# Recursive Peirce

fritz@rodin.wustl.edu (Fritz Lehmann)
```Date: Sat, 13 Nov 93 08:29:26 CST
From: fritz@rodin.wustl.edu (Fritz Lehmann)
Message-id: <9311131429.AA14021@rodin.wustl.edu>
To: cg@cs.umn.edu, interlingua@ISI.EDU
Subject: Recursive Peirce
Cc: PEIRCE-L@ttuvm1.ttu.edu, boley@dfki.uni-kl.de
```
```
Defending purely First-Order model-theoretic semantics for
KIF and Conceptual Graphs, John Sowa recently said:

>Since Fritz mentioned Peirce, I'd like to quote a discussion by Peirce
>in a letter to Victoria, Lady Welby:
>  When we have analyzed a proposition so as to throw into the subject
>  everything that can be removed from the predicate, all that it
>  remains for the predicate to represent is the form of connection
>  between the different subjects as expressed in the propositional form.
>  What I mean by "everything that can be removed from the predicate"
>  is best explained by giving an example of something not so removable.
>  But first take something removable.  "Cain kills Abel."  Here
>  the predicate appears as "--kills--."  But we can remove killing
>  from the predicate and make the latter "--stands in the relation--to--."
>
>This quote is from p. 396 of _Charles S. Peirce: Selected Writings_,
>edited by P. P. Wiener, Dover Publications, NY, 1958.
>
>I would interpret that passage as implying that Peirce would
>sanction the option of reifying relations by "taking them out of
>the predicate" and making them objects subject to quantification
>in a first-order language.  Therefore, that would put Fritz's
>examples taken from Peirce into the category of things that can
>be expressed adequately with a typed first-order logic, as we
>have in CGs.

Yes, absolutely, I'm all for it, except the last sentence,
which looks diametrically wrong to me.  This operation is one
form of Peirce's "hypostatic abstraction" of a relation, reifying
it into an individual object.  But this is an argumment FOR true
higher-order logic and AGAINST the First-Order weak pseudo-
higher-order kludges proposed for KIF and CGs.  Why?  Because
this operation is now applied again, and again ad infinitum.
Here "--kills--" is a dyadic relation which is converted into a
triadic one.  In usual notation, K(c,a) is replaced by R(c,K,a).
The same thing can now be done to R, obtaining R'(R,c,K,a) and so
on, recursively, forever. In fact, all these relations "exist"
merely upon asserting K(c,a).  This example is complicated by
automatically obtains the dyads R'(K,c) and R''(K,a), which again
are expanded recursively forever. [these occur in _addition_ to
R(c,K,a) -- see Marty's article in my 1992 Semantic Networks
collection, Definition 10 on p. 686.]  In CGs this is like
recursively subdividing the links in a chain,  "boxifying" each
"-(R)-" into "-(R')-[R]-(R'')-" at every step, forever.  These
too "exist" immediately upon assertion of the first "R".
Levesque & Brachman (1984) do just this to cram triads into a
Course(ga-1,course100) & Mark(ga-1,85)"  Suddenly ga-1 "exists".
The recursive hyper-exponential infinite proliferation of
individuals indicates a need for higher-order quantification over
all these beasties, hardly the reverse.  We no longer can do the
"limited vocabulary" trick to make the logic First-Order.

This is not just a play-exercise.  Meaningful things are
known about these abstracted objects.  For example, R' and R''
are singleton instances of generic "case relations" (thematic
roles) applied to "Kills";  R' inherits from AGENT and R''
inherits from VICTIM.  (Like the meaningful roles in relations in
Jim Fulton's SUMM.)  However, I am unable to meet Pat Hayes'
challenge to say anything interesting about R'''(50 's)'''.

I think, from an email message he sent a year or two ago,
Sowa suggested that the above abstraction (of the R(c,K,a) triad
>From dyadic relation K(c,a) ) is the main reason Peirce is
concerned with triads.  (This would be supported if the above
letter to Lady Welby were the only source on the subject.)  I
believe this is untrue, and that Peirce was very interested in
fundamental original triads (asserted before any such expansion).
The main one was the "sign-relation" triad relating object,
representamen and interpretant.

The recursive subdivisions of relational graphs (including
"truncations of corners" for triads) in hypostatic abstraction
are closely related to the sums-over-histories employing
recursively expanded Feynman diagrams in many-body and particle
physics.  They can be described with graph-grammars.  The thus-
existing relational "individuals" are logically valid entities in
a system which confers existence upon sets.  They were studied by