Re: Recursive Peirce
cmenzel@kbssun1.tamu.edu (Chris Menzel)
From: cmenzel@kbssun1.tamu.edu (Chris Menzel)
Message-id: <9311132334.AA04725@kbssun1.tamu.edu>
Subject: Re: Recursive Peirce
To: fritz@rodin.wustl.edu (Fritz Lehmann)
Reply-To: cg@cs.umn.edu
Date: Sat, 13 Nov 1993 17:34:41 -0600 (CST)
Cc: cg@cs.umn.edu, interlingua@ISI.EDU
In-reply-to: <9311131429.AA14021@rodin.wustl.edu> from "Fritz Lehmann" at Nov 13, 93 08:29:26 am
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Fritz Lehman wrote:
> Defending purely First-Order model-theoretic semantics for
> KIF and Conceptual Graphs, John Sowa recently said:
> ...
> >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
> choosing the triad abstraction from a dyad. It's simpler to look
> at the dyad abstraction from n-ads, thus, from dyad K(c,a) one
> automatically obtains the dyads R'(K,c) and R''(K,a), which again
> are expanded recursively forever....
> 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.
All of these "higher-order" individuals are definable in type theory
interpreted with Henkin's generalized models; hence they all exist in
such models, are quantified over, etc., as required. But type theory
with Henkin's semantics is essentially first-order. So I don't see
that this particular argument pushes us to true higher-order logic.
Regards,
--Chris
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