# Talk:BPFK Section: Logical Variables

> Re: BPFK Section: Logical Variables > The note misstates the situation for plural variables. A plural variable may > take several objects as value, but whether it takes them one at a time or > t9ogether to fit a predicate depends upon the predicate (place), whether it > calls for distributive or collective use.

The note just says: "A plural variable could take more than one value at once to satisfy a predicate." It doesn't say it _has_ to take them at once.

> Thus, given ways to mark the > requirements of places, {da} etc. could equally well be plural variables -- > as they so often are in practice now (though always with distributive use > assumed).

If da's were not singular, things like these would be false:

pa da broda ijo ge su'o de broda gi ro di poi broda cu du de Exactly one thing1 is a broda iff some thing2 is a broda and every thing3 that is a broda is that thing2.

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Oops! sorry. But why would a plural variable make the statement you give false? Again, a plural variable *may* take several values, but one is the limit case.

Jorge Llambías wrote:

> Re: BPFK Section: Logical Variables > The note misstates the situation for plural variables. A plural variable may > take several objects as value, but whether it takes them one at a time or > t9ogether to fit a predicate depends upon the predicate (place), whether it > calls for distributive or collective use.

The note just says: "A plural variable could take more than one value at once to satisfy a predicate." It doesn't say it _has_ to take them at once.

> Thus, given ways to mark the > requirements of places, {da} etc. could equally well be plural variables -- > as they so often are in practice now (though always with distributive use > assumed).

If da's were not singular, things like these would be false:

pa da broda ijo ge su'o de broda gi ro di poi broda cu du de Exactly one thing1 is a broda iff some thing2 is a broda and every thing3 that is a broda is that thing2.

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pc: > But why would a plural variable make the statement you give false? Again, a > plural variable *may* take several values, but one is the limit case.

Suppose exactly three things broda. Then the right hand side is true (some things are broda and they are all the brodas) but is the right hand side true?

pa da broda ijo ge su'o de broda gi ro di poi broda cu du de Exactly one thing1 is a broda iff some thing2 is a broda and every thing3 that is a broda is that thing2.

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Jorge Llambías wrote: pc: > But why would a plural variable make the statement you give false? Again, a > plural variable *may* take several values, but one is the limit case.

Suppose exactly three things broda. Then the right hand side is true (some things are broda and they are all the brodas) but is the right hand side true?

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Sorry, I was taking quantifiers as always particularizing (the normal {ro} etc, rather than the occasionally useful which goes by subpluralities. With that then, I am not sure what happens, but presumably both sides are false (since some of the subpluralities are not identical with the overall one and there are more than one plurality which brodas).

pc: > Jorge Llambías wrote: >> Suppose exactly three things broda. Then the right hand side >> is true (some things are broda and they are all the brodas) but >> is the right hand side true? >> >> pa da broda ijo ge su'o de broda gi ro di poi broda cu du de >> Exactly one thing1 is a broda iff some thing2 is a broda and >> every thing3 that is a broda is that thing2. > > Sorry, I was taking quantifiers as always particularizing (the normal {ro} > etc, rather than the occasionally useful which goes by subpluralities. With > that then, I am not sure what happens, but presumably both sides are false > (since some of the subpluralities are not identical with the overall one and > there are more than one plurality which brodas).

With the {ro} defined as the dual of {su'o}, (which works best as English "any") then it might work:

Exactly one thing brodas iff some thing brodas and any bordas are that thing.

But with {ro} as the normal plural "all", the one McKay represents with a capital lambda, it does not work. The right hand side just says that there are brodas: Some X brodas and all brodas are that X, which is always true if there are brodas.

So, if for no other reason, we can't take {da} as a plural variable at least until we decide which one of the two "all"s {ro} is.

