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From: Chris Menzel <cmenzel@kbsdec2.tamu.edu> Message-id: <9312190810.AA09783@kbsdec2.tamu.edu> Subject: Higher-order semantics and intended models To: interlingua@ISI.EDU Date: Sun, 19 Dec 1993 02:10:57 +119304028 (CST) Cc: cg@cs.umn.edu X-Mailer: ELM [version 2.4 PL23] Content-Type: text Content-Length: 5198

Norman Foo wrote: > Formally, I think Fritz is right about the necessity of higher order > concepts in semantics. In a 1st-order logic, compactness forces on > us the results that finiteness, standardness for numbers, etc. are > not expressible. If one tries to circumvent these by inventing > predicates for them, these new predicates will in their turn have > non-standard interpretations. Maybe I'm missing something, but the debate between the first-orderists and the higher-orderists seems to me to be a nonstarter; the two positions are symmetrical. The first-orderist is going to insist that she just *means* (bang the table, stamp the foot) the set of natural numbers by the predicate "natural number" together with the usual first-order axiomatization, regardless of the fact that any such axiomatization has nonstandard interpretations. The higher-orderist argues that the issue is moot for him: the semantics for his higher-order quantifiers *guarantees* that by "natural number" (defined in some familiar higher-order manner) he means the natural numbers. But here is where the situation seems directly analogous to the first-orderist's: what guarantees that those higher-order quantifiers are interpreted according to the intended higher-order semantics? What's to prevent them from being interpreted by means of Henkin's generalized "higher-order" models? No additional axioms can force the one interpretation over the other. But then the higher-order theory can just be interpreted as a first-order theory in higher-order guise, and hence is just as open to nonstandard interpretations. The higher-orderist can only reply that when he uses his higher-order quantifiers he *means* (bang the table, stamp the foot) full-blown higher-order quantification; he *intends* to be quantifying over the full power set of his domain. But the higher-orderist's intentions here are nailed down no better (and no worse) than the first-orderists. Insofar as the higher-orderist is allowed to rest content that his quantifiers have their intended meanings, the first-orderist should be no less allowed to rest content that his natural number predicate has *its* intended meaning. They are in the same semantic boat. If this is right, higher-order semantics provides no practical advantage over first-order for knowledge interchange. The sentences involved in a knowledge interchange cannot carry their intended semantics on their sleeve. Nonstandard interpretations are possible either way, and no number of axioms of any order are going to rule them out. But is this so bad? We certainly seem no worse off in the context of knowledge interchange than in everyday communication. Nothing guarantees that you don't have some nonstandard understanding of the term `natural number' when I'm talking to you; we could always systematically reinterpret one another's words to fit our different semantics. So in knowledge interchange we do just as we do every day: as Quine put it, we simply acquiesce in our background language, i.e., we take certain concepts as given and understood and go from there. (I take it this was roughly Tom Gruber's point in a recent post.) If we're first-orderists, we take, e.g., "natural number" to pick out the natural numbers and, on that assumption, define such notions as finitude in the manner that John Sowa pointed out; we just don't mean to include nonstandard "numbers" in the interpretation of "natural number". If we're higher-orderists we take our higher-order quantifiers to range over the power set of our domain and, on that assumption, define the natural numbers, finitude, etc.; we just don't mean anything less than the full power set as the range of our quantifiers. The best we can do in knowledge interchange, then, is to make our intended semantics as explicit as possible, assume our interlocutors understand us and proceed with the interchange. As in everyday communication, that'll probably be plenty good enough. A final point. The above argument seems to leave first-orderism and higher-orderism on the same footing. But that seems misleading. In the first-order case, completeness tells us that our proof theory--the stuff we can program into a computer (modulo well known limitations)--and our notion of logical entailment match up perfectly; anything entailed by some set of assumptions is provable from that set and (by soundness) vice versa. In the higher-order case logical truth and entailment so vastly outstrip any possible proof theory that we don't even get close; we will never, in principle, capture the connection between a set of assumptions and its higher-order logical consequences proof theoretically. Given their parity in other respects, surely this gives us at least some reason to prefer a first-order over a higher-order semantics for an interlingua. --Chris Menzel ================================================================= Christopher Menzel | Internet --> cmenzel@tamu.edu Associate Professor, Philosophy | Phone -----> (409) 845-8764 Texas A&M University | Fax -------> (409) 845-0458 =================================================================