From cbmvax!uunet!computer-lab.cambridge.ac.uk!David.Elworthy Tue Apr 30 23:40:02 1991 Return-Path: Tue, 30 Apr 1991 11:16:54 +0100 To: lojban-list@snark.thyrsus.com Subject: Re: semantics of "most" In-Reply-To: Your message of Mon, 29 Apr 91 12:05:38 -0400. Date: Tue, 30 Apr 91 11:17:08 +0100 From: David Elworthy Message-Id: <"swan.cl.ca.953:30.03.91.10.17.11"@cl.cam.ac.uk> Status: RO I wrote: >> Do you know how >> Lojban handles the semantics of "most"? If it is done by a quantifier similar >> to "exists" or "all", be warned that it can be shown this will not work! I'll >> forward details if you're interested. John Cowan replied: > Please do! Currently, "most" is a number, like "two". I'll try to give an informal description of the problem. If you want to have a formal proof of why you can't express "most" in this way, see Barwise & Cooper: reference below (theorem C12). First, to restate the claim in a more formal way: no quantifier with the same meaning as the English word "most" can be defined in first order logic, i.e. a logic in which there is quantification over individuals drawn from some (finite or infinite) universe. As an aside, it is a standard result that if a quantifier can be defined in first order logic, then it is reducible to some combination of "All" and "Exists" (which I will write here as A and E). For example, the derived quantifier "two x" can be reduced to "Ex.Ey.x /= y" (where /= is "not equal"). So we can translate "Two men walk" as "two x.(man(x) & walk(x))", i.e. "Ex.Ey.(x /= y & man(x) & man(y) & walk(x) & walk(y))". Secondly, I should make it clear what I mean by "most". A paraphrase is "more than half of", i.e. "Most men walk" is equivalent to (1) "more than half of the individuals who are men are individuals who walk." Now, look at how we translate sentences into first order logic and interpret the resulting formulae: English A man walks. FOL Ex.(man(x) & walk(x)) Interp Examine the universe U, and see if you can find at least one individual i from U for which man(i) and walk(i) both hold. English Every man walk. FOL Ax.(man(x) => walk(x)) Interp Examine the universe U, and check that for every individual i from U, if man(i) holds, then walk(i) holds. (And if man(i) does not hold, we don't care if walk(i) does or not.) English Two men walk. FOL 2x.(man(x) & walk(x)) Interp Examine the universe U, and see if you can find at least two different individuals i from U for which man(i) and walk(i) both hold. And if we try it for "most": English Most men walk. ??FOL Most x.(man(x) c walk(x)), where c is a logical connective. Interp Examine the universe U, and see if more than half the individuals i in U are such that the fact of i being a man is related by c to the fact of walking. The question is, then, can we find a connective c, which means that this interpretation is equivalent to (1)? Try the most likely candidates: "&" and "=>". & Examine the universe U, and see if more than half the individuals i in U are such that both man(i) and walk(i) hold. Paraphrase: "more than half the individuals in the universe are men who walk", which is not the same as (1). => Examine the universe U, and see if more than half the individuals i in U are such that if man(i) holds, then walk(i) holds. Paraphrase: "more than half the individuals in the universe are either men who walk, or are not men at all", which is not the same as (1). ... and similarly for all other possible connectives. One solution which is widely used is to adopt what are called "generalized quantifiers". Now, instead of translating a sentence into a quantifier over a variable, and a formula which uses that variable, we express the quantifier as a relation between two sets. Thus English A man walks. GQ The set of men has a non-empty intersection with the set of walkers. English Every man walks. GQ The set of men is a subset of the set of walkers. English Two men walk. GQ The intersection of the set of men and the set of walkers contains at least two individuals. English Most men walk. GQ The intersection of the set of men and the set of walkers contains at least half as many members as the set of men. I don't know if I've explained this terribly clearly. I haven't found any reference which states the problem with "most" (and some other quantifiers) in an intuitive way, but if you want to follow up the formal details, and see how GQ theory works, have a look at: Jon Barwise and Robin Cooper. Generalized Quantifiers and Natural Language. Linguistics and Philosophy Vol.4 (1981), pp. 159-219. Jan van Eijck. Quantification. In A. von Stechow and D. Wunderlich (eds.) Handbook of Semantics. (The copy of the paper I have says "to appear", so I can't tell you the date or publisher, but Reidel or Kluwer would be a fair bet.) -- david elworthy