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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 X-Mailer: ELM [version 2.4 PL20] Content-Type: text Content-Length: 2515

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 ================================================================ Christopher Menzel Internet -> cmenzel@tamu.edu Philosophy, Texas A&M University Phone ----> (409) 845-8764 College Station, TX 77843-4237 Fax ------> (409) 845-0458 ================================================================