tech:logic matters

["John E. Clifford" <pcliffje@CRL.COM>]

Carter (still doing business at the old stand):
St. Anselm's Ontological Proof of the Existence of God is a famous
example of
the existential import of "all".
pc:
An interesting idea that, but I can't think of any reason to believe it,
since I could not find a quantifier in it anywhere that was not
demonstrably instantiated.  No use of import, in a word (or four).

Carter:
If I understand correctly the comments you made about history of logic,
you are
a rock solid Aristotlean whereas And & I cleave to the Stoic school.  Or
maybe
the Frege school.
pc:
Well, the Stoics don't come in, since they are the propositional logic
people as Aristotle is the quantifier guy.  Frege (and some
predecessors) figured out how to use bits of propositional logic to do
bits of the quantifier work as well.

Carter:
Anyway, I found it a very liberating
experience to discover that "all" doesn't have existential import.
Whereas as seen by
you,  that's just wrong -- at your mother's knee you learned that "all"
has
existential import, and And & I are simply wrong.
pc:
We have to do some distinguishing here, since another note suggests
we have all gotten a mass of things mixed together.  The import
status of the English word "all,"  as used in ordinary langauge is
rather unclear, possibly to the level of idiolectic variation (you say
nay, I say yea).  The question was about the *logical* word "all", as
represented by the universal quantifier (however that, in its turn, may
be represented).  That one's status I learned not at my mother's knee
but at Church (Alonzo), and it clearly has existential import, AxFx
entails ExFx, even in Frege.  Part of the claim of Lo??an being a
logical language is that its universal quantifier (_ro_ in the current
version) is the logical one.  And so it has existential import.  This is,
of course, the quantifier in its logical form, prenex and binding a
variable.  This form is derived, historically and now
metalinguistically, from logical form closer to natural language
forms: a quantified noun phrase + singular verb phrase, "All S is P,"
in which, as Geach used to grumble, the "all" should properly have
been "every," a quantifier about whose import there is no serious
doubt (it has it). The derivation (running mainly through the 19th
century) took the form of reducing the subject term to a vacuity,
"thing" roughly, and recreating the subject in various ways in a
complex predicate, finally, with "all," as the antecedent of a material
conditional.  Because of the rules about material conditionals -- true
whenver the antecedent is false,  these new versions of "All S is P,"
i.e., Ax:Sx => Px, were true when there were no S's, while the
original version was false.  Originally, Lo??an had only the strict
modern form for universal (and other quantified) claims: quantifier,
variable, form containing variable (conditional in the case of
universals).  But for all sorts of practical reasons, this situation
changed rapidly to one where quantifiers were used -- with the same
meaning as before -- to modify first existing sumti and eventually
even bridi (similar to the basic English pattern).  At some point, the
conflict between these new Lo??an forms and the very similar
English forms -- as well as the also similar old logical forms -- led to
the introduction of a way of making explicit the form with
existential import of the subject (not just the generic existential
import of the quantifier), _ro da poi..._.  This form was picked partly
because it was about all that was available, partly because it was
more complex and the expectation was that the importing form
would not be so common.  It did give, of course, another target than
_ro da ganai da broda gi ..._ to claim as underlying _ro broda cu..._
and _ro lo broda cu ..._ but these were -- until Cowan's recent move
(for other reasons) -- left as abbreviating only the usual modern
form.  Note, however, that throughout all of this the universal
quantifier as such has continued to have existential import for its
subject, the difference being only what its subject is (things or a
mentioned sort of things).

i,n:
> > They are the quantifiers of natural language ...
You keep insisting on this, but McCawley doesn't appear to think
that it's quite so clear cut.
        ... while 6.3.4b would probably be interpreted as including
        members who incurred no bills among those to whom the 10 percent
        discount is offered:
        ...
        6.3.4 b. Any member who paid all his bills by the fifteenth
        of the month was entitled to a 10 percent discount on
        their publications.
pc:
See earlier.  Notice here that the question is not about "all"  but
"any,"  which notoriously does not have existential import (see
Vendler's article in the Dictionary of Philosophy or the paper it is
based on).  Further, the point here was about what could be handled
by restricted quantifiers (second order predicates of a certain sort)
and "any" clearly fits in that category (subject term is included in
predicate term, usual sense of "included").

