SUO: Predicates Ostensibly Expressing Taxonomies (POET's)
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John, Matthew, All,
I learned most of my logic by osmosis in the course
of my maths courses, but I remember one time taking
a logic course from the philosophy department where
the instructor insisted that we employ the terms
"predicate letter" and "predicate name" instead
of plain old "predicate", and I can remember how
silly I thought that was -- but now I am beginning
to see some of the reasons why. Or maybe I am just
performing some form of anachronistic projection in
the light and from the standpoint of the pragmatic
theory of sign relations that I employ to examine
almost all of these sorts of issues these days.
In this perspective, one can sensibly discuss
the possibility that a sign does not denote,
better said, perhaps, that what appears to be
a sign may not in fact be truly a genuine sign,
a sign proper. Thus, it becomes sensible to ask
whether an "ostensible predicate expression" (OPE)
is truly an "authentic predicate expression" (APE).
Now, this does require one to generalize the theory
from "sign relations" to "sign relational complexes",
but there are precedents in maths for how to do this.
John F. Sowa wrote:
>
> Matthew,
>
> > My understanding of a unary predicate is that it is
> > essentially assigning the argument to the class of
> > the predicate. I.e. it is a classification relation.
>
> One of the basic assumptions of set theory
> is that for every monadic predicate p(x),
> there exists a set S of all x for which
> p(x) is true.
Yes, but there is a catch here.
You could take this as defining
what it is to be a predicate,
that is, a genuine predicate.
> There are several questions that arise from that assumption:
>
> 1. Is that set S a definition of the predicate p?
> If so, then that is a definition of S by extension.
>
> 2. If you claim that a predicate p is completely defined by
> its extension, then any other predicate q that has the same
> extention must be identical to q.
>
> 3. However, the distinction between equality by extension and
> equality by intension has been analyzed since ancient times.
> Are the predicates human(x) and featherlessBiped(x), which
> happen to have the same extension, to be considered as identical?
> Diogenes the Cynic is said to have plucked a chicken and thrown it
> into Plato's academy with the claim "Here is Plato's man!"
>
> 4. Therefore, many people distinguish
> equality by intension (method of definition)
> from equality by extension (set of entities).
>
> 5. There are other points to consider. Near the beginning of the 20th century,
> some people, including Bertrand Russell, discovered that it was possible to
> define predicates that led to contradictions, if one assumed that they could
> define a set. One of them is the following:
>
> p(x) is defined as "x is not a member of x"
>
> This leads to a contradiction if you assume that p(x) can define a set.
> Therefore, all versions of set theory are formulated to prevent such
> definitions from being stated.
Another slightly different angle on this state of affairs
would be to say that an OPE that opes to a contradiction
is not an honest APE, and so does not denote a predicate.
> So to answer your question, yes. Monadic predicates can be used to
> specify sets provided that you formulate their definitions properly.
>
> John Sowa
To turn the style, "provided that they really are predicates".
Highest Reguards,
Jon Awbrey
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