Talk:Tiki forum archive/Forum 5/9048
Source: 2013 Tiki forum 5, thread #9048. Original authors, UTC times, post IDs, and reply-parent IDs are preserved. This is an archive, not an active article discussion.
xorlo and the expansion of bridi
xorlo and the expansion of bridi
Roman Naumann <[email protected]> · 2009-12-05 12:57 UTC · Tiki post #9048
I wonder how bridi expand with xorlo.
Before xorlo {A lo B gerku cu vasxu} ("A of B dogs breath" and "B dogs exist") expanded to {B da poi gerku ku'o A da vasxu}. Where A and B are PA.
Is the (without-xorlo-)expansion correct? How does {A lo B gerku cu vasxu} expand with xorlo?
mu'o mi'e namor
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Jorge Llambías <[email protected]> · 2009-12-05 12:58 UTC · Tiki post #9049 · reply to Tiki post #9048
<[email protected]> wrote: > coi rodo > > I wonder how bridi expand with xorlo. > > Before xorlo > {A lo B gerku cu vasxu} ("A of B dogs breath" and "B dogs exist")
(BTW, what does "B dogs exist" mean to you exactly? Is that the number of dogs alive at the time of the utterance, the number of dogs that have ever existed or will exist on this planet, or something else, depending on the context?)
> expanded to > {B da poi gerku ku'o A da vasxu}. > Where A and B are PA. > > Is the (without-xorlo-)expansion correct?
No, you have "B da poi gerku ku'o" as the x1 of vasxu, and "A da" as the x2. Also, binding the same variable twice doesn't make much sense.
> How does {A lo B gerku cu vasxu} expand with xorlo?
It might be something like this:
lo gerku goi ko'a cu klani li B .i A ko'a cu vasxu The dogs are B in number. A of them breath.
Or we could put everything in one bridi as:
A da poi me lo gerku noi klani li B cu vasxu A of the dogs, which are B in all, breath.
But this is not exactly about xorlo, at least not about "lo". This is how quantifiers work in general. An outer quantifier on any sumti works the same way:
A <sumti> cu broda = A da poi me <sumti> cu broda A of the referents of <sumti> are broda.
And the inner quantifier also works the same for all sumti, it gives the number of referents that the sumti has.
mu'o mi'e xorxes
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xorlo and the expansion of bridi
Roman Naumann <[email protected]> · 2009-12-06 12:24 UTC · Tiki post #9062 · reply to Tiki post #9048
Jorge Llambas wrote: > (BTW, what does "B dogs exist" mean to you exactly? Is that the number > of dogs alive at the time of the utterance, the number of dogs that > have ever existed or will exist on this planet, or something else, > depending on the context?)
I'm not sure what it means, I just remember the 'old' lo implied existential claims.
>> expanded to >> {B da poi gerku ku'o A da vasxu}. >> Where A and B are PA. >> >> Is the (without-xorlo-)expansion correct? > > No, you have "B da poi gerku ku'o" as the x1 of vasxu, and "A da" as > the x2. Also, binding the same variable twice doesn't make much sense.
Sorry, I forgot the prenex. What I meant to say was: {B da poi gerku zo'u A da vasxu} But still, I don't see why you say the variable is being bound twice. According to the example from CLL:16, it should be locally requantified instead of being bound again:
( 14.1) ci da poi mlatu cu blaci .ije re da cu barda
Three Xs which-are cats are white, and two Xs are big.
What does Example 14.1 mean? The appearance of ``ci da quantifies ``da as referring to three things, which are restricted by the relative clause to be cats. When ``re da appears later, it refers to two of the those three things --- there is no saying which ones. Further uses of ``da alone, if there were any, would refer once more to the three cats, so the requantification of ``da is purely local. )
Now, if {B da poi gerku zo'u A da vasxu} is the correct expansion of {A lo B gerku cu vasxu}, does that change with xorlo?
>> How does {A lo B gerku cu vasxu} expand with xorlo? > (...) > Or we could put everything in one bridi as: > > A da poi me lo gerku noi klani li B cu vasxu > A of the dogs, which are B in all, breath.