i,n
> whatever is the denial of _ro_ (?_ronai_? _nairo_? something else
> altogether?).
{naku ro} or {da'a su'o}
pc:
Neither of these work very well, given everybody's habit of pushing
negations around, in the first case, and the existential import in the
second.  When the restricted quantifiers were introduced, the
quantifier set was expanded to contain the fourth corner to the
traditional square -- contradictory to _ro_, subaltern to _no_ and
subcontrary to _su'o_ -- but I have lost all track of the form used.

i,n:
The other thing that needs to be worked out is how the
existential-universal interacts with (bridi) negation.
You may consider it to be a trivial exercise for the
reader, but it's a significant part of the negation
paper, and virtually part of the definition of what
{na} and {naku} mean.
pc:
Why we had four quantifiers in the final set: negations carried to the
diagonally opposite quantifier with all else unchanged.

i,n:
Given that we want to be able to express both "one and all"
(universal with existential import) and "any and all"
(universal which may be vacuous) in simple forms such
as the above, we need a quantifier for each.  I previously
offered you something like {ro su'o} as an existential-
universal to contrast with plain {ro} as a possibly-vacuous
universal, but you weren't impressed.  You did not however
offer me a possibly-vacuous universal in return, except
as a circumlocution, which the above seems to indicate
you agree is undesirable.  I am therefore forced to
propose my own, {ro su'o no}, which is not particularly
pretty, but I can always hope that usage will eventually
establish that a naked {ro} is at least ambiguous between
the two possibilities, as it is in English, and preferably
that {ro su'o no} is the default interpretation, at least
barring pragmatic indications to the contrary.
(I have no particular objections to particular constructions
such as {ro lo broda} carrying existential import, providing
it can be explained in such a way that this is not part of
the meaning of {ro}, but arises from the context as a whole,
for instance by a default {su'o} inner quantifier.  If it could
also be explained using your definitions, and still end up
with the same meaning, then we might both be satisfied, but I
won't hold my breath.  In any case, I think we need some explicit
quantifiers, such as discussed above, up our sleeves, to override
whatever implications might arise from the context.)
pc:
We have all of these things -- in various degrees of complexity and
with various theories about what expression means what -- already.
The only thing we do not have is a universal affirmative quantifier
that does not have inherent existential import.   Since I cannot quite
imagine why anyone would want to talk about a universe which was
totally empty (or, indeed, what one could say about it), this lack
seems far too minor to be worth hassling about.  Even _ro su'o no_
(which does not make much sense to me, as indeed does not _ro
su'o_)  seem way too short an expression for the purpose, which has
not yet turned up in any text.

&:
So how do you make sense of {no lo ro broda}? And must {lo no broda
e lo ro broda cu brode} be false?
pc:
{no lo ro broda} "none of the broda," i.e., there are some but none of
them fit whateve goes on thereafter.
{lo no broda e lo ro broda cu brode} is false because it
simultaneously asserts that the class of broda has some members and
that it does not (another reason, I think, for being sceptical about
those internal quantifiers, but that is another issue).

&:
"Most" can be done as a predicate
taking sets as arguments. (I recognize that you're being more orthodox
than me here, but I'm too much of an outsider to understand why the
orthodoxy is the orthodoxy.)
pc:
Hey, that is exactly what I just said quantifiers are in the general
theory.  That is, you recommend treating it just like the others (or the
general theory recommends treating the others just like "most").

Colin:
structures like
        re lo xirma
and
        re lo ci xirma
would appear to be anomolous
pc:
Not anomolous as such but just as representing quantifier expressions
directly.  Many of them turn out to be reducible to quantifier expressions
but it takes a couple of steps.  Of course, _re lo ci xirma_ also has that
extra bit about how many xirma there are altogether and that is a
different factor.
pc>|83