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I'm not sure that I understand the quantifiers here in the way you do, which seems to range them over plural packets rather than just over several things. {pa da broda} says that one thing brodas, not that one plurality of unspecified size does. I don't think that the whole other {ro} is accompanied by a whole other PA. The plurality of the reference comes out mainly in the case of {su'o da} where a plurality of any size may answer the call. Your apporach (as I understand it) is singularist, making a plurality somehow a single thing, in the example a triad of some sort. I agree that the {naku su'o naku} universal is "any," since "any" (and "all") are not importing and neither is nsn. But it is not the normal "all" ("every" importing) of McKay's system. We have the distinction (not the only one perhaps but an identifying one) in the old {ma'u ro} (importing) and {ni'u ro} (not importing). If we were doing McKay, we would take {ma'u ro} as the norm and so unmarked and then mark {ni'u ro} when it is needed.

Jorge Llambías wrote:

pc: > Jorge Llambías wrote: >> Suppose exactly three things broda. Then the right hand side >> is true (some things are broda and they are all the brodas) but >> is the right hand side true? >> >> pa da broda ijo ge su'o de broda gi ro di poi broda cu du de >> Exactly one thing1 is a broda iff some thing2 is a broda and >> every thing3 that is a broda is that thing2. > > Sorry, I was taking quantifiers as always particularizing (the normal {ro} > etc, rather than the occasionally useful which goes by subpluralities. With > that then, I am not sure what happens, but presumably both sides are false > (since some of the subpluralities are not identical with the overall one and > there are more than one plurality which brodas).

With the {ro} defined as the dual of {su'o}, (which works best as English "any") then it might work:

Exactly one thing brodas iff some thing brodas and any bordas are that thing.

But with {ro} as the normal plural "all", the one McKay represents with a capital lambda, it does not work. The right hand side just says that there are brodas: Some X brodas and all brodas are that X, which is always true if there are brodas.

So, if for no other reason, we can't take {da} as a plural variable at least until we decide which one of the two "all"s {ro} is.

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pc: > I'm not sure that I understand the quantifiers here in the way you do, which > seems to range them over plural packets rather than just over several things.

No, that's not what I'm doing.

> {pa da broda} says that one thing brodas, not that one plurality of > unspecified size does.

Yes.

> I don't think that the whole other {ro} is accompanied > by a whole other PA.

I don't think so either. But there are two distinct plural ro's that reduce to the same thing for the singular variable case. The two plural ro's correspond to "all brodas" and "any brodas". The first is just another proportional quantifier, and the second is the dual of {su'o}.

> The plurality of the reference comes out mainly in the > case of {su'o da} where a plurality of any size may answer the call. Your > apporach (as I understand it) is singularist, making a plurality somehow a > single thing, in the example a triad of some sort.

Nope.

> I agree that the {naku su'o naku} universal is "any," since "any" (and "all") > are not importing and neither is nsn.

Importingness is a secondary issue here. The proportional "all" has two versions, importing and not importing. And then there's the dual of su'o (which is also not importing, but that's not its important feature).

> But it is not the normal "all" > ("every" importing) of McKay's system.

McKay's system has two (or three) "all"s.

> We have the distinction (not the only > one perhaps but an identifying one) in the old {ma'u ro} (importing) and > {ni'u ro} (not importing). If we were doing McKay, we would take {ma'u ro} > as the norm and so unmarked and then mark {ni'u ro} when it is needed.

Actually McKay marks the importing rather than the not importing, but that's not that relevant. In addition to the importing and not importing proportional ro's, we would need another one, the dual of su'o, "any".

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A>I don't get your point then.

B> But a handy one, since we know how to handle it and not this other thing, whatever it is (and non-importing is the immediate consequence of the nsn formula).

C> I only count two (and note he is mistakena about the need for non-importing regular lambda "all" since he screwed up an example.

D> As I just said. McKay is mildly perverse in thinking that a restricted universal quantifier is non-importing (yet different from nsn). Too much bad 19th century "logic" in his lineage, I suppose.

