Re: quantifiers

[jorge@PHYAST.PITT.EDU]

> xorxes:
> . Even with the set interpretation [two orders of particular quantifiers]
> are equivalent (as they should be, since the talk of sets is purely
> clarificational, it doesn't change the meaning): there is a set of
> at least one dog such that for every dog of that set there is a set
> of at least one man such that each man of that set pets the dog. It
> all comes out as saying that there are at least one dog and one man
> such that the man pets the dog. (In this case, talk of sets muddles
> everything, but it comes out to be the same in the end.)
> pc:
> I don't see it: how do we equivalently push particular quantifiers across
> universals to convert from  Ex(x/=0 & x c {man} & Ay(y e x => Ez(z/=0
> & z c {dog} & Aw(w e z => y pets w)))) to Ez (z/=0 & z c {dog} &
> Aw(w e z => Ex(x/=0 & x c {man} & Ay(y e x => x pets y)))).  Maybe
> we can manage using some principles of set theory (which may be
> appropriate, given we have introduced sets) but the general pattern does
> not transform in logic.

I don't understand the objection. If sets are the apropriate way of
describing {lo broda}, then the principles of set theory must be
allowed in the description. If only logic is allowed (somehow excluding
priciples of set theory) then only logic should be used in explaining
what {lo broda cu brode lo brodi} means.

> But xorxes may have some other notion of
> equivalence or subordination here.

If you think that {lo broda cu brode lo brodi} is _not_ equivalent
to {lo brodi cu se brode lo broda} in any interesting sense, then
please explain, preferably giving an example. I don't think I'm using
any weird notion of equivalence or subordination here.

> xorxes:
> Given two quantifiers Q1 and Q2, there are three possibilities of
> subordination:
> (i)     Q1(Q2)  i.e. Q2 subordinate to Q1
> (ii)    Q2(Q1)  i.e. Q1 subordinate to Q2
> (iii)   Q1-Q2   i.e. Q1 and Q2 are coordinate.
> In the case of the men and dogs, (iii) is the round robin,
> and (i) and (ii) are three independent dogs for each man and
> three independent men for each dog, respectively.
> But when the quantifier is {ro} or {su'o}, these three cases
> are degenerate.
> If Q1 is {ro}, then (ii) and (iii) are the same.
> If Q2 is {ro}, then (i) and (iii) are the same. And so if both
> Q1 and Q2 are {ro}, then all (i), (ii) and (iii) are the same,
> which is right, since multiple {ro}s commute.
> If Q1 is {su'o}, then (i) and (iii) are the same.
> If Q2 is {su'o}, then (ii) and (iii) are the same. And so if
> both Q1 and Q2 are {su'o}, then all (i), (ii) and (iii) are
> the same.

pc:
> What is the other quantifier in each case?

Anything.

> the other of _ro_ - _su'o_?

Could be. In which case, notice that no new degeneracies appear:
ro-su'o makes (ii) and (iii) the same, while su'o-ro makes (i)
and (iii) the same, under either the rules for {ro} or those for
{su'o}.

> Or
> might it be a numeric?

Yes, for the general case.

> I take it that "subordinate" here means that the
> choice of the instantiations of the subordinate is a function of/depends
> on the choice for the superordinate.

Right.

> I guess my problem is what does
> "the same" mean?

That they are both true in the same circumstances. There is no situation
that makes one true and the other false.

> While AxAyFxy is equivalent to AyAxFxy (case iii
> with _ro_?)

All three cases are equivalent with ro-ro.

> AxEyFxy does not entail (and so is not equivalent to)
> EyAxFxy, so the two do not seem to be coordinate.

I didn't say they were. The first is (i) and the other is (ii)=(iii).


> Similarly, ExAyFxy
> is not entailed by AyExFxy and so they are not equivalent nor
> coordinate.

I agree, of course. Also:

ExEyFxy = EyExFxy, and in this case as with ro-ro (i)=(ii)=(iii).


> Or do you have some other sense of coordinate in mind?  I
> do not understand this passage at all.

I was talking of two quantifiers being coordinate, not two expressions.

> After a day of (occasional) mulling what I come up with is that this is
> another way of saying that (where "affect" is about allowing different
> choices with different instances)
>         1. Universals are not affected by superordinates (and so are
> coordinate with them?)
>         2. Universal otherwise affect what is subordinate to them
>         3.  Particulars do not affect subordinates (and so are coordinate
> with them?)
>         4.  Particulars are otherwise affected by any superordinate
>         5.  Numerics affect subordinate numerics
> These are all true and mean that there are several cases where (assuming
> some connection between order and subordination)  order does not
> matter.  But it also leaves a lot of cases where it does and wouldn't it be
> nice to have a handy way of dealing with them and perhaps a uniform
> way for all the others as well?  So far as I can see, we do not at the
> moment.

I don't disagree with any of that.

Jorge