Re: 3 dogs, 2 men, many arguments

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John Cowan

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Oct 19, 1999, 11:42:48 AM10/19/99
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xod scripsit:
>
> From: xod <x...@bway.net>
>
> I take issue with Chapter 16, section 7: Grouping of Quantifiers.
>
> http://www.animal.helsinki.fi/lojftp/reference-grammar/chap16.html#s7
>
> The meaning of "ci gerku cu batci re nanmu" is taken to mean "each of
> three dogs bite two men", leaving the number of men not necessarily 2
> but any value between and including 2 and 6. The result is that the 2 is
> taken less literally than the 3 because it is declared later.

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.


> 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."

--
John Cowan co...@ccil.org
I am a member of a civilization. --David Brin

Jorge Llambias

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Oct 20, 1999, 5:28:15 PM10/20/99
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la xod cusku di'e

> >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.
>
>It seems to me that if G(x) = 2y F(x, y) then G is a function
>of (x, y) and not (x) alone.

No, G does not depend on any value of y, it is only a
function of x. Replace y in that expression with any other
bound variable and you will see that G only depends on x.
y is not a free variable in the expression 2y: F(x,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."

That could have been the convention: take all the existentials
first and all the universals later when dealing with more than
one numeric quantifiers.

>This is the symmetrical interpretation, free of the malglico of default
>restricted scope.

I don't see that one interpretation is more or less malglico
than the other. What you gain in symmetry you lose in
the ease of formula reduction. You would also need to
specify what to do when you have for example {ro} and
a number in one expression, {ro} and {su'o} and a number,
etc.

In practical terms, I don't see how it matters much one way
or the other, since we hardly ever will want to say any of
those things. When speaking of groups of things it is much
more common to refer to them collectively, in which case
this problem doesn't even arise.

co'o mi'e xorxes


xod

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Oct 19, 1999, 12:15:23 AM10/19/99
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I take issue with Chapter 16, section 7: Grouping of Quantifiers.

http://www.animal.helsinki.fi/lojftp/reference-grammar/chap16.html#s7

The meaning of "ci gerku cu batci re nanmu" is taken to mean "each of
three dogs bite two men", leaving the number of men not necessarily 2
but any value between and including 2 and 6. The result is that the 2 is
taken less literally than the 3 because it is declared later.

This is malglico, reflecting subject/object asymmetry in English. I
suggest a more Lojbanic alternative.

Does "ci gerku" mean "each of three different dogs", or does it mean "3
dog instances", a condition which could just as easily be satisfied by one
dog three times?

The statement "da broda de" should, by default, be symmetrical between da
and de.

Unless further specified, "ci gerku cu batci re nanmu" can only be
satisfied by a saturated relationship: each of exactly three dogs bites
each of exactly two men at least once.


-----
Perpetual Progress, Self-Transformation, Practical Optimism, Intelligent Technology,
Open Society, Self-Direction, and Rational Thinking.

http://extropy.com/


Jorge Llambias

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Oct 19, 1999, 9:42:31 AM10/19/99
to
la xod cusku di'e

>The meaning of "ci gerku cu batci re nanmu" is taken to mean


>"each of three dogs bite two men", leaving the number of men not
>necessarily 2 but any value between and including 2 and 6. The
>result is that the 2 is taken less literally than the 3 because it is
>declared later.

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 &). 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.

xod

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Oct 25, 1999, 7:43:21 PM10/25/99
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At 2:28 PM -0700 10/20/99, Jorge Llambias wrote:
>From: "Jorge Llambias" <jjlla...@hotmail.com>

>
>No, G does not depend on any value of y, it is only a
>function of x. Replace y in that expression with any other
>bound variable and you will see that G only depends on x.
>y is not a free variable in the expression 2y: F(x,y)


I no longer understand this notation. Let me try:

3x 2y F(x, y) ci da poi gerku re de poi nanmu zo'u da batci de
G(x) = 2y F(x, y) broda cei re de poi nanmu zo'u da batci de

Now, by "G only depends on x " you mean broda has no place for da?


>>"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."
>
>That could have been the convention: take all the existentials
>first and all the universals later when dealing with more than
>one numeric quantifiers.


That's not quite what I was getting at.