Jorge Llambías wrote: pc: > I'm not sure that I understand the quantifiers here in the way you do, which > seems to range them over plural packets rather than just over several things.

A>No, that's not what I'm doing.

> {pa da broda} says that one thing brodas, not that one plurality of > unspecified size does.

A>Yes.

> I don't think that the whole other {ro} is accompanied > by a whole other PA.

A>I don't think so either. But there are two distinct plural ro's that reduce to the same thing for the singular variable case. The two plural ro's correspond to "all brodas" and "any brodas". The first is just another proportional quantifier, and the second is the dual of {su'o}.

> The plurality of the reference comes out mainly in the > case of {su'o da} where a plurality of any size may answer the call. Your > apporach (as I understand it) is singularist, making a plurality somehow a > single thing, in the example a triad of some sort.

A>Nope.

> I agree that the {naku su'o naku} universal is "any," since "any" (and "all") > are not importing and neither is nsn.

B>Importingness is a secondary issue here. The proportional "all" has two versions, importing and not importing. And then there's the dual of su'o (which is also not importing, but that's not its important feature).

> But it is not the normal "all" > ("every" importing) of McKay's system.

C>McKay's system has two (or three) "all"s.

> We have the distinction (not the only > one perhaps but an identifying one) in the old {ma'u ro} (importing) and > {ni'u ro} (not importing). If we were doing McKay, we would take {ma'u ro} > as the norm and so unmarked and then mark {ni'u ro} when it is needed.

D>Actually McKay marks the importing rather than the not importing, but that's not that relevant. In addition to the importing and not importing proportional ro's, we would need another one, the dual of su'o, "any".

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pc: > A>I don't get your point then.

My point is that:

pa da broda ijo ge su'o de broda gi ro di poi broda cu du de

is true when the da's are singular variables, but false when they are plural variables (taking ro as McKay's lambda quantifier).

In other words, assuming there are broda, {su'o da poi broda cu du ro de poi broda} is always true with plural variables, but normally not true with singular variables.

[McKay's system has two (or three) "all"s.] > C> I only count two (and note he is mistakena about the need for > non-importing regular lambda "all" since he screwed up an example.

Count again. He represents one of them with an inverted A, that's the dual of inverted E. The other two are represented with capital lambda, one with and one without a superscript that indicates existential import. But this latter distinction is not that interesting, the two interesting cases are the inverted A versus the lambda.

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The "need" for the two lambdas comes about because he thinks that "All the students in my class who are Italian are chemistry majors" follows from "All the students in my class are chemistry majors," even when none of the students are Italian. And there is a sense in which it does, but not what he seems to think. That is, he wants "[ax:x is a student in my class and an Italian Ax:x is a student in my class and an Italian]x is a chemistry major" to follow from "[ax: x is a student in my class Ax: x is a student in my class] x is a chemistry major." But, with importing A, you can expand the subject only when the expanded expression has an instance, so not in this case. On the other hand, it always works to go from "[ax:Fx Ax:Fx] Gx" to "[ax:Fx Ax:Fx] if Hx then Gx" (indeed, expanding the subject moves through this as an intermediate step), and this can be read as "All fs which are H are G" or so as well as the earlier version.

The lambda, even as importing is curiously defined, taking acoount of the anomolous cases rather than eliminating them from the get-go. If it were just what we would write in unrestricted quantifiers as "~Ex(Fx & ~Gx) & ExFx" (or, at the end, "Ex(Fx & Gx)") it does not appear that the anomolous cases would arise. Further, all the cases could be handled using importing quantifiers.

