Re: Roles, again

sowa@west.poly.edu (John F. Sowa)
Date: Sat, 16 Sep 1995 22:36:01 +0500
From: sowa@west.poly.edu (John F. Sowa)
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        phayes@cs.uiuc.edu, srkb@cs.umbc.edu
Subject: Re: Roles, again
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Fritz wrote

>     A. There is the formal algebraic version of Peircean Firstness,
>Secondness and Thirdness -- which John Sowa has not even mentioned in
>any of his speeches and papers to date.  This was in Burch's book, and
>consists of the fact that some concepts cannot be properly defined using
>only monadic predicates, and some cannot be properly defined using
>only monadic and dyadic predicates, but require at least one truly
>triadic predicate.  At this very deep and formal level, one must opine
>with great caution since there are traps for the unwary. 

I don't disagree with Fritz.  But one reason why I have hesitated to emphasize
Peirce's treatment in terms of the valence (adicity, arity...) of the
relations is that that approach too quickly gets bogged down in irrelevant
syntactic issues.  (Witness the past few dozen notes in this discussion.)

One trap for the unwary is a failure to distinguish obligatory arguments,
optional arguments, and implicit arguments.  For example, we could say that
the English word "mother" may be represented by either a dyadic predicate
motherOf(x,y) or by a monadic predicate or type label mother(x).  However,
the monadic predicate doesn't imply that the concept of mother is an
example of Firstness because, as Pat has pointed out, there is an implicit
individual brought in by the following axiom:

   (Ax)(mother(x) -> (Ey)motherOf(x,y)).

Such issues are syntactic matters that are very far from what Peirce was
trying to get at.

I also sympathize with Pat's impatience at all this numerology, which sounds
vaguely like Martin Gardner's old visits with Dr. I. J. Matrix.  (And by the
way, I was first led to Peirce's existential graphs by one of Martin G's
columns in the Scientific American in 1978.  Gardner mentioned Don Roberts'
book on P's existential graphs, which I then asked the IBM library to order,
since I wasn't quite sure whether it was worth spending my own cash to buy.)

In fact, Peirce himself said that when he first stumbled on the patterns of
threes while studying Kant's categories (which Kant arranged in four triads),
he was very suspicious of the idea.  He said that he resisted the idea himself
for a long time because it smacked too much of cenoPythagorean mysticism.
But after finding example after example of these triads, he eventually felt
compelled to admit that there seemed to be something important in it.

After I read Don R's book, I found that Peirce's graph logic formed the best
foundation I had ever seen for a logic for semantic networks.  In fact, I still
believe that Peirce's rules of inference are the simplest and most elegant rulesever discovered by anyone for any version of logic.  At the same time, I also
read a bit about Peirce's categories, but I wasn't very impressed by them.
For about a dozen years, in fact, I was very unimpressed by them.  At the
Peirce Sesquicentenial Congress at Harvard in 1989, I became somewhat more
sympathetic to them.

What finally convinced me of the value of Peirce's categories was that I
found that they gave me insights into writings that I had not previously
been able to decipher.  One such book is Whitehead's _Process and Reality_.

I hesitate to mention the others because I know that they will immediately
raise red flags (or hackles or whatever else such things raise) for people
like Pat.  But here goes, anyway:  Husserl, Heidegger, and the later
Wittgenstein.

At this point, we should wait a decent interval for Pat to calm down.
So I will sign off for now and perhaps say a bit more later.

John