I should have been more specific about what kind of expansion I had in mind. The thing I'm looking for is a canonical form of {A lo B <sumti> cu broda} with a proper prenex. That canonical form should not contain {lo}. (Something a computer could use for doing some logic on it)
> But this is not exactly about xorlo, at least not about "lo". This is > how quantifiers work in general. An outer quantifier on any sumti > works the same way: > > A <sumti> cu broda > = A da poi me <sumti> cu broda > A of the referents of <sumti> are broda.
Isn't A an _inner_ quantifier here? Without xorlo, there should be {ro} as an implicit outer quantifier.
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xorlo and the expansion of bridi
Jorge Llambías <[email protected]> · 2009-12-06 12:27 UTC · Tiki post #9063 · reply to Tiki post #9048
> Thanks for your reply. > > Jorge Llambías wrote: >> >> (BTW, what does "B dogs exist" mean to you exactly? Is that the number >> of dogs alive at the time of the utterance, the number of dogs that >> have ever existed or will exist on this planet, or something else, >> depending on the context?) > > I'm not sure what it means, I just remember the 'old' lo implied existential > claims.
I just wanted to point out that the 'old' meaning is not really that different from the new one, at least as far as it can be understood. The inner quantifier gives the number of referents that the sumti has. This was always the case. The old exposition seemed to assume that any given sumti headed with 'lo' has a certain number of referents independently of the context in which it is used, but that doesn't make a lot of sense. It is hard to come up with sumti that are so context independent.
> What I meant to say was: {B da poi gerku zo'u A da vasxu}
> But still, I don't see why you say the variable is being bound twice.
In "A da vasxu", the variable "da" is bound by the quantifier "A". It is equivalent to "A da zo'u da vasxu".
"B da poi gerku zo'u A da vasxu" is equivalent to B da poi gerku zo'u A da zo'u da vasxu", with apparently two bindings for the same variable.
> According to the example from CLL:16, it should be locally requantified > instead of being bound again: > > ( 14.1) ci da poi mlatu cu blaci .ije re da cu barda > Three Xs which-are cats are white, and two Xs are big. > > What does Example 14.1 mean?
In normal predicate logic, it's equivalent to "ci da poi mlatu cu blabi .ije re de cu barda". The two variables are independent of one another. In CLL-logic it means something else, but since Lojban is supposed to be based on predicate logic, I prefer to stick to that. The CLL-logic of quantifiers is not completely coherent or consistent.
> The appearance of ``ci da quantifies ``da > as referring to three things, which are restricted by the relative clause to > be cats.
The way I would say it is that the variable "da" takes values from the things that are cats, and (exactly) three of those values satisfy the predicate "blabi" (which also means all but three of those values don't satify it). If there is any reference going on here at all it is to all cats, not just to the three that do satisfy the predicate "blabi". If we were to say "no da poi mlatu cu blabi" then "da" again takes values from the same set of referents, but now the number of them that are said to satisfy blabi is zero. Would we say that "da" doesn't refer to anything in this case?
> When ``re da appears later, it refers to two of the those three > things --- there is no saying which ones.
That's the CLL story, yes. It's appealing in simple cases like this, but it doesn't work in general. (Or at least nobody has given any satisfactory account of how these "double quantifications" should work in general.)
>>> How does {A lo B gerku cu vasxu} expand with xorlo?
>
> I should have been more specific about what kind of expansion I had in mind.
> The thing I'm looking for is a canonical form of {A lo B <sumti> cu broda}
> with a proper prenex.
"A lo B <sumti> cu broda" or "A lo B <selbri> cu broda"?
Both are grammatical, at least if <sumti> doesn't have another quantifier. I assume you want an expansion of "A lo B broda cu brode". The proper expansion would be:
A da poi ke'a me lo B broda zo'u da brode
>That canonical form should not contain {lo}.
OK, but this is independent of the quantifiers.
I expand "lo broda" to "zo'e noi ke'a broda".
And an "inner quantifier", which is not a logical quantifier, can be expanded thus: "lo B broda" = "lo broda noi ke'a klani li B".