Having said that, I should note that, while this solves the problem for "all," it does not do so for the "proportional quantifiers" (other than "all" if you want to include it in this class). They still seem to require a totality that is not available with non-cumulative predicates. At this point it seems to me that McKay should take heed of what singularists do with non-dstributive predicates, namely turn them into relations and consider the pluralities involved pairwise. Thus, "I are shipmates" becomes (roughly) "for all j, k among I, j is a shipmate of k and k of j." The the non-cumulative cases (and in deed the non-distributive ones generally) become, taking "most" as an example, "most x that engage in F (with someone) are G (with someone or with someone — or all or most or whatever fits --they F with). I think that is what he is shooting for but is hampered from getting there by his presuppositions about how proportional quantifiers work (though that pattern could be made to work with this reading and a bit of work-- the sortal has been distributed).

Jorge Llambías wrote:

pc: > A>I don't get your point then.

My point is that:

pa da broda ijo ge su'o de broda gi ro di poi broda cu du de

is true when the da's are singular variables, but false when they are plural variables (taking ro as McKay's lambda quantifier).

In other words, assuming there are broda, {su'o da poi broda cu du ro de poi broda} is always true with plural variables, but normally not true with singular variables.

[McKay's system has two (or three) "all"s.] > C> I only count two (and note he is mistakena about the need for > non-importing regular lambda "all" since he screwed up an example.

Count again. He represents one of them with an inverted A, that's the dual of inverted E. The other two are represented with capital lambda, one with and one without a superscript that indicates existential import. But this latter distinction is not that interesting, the two interesting cases are the inverted A versus the lambda.

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pc: > The lambda, even as importing is curiously defined, taking acoount of the > anomolous cases rather than eliminating them from the get-go. If it were > just what we would write in unrestricted quantifiers as "~Ex(Fx & ~Gx) & > ExFx" (or, at the end, "Ex(Fx & Gx)") it does not appear that the anomolous > cases would arise. Further, all the cases could be handled using importing > quantifiers.

But the point is that lambda is NOT the dual of E.

Lambda is for cases where all do but at the same time some don't. All the students surround the building, but at the same time some of the students (the Italians) don't surround it, as there's only two of them.

> Having said that, I should note that, while this solves the problem for > "all,"

It does not.

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A> Nor is my proposed alternative, since the dual of E would be non-importing. Lambda is not for the cases where all do but some do not. Those cases arise independently of lambda, namely from trying to expand the subject (without an existence proof) or when the sortal is collective (that that move does not work is part of the definition of "collective"). So those cases will be treated the same under my quantifier as under lambda, but the anomolous cases won't arise.

Nostalgia! Familiarity is a pain! When I saw the stock definition of "1 F is G," I did not pay any attention to it. But it is wrong for McKay's version, which is "[ei: I is F EI: I is F] I is G and I is 1 in number." The last clause is just "I is an individual" ("E J: J among I I among J") (the other numbers can be built on this inductively, given a defined "I is n in number," "I is n+1 in number is just "[ej: J among I EJ: J among I] J is n in number & E K: K among I and K not among J K is 1 in number"). The nearest McKay comes to what you suggest is "the Fs that are G are one in number," where "the Fs that are G" is something like what you offer: "[ei:I is F EI:I is F] I is G & A J: J is F and G J among I" (or so).

Jorge Llambías wrote:

pc: > The lambda, even as importing is curiously defined, taking acoount of the > anomolous cases rather than eliminating them from the get-go. If it were > just what we would write in unrestricted quantifiers as "~Ex(Fx & ~Gx) & > ExFx" (or, at the end, "Ex(Fx & Gx)") it does not appear that the anomolous > cases would arise. Further, all the cases could be handled using importing > quantifiers.

A>But the point is that lambda is NOT the dual of E.

Lambda is for cases where all do but at the same time some don't. All the students surround the building, but at the same time some of the students (the Italians) don't surround it, as there's only two of them.

> Having said that, I should note that, while this solves the problem for > "all,"

It does not.

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pc: > Lambda is not for the cases where all do but some do not.

Huh? It's for the cases where all do even though some may not. That's just what McKay uses it for! As in "All students surround the building". The dual of E on the other hand, is for the cases where any and all do (and therefore it is not the case that some do not: ~E~.)

