fuzzy proponents. Their answer is that they do take
> such matters into account in their applications. That
> is why they are able to demonstrate useful applications
> of their theory.
>
> That response, however, turns fuzzy practice into an
> art rather than a science. Since the theory does not
> show the relationships explicitly, there is no "official"
> interpretation of what the truth numbers are supposed to
> mean. By long experience, some fuzzy artists get better
> results than others because they have more experience
> in applying the mysterious numbers to an application.
>
> This all gets back to my point about the fallacy of
> misplaced fuzziness. T and F are the result of
> a binary relation between a theory and the model.
> When you bring the real world into account, you
> need a triadic relation between the theory, the
> model, and the world.
Yes, I think I agree with that. Or maybe I say
reality, phenomenon, and model, but only because
I consider model_of_a_theory as bound together.
> You can add the interpreter to that triad to make
> it a tetrad -- but then, as Peirce observed, you
> could break the tetrad into two triads. In any
> case, you have to go beyond two.
>
> The fuzzy theorists have been trying to go beyond
> T and F while sticking with just a dyadic relation
> between a proposition and the world. That doesn't
> work as a science, and it only works as an art
> when the artist makes some unstated assumptions.
I think I agree with that.
> Bottom line: The fuzzy people need to study Peirce in order
> to learn how to make the implicit relationships explicit.
Yes, indeed.
Jon Awbrey
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