SUO: Re: Predicates Ostensibly Expressing Taxonomies (POET's)

["John F. Sowa" <sowa@bestweb.net>]

Jon et al.,

>.... 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.

Yes, indeed.  The lattice of theories, which has often been
mentioned on this list, is an example.  For a brief reminder,
following is the discussion from Section 6 of my causality
paper:  http://www.bestweb.net/~sowa/ontology/causal.htm#s6

Instead of looking at the full paper, which takes a long
time to download, the following diagram is sufficient to
illustrate the point:

   http://www.bestweb.net/~sowa/ontology/theories.gif

Every theory in that lattice is constructed from symbols
in the same dialect of FOL, and all of the predicate names
are formed from strings of letters from the same alphabet.
But each theory has a different collection of axioms and
theorems.  Therefore, a predicate name p in one theory
cannot denote the same predicate as the name p in another
theory which has different axioms for p.

Furthermore, the predicates are not formal parts of the
theories, but of the models of those theories.  The top
theory has all possible models, and the bottom theory has
no models at all.  Every theory between top and bottom has
one or more models, and any predicate name in any theory
may denote different predicates in different models.
   
I must also confess that I accidentally typed S instead of p
in the second line of the following point in my previous note:

>>  1. Is that set S a definition of the predicate p?
>>     If so, then that is a definition of S by extension.

I intended to say that the set S defines the predicate p
by extension (and the predicate p could be used to define S
by intension).

In any case, I inadvertently illustrated another very important
principle:  the use of variables in mathematics can be much
more precise and unambiguous than the use of pronouns in
ordinary language.  That is why I used the letters p and S
instead ordinary pronouns.  But by my typo, I ended up saying
something very precise that is not what I intended.

That is a fundamental weakness of every formal notation,
including every programming language:  what they say so
precisely may be totally different from what the author
intended to say.  That is why I recommend bilingual
specifications in both SUO-KIF and SUO-CE -- they help
catch possible typos by the author and misinterpretations
by the reader.

John Sowa