> Those cases arise independently of lambda, namely from trying to expand the > subject (without an existence proof) or when the sortal is collective (that > that move does not work is part of the definition of "collective"). So those > cases will be treated the same under my quantifier as under lambda, but the > anomolous cases won't arise.

No, no. Your quantifier, namely "~Ex(Fx & ~Gx) & ExFx" fails for this case, because when all students surround the building it is easy to find some students that don't surround the building. This has nothing to do with existential import.

> Nostalgia! Familiarity is a pain! When I saw the stock definition of "1 F > is G," I did not pay any attention to it. But it is wrong for McKay's > version, which is "[ei: I is F EI: I is F] I is G and I is 1 in number."

I was not giving McKay's version of anything. I was giving an example of a sentence that is true with singular variables and false with plural variables. Anyway, McKay's numeric quantifiers are not "exact", so when he says 3X:FX it may be the case that more than three things are F.

> The last > clause is just "I is an individual" ("E J: J among I I among J")

I guess you mean "[A J: J among I] I among J".

> (the other > numbers can be built on this inductively, given a defined "I is n in number," > "I is n+1 in number is just "[ej: J among I EJ: J among I] J is n in number & [http://www.lojban.org/tiki/E%20K:%20K%20among%3Cbr%20/%3E%3E%20I%20and%20K%20not%20among%20J E K: K among > I and K not among J] K is 1 in number").

Yes.

> The nearest McKay comes to what you > suggest is "the Fs that are G are one in number," where "the Fs that are G" > is something like what you offer: "[ei:I is F EI:I is F] I is G & A J: J is F and G J > among I" (or so).

What did I suggest?

All I did was give a sentence that is true with singular variables and false with plural variables, I never even attempted to say "exactly one thing is a broda" using plural variables.

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A> No, E is enough (it turns out in this case that this is also A, of course). B> This does have to be A, otherwise this would be trivially true for any number greater than n.

c> The original was "If da's were not singular, things like these would be false:

pa da broda ijo ge su'o de broda gi ro di poi broda cu du de Exactly one thing1 is a broda iff some thing2 is a broda and every thing3 that is a broda is that thing2." So, assuming you are using "1x" in the usual way, not McKay's, what effect does these variable being plural have on the above. If more than one thing brodas then {pa da broda} is false in both cases and the second clause on the right is false as well. On the other hand, if there is only one broda then all three components come out true. Unless you mean "1x" in McKay's sense and you are right that that is only an "at least" claim (or, more accurately, a claim about the critters of interest without regard to what happens a zillion mile away or even here but out of focus). "There is I of broda and it is a monad just in case there is I of broda and every J of broda, J is among I and I among J" (that is, the two are identical and true if and only if I (and J) are monads). It is a bit harder to do for "2x" and on.

Jorge Llambías wrote:

> Nostalgia! Familiarity is a pain! When I saw the stock definition of "1 F > is G," I did not pay any attention to it. But it is wrong for McKay's > version, which is "[ei: I is F EI: I is F] I is G and I is 1 in number."

I was not giving McKay's version of anything. I was giving an example of a sentence that is true with singular variables and false with plural variables. Anyway, McKay's numeric quantifiers are not "exact", so when he says 3X:FX it may be the case that more than three things are F.

> The last > clause is just "I is an individual" ("E J: J among I I among J")

A>I guess you mean "[A J: J among I] I among J".

> (the other > numbers can be built on this inductively, given a defined "I is n in number," > "I is n+1 in number is just "[ej: J among I EJ: J among I] J is n in number & [http://www.lojban.org/tiki/E%20K:%20K%20among%3Cbr%20/%3E%3E%20I%20and%20K%20not%20among%20J E K: K among > I and K not among J] K is 1 in number").

B>Yes.

