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.
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