Putting all three expansions together (which are all inependent of one another) we get:
A lo B broda cu brode = A da poi ke'a me zo'e noi ke'a broda gi'e klani li B zo'u da brode
We have shifted the outer quantifier A (a logical quantifieer) to a proper prenex, we have transformed the "inner quantifier" B to an ordinary number "li B" used as an ordinary argument, and we have got rid of "lo".
B cannot be transformed into a proper prenexed quantifier because it was never a true quantifier to begin with, it's just a cardinality.
>> A <sumti> cu broda
>> = A da poi me <sumti> cu broda
>> A of the referents of <sumti> are broda.
>
> Isn't A an _inner_ quantifier here?
No, an "inner quantifier" (which is not a logical quantifier) always goes after the gadri.
An inner quantifier gives the number of referents of a sumti. It is only concerned with a sumti, not with a whole bridi.
An outer quantifier binds a variable and quantifies a whole bridi, it says for how many values of the variable the bridi in which that variable appears is true.
mu'o mi'e xorxes
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xorlo and the expansion of bridi
Roman Naumann <[email protected]> · 2009-12-06 12:27 UTC · Tiki post #9064 · reply to Tiki post #9048
> In normal predicate logic, it's equivalent to "ci da poi mlatu cu > blabi .ije re de cu barda". The two variables are independent of one > another. In CLL-logic it means something else, but since Lojban is > supposed to be based on predicate logic, I prefer to stick to that. > The CLL-logic of quantifiers is not completely coherent or consistent.
In {ci da poi mlatu cu blaci .ije re da cu barda} [according to your (more mathematical) use of logic, not the CLL use], is {da} only to be considered an independent variable, because there is a leading PA before it?
(If not:) If reusing a variable in a connected bridi generally introduces a new, independend variable, expressions like {lo gerku cu vasxu gi'e bajra}, which expand to {da poi gerku zo'u da vasxu ije da bajra} would mean {da poi gerku zo'u da vasxu ije de poi gerku zo'u de bajra}, which means, we could be talking about different dogs. That would collide with my understanding of {gi'e}.
>> The appearance of ``ci da quantifies ``da >> as referring to three things, which are restricted by the relative clause to >> be cats. > > The way I would say it is that the variable "da" takes values from the > things that are cats, and (exactly) three of those values satisfy the > predicate "blabi" (which also means all but three of those values > don't satify it). If there is any reference going on here at all it is > to all cats, not just to the three that do satisfy the predicate > "blabi". If we were to say "no da poi mlatu cu blabi" then "da" again > takes values from the same set of referents, but now the number of > them that are said to satisfy blabi is zero. Would we say that "da" > doesn't refer to anything in this case?
That's an interesting question. I would say it refers to all referents in the set, but the bridi is negated. At least the statement still has an implication on all referents in the set, your explanation includes that, the CLL one does not. (..or was that a rethorical question of yours..?)
>> When ``re da appears later, it refers to two of the those three >> things --- there is no saying which ones. > > That's the CLL story, yes. It's appealing in simple cases like this, > but it doesn't work in general. (Or at least nobody has given any > satisfactory account of how these "double quantifications" should work > in general.)
I understand.
> "A lo B <sumti> cu broda" or "A lo B <selbri> cu broda"? > > Both are grammatical, at least if <sumti> doesn't have another > quantifier. I assume you want an expansion of "A lo B broda cu brode".
You are right about that.
> We have shifted the outer quantifier A (a logical quantifieer) to a > proper prenex, we have transformed the "inner quantifier" B to an > ordinary number "li B" used as an ordinary argument, and we have got > rid of "lo".
Indeed and thanks. That's the form I was looking for. I think it could be useful for letting the computer reason about lojban.
> B cannot be transformed into a proper prenexed quantifier because it > was never a true quantifier to begin with, it's just a cardinality.
I understood that now. The inner quantifier is just incidential information about the cardinality of the set-of-things we are talking about.