> The nearest McKay comes to what you > suggest is "the Fs that are G are one in number," where "the Fs that are G" > is something like what you offer: "[ei:I is F EI:I is F] I is G & A J: J is F and G J > among I" (or so).

C>What did I suggest?

All I did was give a sentence that is true with singular variables and false with plural variables, I never even attempted to say "exactly one thing is a broda" using plural variables.

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pc: > A> No, E is enough (it turns out in this case that this is also A, of > course).

I'm sorry, but: "[E J: J among I] I among J" is not the same as "I is an individual".

In fact "[E J: J among I] I among J" is true for any I. If I are three individuals, then "[E J: J among I] I among J" is true.

On the other hand, "[A J: J among I] I among J" does mean that "I is an individual".

> B> This does have to be A, otherwise this would be trivially true for any > number greater than n.

"This" was:

(the other numbers can be built on this inductively, given a defined "I is n in number," "I is n+1 in number is just "[ej: J among I EJ: J among I] J is n in number & [http://www.lojban.org/tiki/E%20K:%20K%3Cbr%20/%3Eamong%20I%20and%20K%20not%20among%20J E K: K among I and K not among J] K is 1 in number").

What has to be A there? I think that's fine as it is.

> c> The original was
> "If da's were not singular, things like these would be false:
>
> pa da broda ijo ge su'o de broda gi ro di poi broda cu du de
> Exactly one thing1 is a broda iff some thing2 is a broda and
> every thing3 that is a broda is that thing2."

Right.

> So, assuming you are using "1x" in the usual way, not McKay's, what effect > does these variable being plural have on the above.

It depends on which of the two universal quantifiers we take {ro} to represent.

If {ro} is McKay's lambda, then {ge su'o de broda gi ro di poi broda cu du de} just means {su'o da broda}.

If {ro} is the dual of {su'o}, (McKay's inverted A) then it does work. But then how do we say "all students surround the building"?

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A> Well, I am being dense here, but this looks like the first case ("I is an individual")If there are n+a zillion things among I, then there will be n things among I and an individual among I but not among those those n things, indeed, a zillion such. The formula works for "at least n+1" but that is not what was sought.

B> Some I are broda and some J are the broda, some K are identical to I and K is all of J. So, if there is 1 broda, then this is true. If there are two broda, the the first clause is true: pick any of x, y, x+y. If we pick x or y, then this is false, since K (i.e I) is not all of J. If we pick x+y then it is true. So it is not equivalent to 1x:x broda. Sorry it took so long.

C> One of the points here was to get rid of all these "all"s. All this going around has helped me sort through a mess of stuff (making public mistakes along the way), but this last bit puts the finishing touches on one more try: [qx:Fx QX:Fx]GX depends upon the notion of *the* Fs or rather the F's, where F' is the distributive correlate of F, "is involved in F". Then, if a are the F's iff ~[ei:I are F EI:I are F] ~I among a & ~[ej: J among a & J individual EJ: J among a & J individual] ~[file:///K:/K%20are%20F K:K are F] J among K. And some existence things I need to sneak in there somehow — or just move over to normal universals. Then the quantified sentence above is just [ei: I are the F & G)'s EI: I are the F & G)'s][ej: J are the F's EJ: J are the F's] I is Q of J. Some fuzzy stuff here still, so plese knock this doen so it comes into focus.

Jorge Llambías wrote:

> B> This does have to be A, otherwise this would be trivially true for any
> number greater than n.

"This" was:

(the other numbers can be built on this inductively, given a defined "I is n in number," "I is n+1 in number is just "[ej: J among I EJ: J among I] J is n in number & [http://www.lojban.org/tiki/E%20K:%20K%3Cbr%20/%3Eamong%20I%20and%20K%20not%20among%20J E K: K among I and K not among J] K is 1 in number").

A>What has to be A there? I think that's fine as it is.