>>This is the symmetrical interpretation, free of the malglico of default
>>restricted scope.
>
>I don't see that one interpretation is more or less malglico
>than the other. What you gain in symmetry you lose in
>the ease of formula reduction. You would also need to
>specify what to do when you have for example {ro} and
>a number in one expression, {ro} and {su'o} and a number,
>etc.


Why is "formula reduction" a value? Is that the way we think and speak?


>In practical terms, I don't see how it matters much one way
>or the other, since we hardly ever will want to say any of
>those things. When speaking of groups of things it is much
>more common to refer to them collectively, in which case
>this problem doesn't even arise.


As for "collectively", what do you mean? Masses where a single member's
validity is enough? Where if at least one of the 3 dogs bites only one of
the 2 men, the sentence is true?

How would you state my sample sentence "There exist exactly 3 dogs, and
there exist exactly 2 men, such that: every dog bites every man at least
once." Do you really think this is an unlikely sentence to utter?

In these sentences we are defining a relationship between every element of
some sets. The prenex declares the composition of the sets; the bridi
defines the relationship. Why should there be another, implicit
relationship between the sets? And why handle sets enumerated by ro and
su'o differently?

The condition of element-set mapping (each element of a set listed in the
prenex in position n maps to a set of type listed as n+1) should be marked.
The implictness of this is malglico and needs to be made explicit in a
logical language.

xod

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Oct 20, 1999, 4:35:54 PM10/20/99
to
>From: "Jorge Llambias" <jjlla...@hotmail.com>

>
>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 <co...@locke.ccil.org>
>
>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.


Jorge Llambias

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Oct 26, 1999, 5:32:51 PM10/26/99
to
la xod cusku di'e

>I no longer understand this notation. Let me try:


>
>3x 2y F(x, y) ci da poi gerku re de poi nanmu zo'u da batci de
>G(x) = 2y F(x, y) broda cei re de poi nanmu zo'u da batci de
>
>Now, by "G only depends on x " you mean broda has no place for da?

It has no place for de actually, because broda would be a
one-place predicate. (Your second line is not grammatical
Lojban, but I think I understand what you mean. The first
line is a full expression, a sentence, the second line is only
meant to be the definition of a predicate, G(x), it doesn�t
state anything.)

>Why is "formula reduction" a value? Is that the way we think and speak?

I don't know and I don't think anybody knows just
what is the way we think and speak. What I called
"formula reduction" is just one way to analyse what
we say.

>As for "collectively", what do you mean? Masses where a single member's
>validity is enough?

No, that is definitely not my view of masses. For example,
when I say that a mass of three dogs weighs 20 kg I don't
mean that only one of the dogs may weigh that. I mean that
they weigh 20kg as a whole.

>Where if at least one of the 3 dogs bites only one of
>the 2 men, the sentence is true?

No, that's not what I mean. I mean that it would be under
extremely rare circumstances that we have a situation where
there are three dogs and two men such that each of the
dogs bites each of the men.

>How would you state my sample sentence "There exist exactly 3 dogs, and

>there exist exactly 2 men, such that: every dog bites every man at least
>once."

One of the suggestions in another round of this discussion
was {ci da poi gerku e re de poi nanmu zo'u da batci de}.
This is grammatical, but not with an officially sanctioned
meaning, as far as I know.

>Do you really think this is an unlikely sentence to utter?

Extremely unlikely. I don't think I have ever been in such
a situation, nor remember anyone ever telling me about
something like that.

>In these sentences we are defining a relationship between every element of
>some sets. The prenex declares the composition of the sets; the bridi
>defines the relationship.

Many bridi are not relationships between all members of one
set and all members of another set.

>Why should there be another, implicit relationship between the sets? And
>why handle sets enumerated by ro and
>su'o differently?

{ro} and {su'o} do not enumerate the sets. In fact all
sets have ro members, so ro is useless as an enumerator.
In their function as quantifiers, numbers are not primarily
enumerating either. I'm sure I'm using all the wrong
technical words, but if you don't agree that the order in
which ro and su'o appear is of great significance, then
we have a much more basic disagreement than with
numbers. {ro da prami su'o de} (everyone loves at least
someone) does not mean the same as {su'o de se prami
ro da} (at least someone is loved by everyone). This is
basic logic, not particularly Lojbanic.

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