> An inner quantifier gives the number of referents of a sumti. It is > only concerned with a sumti, not with a whole bridi. > > An outer quantifier binds a variable and quantifies a whole bridi, it > says for how many values of the variable the bridi in which that > variable appears is true.
Well explained. By the way, is there any document about lojban logic on the web which sticks closer to what you explaned/to logic than the CLL?
mu'o mi'e namor
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xorlo and the expansion of bridi
John E Clifford <[email protected]> · 2009-12-06 12:27 UTC · Tiki post #9065 · reply to Tiki post #9048
From: Roman Naumann <[email protected]> To: [email protected] Sent: Thu, October 29, 2009 3:47:59 PM Subject: [lojban] Re: xorlo and the expansion of bridi
Jorge Llambías wrote: > In normal predicate logic, it's equivalent to "ci da poi mlatu cu > blabi .ije re de cu barda". The two variables are independent of one > another. In CLL-logic it means something else, but since Lojban is > supposed to be based on predicate logic, I prefer to stick to that. > The CLL-logic of quantifiers is not completely coherent or consistent.
In {ci da poi mlatu cu blaci .ije re da cu barda} [according to your (more mathematical) use of logic, not the CLL use], is {da} only to be considered an independent variable, because there is a leading PA before it?
(If not:) If reusing a variable in a connected bridi generally introduces a new, independend variable, expressions like {lo gerku cu vasxu gi'e bajra}, which expand to {da poi gerku zo'u da vasxu ije da bajra} would mean {da poi gerku zo'u da vasxu ije de poi gerku zo'u de bajra}, which means, we could be talking about different dogs. That would collide with my understanding of {gi'e}.
Well, no. Although Lojban mucks up here a bit by making it look like the prenex only goes with the first chunk, it does, in fact, cover the whole. The error comes in forgetting the step in the expansion {da poi gerku zo'u da vasxu gi'e bajra}, which does not lead to the next step logically, though appears to CLL wise. The scope is set at the beginning and does not change as the sentences within it develop.
>> The appearance of ``ci da quantifies ``da >> as referring to three things, which are restricted by the relative clause to >> be cats. > > The way I would say it is that the variable "da" takes values from the > things that are cats, and (exactly) three of those values satisfy the > predicate "blabi" (which also means all but three of those values > don't satify it). If there is any reference going on here at all it is > to all cats, not just to the three that do satisfy the predicate > "blabi". If we were to say "no da poi mlatu cu blabi" then "da" again > takes values from the same set of referents, but now the number of > them that are said to satisfy blabi is zero. Would we say that "da" > doesn't refer to anything in this case?
That's an interesting question. I would say it refers to all referents in the set, but the bridi is negated. At least the statement still has an implication on all referents in the set, your explanation includes that, the CLL one does not. (..or was that a rethorical question of yours..?)
xorxes' point also points out why translating out of 'lo' language into quantifiers is not preservative: you lose information at every step if you are not very careful and just a bit sneaky (this was true of "old" lo as well as xorlo, but more for xorlo where they differ -- which is not here, by the way).
>> When ``re da appears later, it refers to two of the those three >> things --- there is no saying which ones. > > That's the CLL story, yes. It's appealing in simple cases like this, > but it doesn't work in general. (Or at least nobody has given any > satisfactory account of how these "double quantifications" should work > in general.)
I understand.
> "A lo B <sumti> cu broda" or "A lo B <selbri> cu broda"? > > Both are grammatical, at least if <sumti> doesn't have another > quantifier. I assume you want an expansion of "A lo B broda cu brode".
You are right about that.
> We have shifted the outer quantifier A (a logical quantifieer) to a > proper prenex, we have transformed the "inner quantifier" B to an > ordinary number "li B" used as an ordinary argument, and we have got > rid of "lo".
Indeed and thanks. That's the form I was looking for. I think it could be useful for letting the computer reason about lojban.
> B cannot be transformed into a proper prenexed quantifier because it > was never a true quantifier to begin with, it's just a cardinality.
I understood that now. The inner quantifier is just incidential information about the cardinality of the set-of-things we are talking about.