> c> The original was
> "If da's were not singular, things like these would be false:
>
> pa da broda ijo ge su'o de broda gi ro di poi broda cu du de
> Exactly one thing1 is a broda iff some thing2 is a broda and
> every thing3 that is a broda is that thing2."

Right.

> So, assuming you are using "1x" in the usual way, not McKay's, what effect > does these variable being plural have on the above.

It depends on which of the two universal quantifiers we take {ro} to represent.

B>If {ro} is McKay's lambda, then {ge su'o de broda gi ro di poi broda cu du de} just means {su'o da broda}.

C>If {ro} is the dual of {su'o}, (McKay's inverted A) then it does work. But then how do we say "all students surround the building"?

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pc: > A> Well, I am being dense here, but this looks like the first case ("I is an > individual")If there are n+a zillion things among I, then there will be n > things among I and an individual among I but not among those those n things, > indeed, a zillion such. The formula works for "at least n+1" but that is not > what was sought.

You're right, you do need the "all" there.

> C> One of the points here was to get rid of all these "all"s.

I don't think we can get rid of them. There are two "all"s that differ significantly (each of them can have an existential and a non-existential variant, but that is generally irrelevant).

The two different "all"s are the one used in "all students surround the building" and the one used in "all companies that compete share common interests". "Any" can substitute for it in this second case "any companies that compete share common interests".

But I don't see how the point can be to get rid of them. Both have clear and distinct uses.

> All this > going around has helped me sort through a mess of stuff (making public > mistakes along the way), but this last bit puts the finishing touches on one > more try: [qx:Fx QX:Fx]GX depends upon the notion of *the* Fs or rather the F's, > where F' is the distributive correlate of F, "is involved in F".

I'm not sure why you need F'. Anyway, "the Fs" are needed for proportional quantifiers. Non-proportional ones are meaningful even when "the Fs" is not well defined.

> Then, if a are the F's iff ~[ei:I are F EI:I are F] ~I among a & ~[ej: J among a & J

> individual EJ: J among a & J
> individual] ~[file:///K:/K%20are%20F K:K are F] J among K.

What's the second part for? What's the quantifier that binds K? I would have thought that a are the Fs iff ~[ei:I are F EI:I are F] ~I among a. Is that not enough?

> And some existence things I need to sneak > in there somehow — or just move over to normal universals. Then the > quantified sentence above is just [ei: I are the F & G)'s EI: I are the F & G)'s][ej: J are the F's EJ: J are the F's] > I is Q of J. Some fuzzy stuff here still, so plese knock this doen so it > comes into focus.

Why not simply [EI: I are G][EJ: J are the Fs] I are Q of J, which is what McKay gives for proportional quantifiers? I think it amounts to the same thing when "the F&Gs" is well defined.

Either way, it won't work for {no}, for {su'eci}, or for A (the dual of E). But as McKay shows these are definable in terms of the others.

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A> I think I have managed — with a little more fiddling. The non-importing versions have not been shown necessary.

B>Aesthetics; it is a neater system with only one universal quantifier. Also, the lambda quantifer gives anomolous cases (that is, universal claims that are not covered by thatuniversal quantifier) and getting rid of them is surely a good idea (although pointing them out is also useful for the study of English, say).

C> Because there are sometimes (i.e., with noncumulative F) no "the Fs" yet the proportional quantifiers need a whole to work off of. Yes, I should separate the proportional ones from the others, since the whole reference to the Fs is unneccesary there,

D> The second part gets that only F's are in (each hsa to be in some group that Fs), while the first gets them all in. It should read "[EK:"

E> Yes, I seem to have actually defined the stronger notion which deal with all the FGs, not merely the locally interesting group. Of course you need this fuller form to do a bang-up job when Q = "all", but by inference even that can make do with only an "E" and without the "the F' G's." Nice, since the steps for "the" are messy in detail. Thanks. This should amount to the same thing when "the F and G" are well-defined, but it covers (is meant to, anyhow) the remaining cases as well in a single formula.

