From xod@xxxx.xxxx Wed Oct 20 13:35:54 1999 X-Digest-Num: 261 Message-ID: <44114.261.1407.959273825@eGroups.com> Date: Wed, 20 Oct 1999 16:35:54 -0400 From: xod From: "Jorge Llambias" > >I wouldn't say less literally. What happens is that the 2 has >in this case a narrower scope than the 3, because it is >declared later. > >Number quantifiers can be understood in terms of the more >basic existential and universal quantifiers. For example, >2x: F(x) could be rewritten as Ex Ey: x<>y & F(x) & F(y). > >This expansion will always involve both existential and >universal quantifiers (here the universal is in the form >of &). What is the significance of this theorem? The order in which these quantifiers appear is >what determines their scope. > >There are many ways in which 3x 2y F(x,y) could have >been given meaning. The one chosen is to take it as >3x G(x), where G(x) = 2y F(x,y), and there we can see >why the scope of the second quantifier is narrower. > >The way you propose would involve having a separate >expansion for the quantification (3x 2y) that could not be >reduced to the single variable case. > >co'o mi'e xorxes It seems to me that if G(x) = 2y F(x, y) then G is a function of (x, y) and not (x) alone. Making x, y asymmetrical makes one dependent on the other. For a function, the variables should be free. >From: John Cowan > >xod scripsit: >> >Not less literally, merely as a matter of distribution. >Quantifiers are implicitly understood left to right, which is a bias >in favor of the direction of the Latin alphabet, not in favor of >English particularly. For each dog, there are two men bitten; >the two men must be distinct from each other, or they would be only >one man, but nothing is said about whether the two men are the same >or different from those bitten by other dogs. But why doesn't the sentence equally imply that "for each man, there are 3 dogs biting him."? > > >> The statement "da broda de" should, by default, be symmetrical between da >> and de. > >No, because it means su'oda su'ode zo'u da broda de, and the order >of quantifiers matters. "For at least one X, there exists at least one >Y such that X bites Y." > "For at least"..."there exists" indicates a dependency of existence. I think this fact should be made explicit, and without such a marking, it should mean: "There exists exactly 3 dogs, and there exists exactly 2 men, such that: each/any dog bites each/any man at least once." This is the symmetrical interpretation, free of the malglico of default restricted scope. ----- Perpetual Progress, Self-Transformation, Practical Optimism, Intelligent Technology, Open Society, Self-Direction, and Rational Thinking. http://extropy.com/