> An inner quantifier gives the number of referents of a sumti. It is > only concerned with a sumti, not with a whole bridi. > > An outer quantifier binds a variable and quantifies a whole bridi, it > says for how many values of the variable the bridi in which that > variable appears is true.
Well explained. By the way, is there any document about lojban logic on the web which sticks closer to what you explaned/to logic than the CLL?
mu'o mi'e namor
I think that my starts in that direction got taken down a long time ago as being both incoplete and in some cases inaccurate for one side or the other. There are some minor notes at pckipo.blogspot.com under LoCCan. But someone else may have taken up the slack.
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you're really stuck, send mail to [email protected] for help.xorlo and the expansion of bridi
Jorge Llambías <[email protected]> · 2009-12-06 12:27 UTC · Tiki post #9066 · reply to Tiki post #9048
> > In {ci da poi mlatu cu blaci .ije re da cu barda} [according to your (more > mathematical) use of logic, not the CLL use], is {da} only to be considered > an independent variable, because there is a leading PA before it?
There's also the possible problem of ".ije". This one would be unproblematic:
ci da poi mlatu zo'u ge da blabi gi da barda "There are three cats x such that: both x is white and x is big."
This does not preclude there being any number of white cats that are not big or any number of big cats that are not white.
If what you want to say is that there are three cats that are white, and that all the cats that are white are big, then that will be something different. And if you want to say that there are three cats that are white, and that all and only the cats that are white are big, that will be different again.
> (If not:) If reusing a variable in a connected bridi generally introduces a > new, independend variable, expressions like {lo gerku cu vasxu gi'e bajra}, > which expand to {da poi gerku zo'u da vasxu ije da bajra} would mean {da poi > gerku zo'u da vasxu ije de poi gerku zo'u de bajra}, which means, we could > be talking about different dogs. That would collide with my understanding of > {gi'e}.
The correct expansion is:
da poi gerku cu vasxu gi'e bajra = da poi gerku zo'u da vasxu gi'e bajra = da poi gerku zo'u ge da vasxu gi da bajra
so your understanding of "gi'e" is correct.
"da poi gerku zo'u da vasxu ije de poi gerku zo'u de bajra" is actually ungrammatical, because "ije" cannot connect two bridi with prenexes. The first prenex is in fact a prenex for the whole thing, and you can't have a second prenex after "ije". This is quite unintuitive and was a bad choice, in my opinion, I think there even is an example in CLL that violates this, but that's how it was defined in the formal grammar. If you stick to forethough connectives "ge ... gi ...", there can be no doubt about the scopes of the different prenexes, so it is better to use those for clarifications. The relative scopes in that case are transparent.
>>> The appearance of ``ci da quantifies ``da >>> as referring to three things, which are restricted by the relative clause >>> to be cats. >> >> The way I would say it is that the variable "da" takes values from the >> things that are cats, and (exactly) three of those values satisfy the >> predicate "blabi" (which also means all but three of those values >> don't satify it). If there is any reference going on here at all it is >> to all cats, not just to the three that do satisfy the predicate >> "blabi". If we were to say "no da poi mlatu cu blabi" then "da" again >> takes values from the same set of referents, but now the number of >> them that are said to satisfy blabi is zero. Would we say that "da" >> doesn't refer to anything in this case? > > That's an interesting question. I would say it refers to all referents in > the set, but the bridi is negated. At least the statement still has an > implication on all referents in the set, your explanation includes that, the > CLL one does not. (..or was that a rethorical question of yours..?)
Yes, it was sort of a rhetorical question to show why I wouldn't say that da refers to three things in the case of "ci da". If da refers to anything, it refers to all the values that the quantifier ranges over, not just to those that make the bridi true.
[...] > I understood that now. The inner quantifier is just incidential information > about the cardinality of the set-of-things we are talking about.
Exactly.
[...] > Well explained.
Thank you! The usual situation is that nobody understands what I'm saying when I try to explain these things. :)
> By the way, is there any document about lojban logic on the > web which sticks closer to what you explaned/to logic than the CLL?
Nothing I can think of. The CLL is correct for the most part, but it has some glitches here and there.
mu'o mi'e xorxes
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