F> Yes; even McKay has to do "no" separately, since it involves the denial of some or all of those E's he has built in. I aam less sure about the dual of E, which does seem to work out — but my score with working things out is low right now. I don't see the problem with "at most" unless you mean it to cover the "no" case — and if you do, then it is just "not at least" which is admittedly another new pattern. But McKay has to deal with all those, too, so I seem to be one up on him (and he does not even describe how he will do the other proportional quantifiers with noncumulative predicates). Does he reduce the dual of E other than by using the dual formula?

Jorge Llambías wrote:

pc: > A> Well, I am being dense here, but this looks like the first case ("I is an > individual")If there are n+a zillion things among I, then there will be n > things among I and an individual among I but not among those those n things, > indeed, a zillion such. The formula works for "at least n+1" but that is not > what was sought.

You're right, you do need the "all" there.

> C> One of the points here was to get rid of all these "all"s.

A>I don't think we can get rid of them. There are two "all"s that differ significantly (each of them can have an existential and a non-existential variant, but that is generally irrelevant).

The two different "all"s are the one used in "all students surround the building" and the one used in "all companies that compete share common interests". "Any" can substitute for it in this second case "any companies that compete share common interests".

B>But I don't see how the point can be to get rid of them. Both have clear and distinct uses.

> All this > going around has helped me sort through a mess of stuff (making public > mistakes along the way), but this last bit puts the finishing touches on one > more try: [QX:Fx]GX depends upon the notion of *the* Fs or rather the F's, > where F' is the distributive correlate of F, "is involved in F".

C>I'm not sure why you need F'. Anyway, "the Fs" are needed for proportional quantifiers. Non-proportional ones are meaningful even when "the Fs" is not well defined.

> Then, if a are the F's iff ~[EI:I are F] ~I among a & ~[EJ: J among a & J > individual] ~[K:K are F] J among K.

D>What's the second part for? What's the quantifier that binds K? I would have thought that a are the Fs iff ~[EI:I are F] ~I among a. Is that not enough?

> And some existence things I need to sneak > in there somehow — or just move over to normal universals. Then the > quantified sentence above is just [EI: I are the F & G)'s][EJ: J are the F's] > I is Q of J. Some fuzzy stuff here still, so plese knock this doen so it > comes into focus.

E>Why not simply [EI: I are G][EJ: J are the Fs] I are Q of J, which is what McKay gives for proportional quantifiers? I think it amounts to the same thing when "the F&Gs" is well defined.

F>Either way, it won't work for {no}, for {su'eci}, or for A (the dual of E). But as McKay shows these are definable in terms of the others.

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In the light of this regrettably overlong discussion (my apologies again), the original sentence raises some interesting points. The natural reading of {pa da broda} as a plural quantifier in McKay's system would be in effect "there are broda which are one in number," which would be true whenever there are broda. On the other hand, the Lojban intention is "the broda are one in number" which is a different claim altogether and clearly not equivalent to your right-hand-side formula. This raises the interesting possibility of finding a significant (i.e., semantic rather than pragmatic) difference between {da poi broda} and {lo broda}: one of them is global (about the broda), the other local (about some broda of interest). The logically natural move would be to take the quantifier expressions as global and the descriptors as local, but everyone claims that Lojban usage goes the opposite way. This is unfortunate, since the local case is the one that needs the kinds of things that {lo} allows (partitive quantification, size independent of the number of broda and also dependent) and the universal does not (it shifts to the local for all those moves). Of course, these distinctions may not make as much difference if we introduce modals for "generally" and the like.

Re: BPFK Section: Logical Variables The note misstates the situation for plural variables. A plural variable may take several objects as value, but whether it takes them one at a time or t9ogether to fit a predicate depends upon the predicate (place), whether it calls for distributive or collective use. Thus, given ways to mark the requirements of places, {da} etc. could equally well be plural variables — as they so often are in practice now (though always with distributive use assumed).