Message-Id: <m0sdigQ-0000ZHC@xiron.pc.helsinki.fi>
Date:         Wed, 2 Aug 1995 14:36:58 EDT
Reply-To:     jorge@PHYAST.PITT.EDU
Sender:       Lojban list <LOJBAN@CUVMB.BITNET>
From:         jorge@PHYAST.PITT.EDU
Subject:      Re: quantifiers
To:           Veijo Vilva <veion@XIRON.PC.HELSINKI.FI>
Content-Length: 8209
Lines: 163

pc:
>         xorxes suggest that we should understand ro broda cu brode as Ax(x
> broda => x brode).  OK.  But then we have to break either the connection
> between ro broda and ro lo broda or between ro lo broda and ro da poi
> broda.

I think {ro broda} = {ro lo broda} should be kept, I really meant to talk
about {ro lo broda}. The {ro broda} notation is much less flexible, and
I think it's just a convenience, I don't think it should have any special
significance.

> For ro da poi broda cannot be Ax(x broda => if da poi broda is to
> be either the corresponding Ex( x broda & or (and this turns out to be
> equivalent) Ex st x broda;

I never proposed that. I'm also not saying that {ro lo broda} should
be Ax(x broda => . That last thing is not a complete expression that
can function as a sumti, while {ro lo broda} is.

> (there are eight possible
> positions in categoric logic that would deny this, but (I think
> literally) no one holds any of them).  Unless, of course, poi is to take
> on fairly severe ambiguity (which gets worse if we bring in numerical and
> plurative quantifiers).

I'm afraid I don't see the ambiguity. I read {ro da poi broda cu brode}
as "every/each/all x which is a broda is also a brode". The only ambiguity
can be whether this is also saying that there is at least one x which is a
broda. The simplest logically is that it is not saying that.

> it comes as quite a shock to students that "All S is P" is true when there
> are no Ss (and a worse one if they hear that it is BECAUSE there are no
> Ss).

Actually, even children are well aware of that, even if it is hard to
recognize it in the logicians notation. Or do you think this dialogue
could not happen?

  child: Can I watch TV?
  parent: Did you put away all the toys you were playing with?
  child: Yes.
  parent: Ok, you can watch TV.
(later)
  parent: What's this? Your room is a mess, didn't you tell me that
          you had put away your toys?
  child: Yes, the toys that I was playing with, but I wasn't really
         playing with any.

Which means essentially that the child was taking "all" without
existential import when it was advantageous to do so. Of course, the
child will not understand that "All S is P" is true when there are no Ss
if you put it like that, but that doesn't mean that they don't in fact
understand the principle behind it. In fact, the child understood the
parent perfectly well, and yet uses this very technicality of the
quantifier to get away with it.

> The restrictive sense is the natur al one, in short (the "All S is
> P" form is largely an artificial logicians device, based either upon a
> totally different reading of these forms or an attempt to cover up the
> unnaturalness of the modern reading, forcing the collectivist "all" to do
> the work for what was in Greek and Latin the equivalent of "every").

I'm sure that the natural interpretation is to assume existential import,
but I think that comes from another source, not from the natural meaning
of the quantifier.

If I say "Every person that I met this mornig was wearing a blue hat", you
will naturally assume that I met at least one, and more likely more than
one person. But in my opinion this is not so much from what I said, but
from what I could have said instead but didn't. If I didn't meet anyone
this morning, then my sentence (even if taken as true) is not informative.
The talk of blue hats is even misleading. We tend to use language with some
purpose, not just to state irrelevant and misleading truths. If I met only
one person, then I would more likely have said "the person", instead of
"every person", so you will assume that I met more than one, since otherwise
there would be no reason to use "every" instead of "the". But when we
want to be misleading (as the child above), we are quite happy to take
recourse in the true deep meaning of quantifiers to state misleading truths.
And even children are very capable of doing this, it is not necessary
to have studied formal logic.

> It
> is also the logically simpler one, since what I have written as Ax st x
> broda is a single symbol, a variable binding connective, while Ax(x broda
> => is two level of symbols, a quantifier and a sentential connective.

This is just notation, I don't see any deep significance to it.
Ax st x broda can just as well be taken to have no existential import.
It is not a full sentence anyway.

> These are, of course, only notational devices (although Lojban follows
> them in making ro da poi broda simpler syntactically than ro da broda
> nagi'a).  The supposed greater complexity of negation shifting with
> restrictive quanti fiers could be dealt with by introducing from categoric
> logic the O quantifier to match the existing A,I, and E (ro, su'o and no
> -- ? has that last one changed again?).

I'm not sure I follow. {ro} is A, {su'o} is E, what is I, {pa}?
And what is O, is that {no}?

>         Incidentally, ro lo broda always was ro lo su'o broda, since ro
> ALWAYS implies su'o -- it is the "if" that gets the empty set in,
> remember.

That's not what the grammar papers say. They have inner {su'o} for {le},
which makes a lot of sense to me, because there I do want existential
import. But it has the innocuous {ro} as the inner for {lo}.

>         But, if you want the modern reading, Lojban has it and exactly the
> way modern logic does, so there is no loss.

You said modern logic takes "All S is P" to have no existential import.

>         The "artificial term" is just the device for making subselection
> in quantified expressions, rather than requantifying the same variable.
> So, for example, to deal with "There are three broda and two of them are
> brode" one habit was to say ci da zo'u da broda ije re da brode, while the
> proposal is to say ci da zo'u da broda ije re de poi de xu'u da cu brode
> (or so).  This latter would expand to ExEyEz ( x broda & y broda & z broda
> & Ew Ev ( ((w=x & v=y) or (w=y & v=z) or (w=x & v=z)) & w brode & v brode
> ))

Just as a comment, you are taking {ci da} to mean "at least three x".
The canon is that bare numbers are exact, so that {ci da} would mean
"at least 3 and at most 3 x". I would prefer them to be of the "at least"
variety, but traditionally they are not.

> Certainly, this form ought to allow for as many as four dogs being
> involved, which has been the crucial point.  I would urge that for 4 is
> represented in Lojban with something in the general area: re prenu, re
> gerku zo'u py pencu gy.  That would then settle the whole mess in a
> systematic way (and, of course, could be reached as afterthought using
> leapers).

I think that still gets you the nested form. {re prenu re gerku cu pencu}
should be equivalent to the prenexed {re prenu re gerku zo'u py pencu gy}.
For the other I propose {re prenu e re gerku zo'u py pencu gy}. The {e}
forces the two persons and two dogs to be at the same level.

>         What is abstruse about version 4?  It is different and more narrow
> than 3 but hardly difficult to understand, since it is almost as often
> what one means or understands by the English version "Two men patted two
> dogs."

Not really. When you say that in English, you don't usually mean to describe
exactly four relationships. It usually means {lu'o re nanmu cu pencu lu'o
re gerku}, a single relationship between a mass of two men and a mass of
two dogs. This allows for one man touching both dogs and the other touching
only one of them, for instance, which would be covered by the English version.
The English version would be unlikely if the exact claim was wanted. A much
longer paraphrase like "there were two dogs and two men, and each man
touched each dog" would probably be used.

>         Is lovi and the like still legal chat?  Dealing with the whole
> range of lo broda gets cumbersome (and people regularly resist doing it:
> witness the internal quantifier errors which almost all of us make
> occasionally); on the other hand, le broda seems to force us to think
> (incorrectly) that we are talking about non-brodas.

It doesn't force me to think that. I always assume that {le broda} means
we are talking about a broda. {lo vi broda } is still legal also.

Jorge