Message-Id: <m0sdUsP-0000ZHC@xiron.pc.helsinki.fi>
Date:         Tue, 1 Aug 1995 20:20:35 -0700
Reply-To:     "John E. Clifford" <pcliffje@CRL.COM>
Sender:       Lojban list <LOJBAN@CUVMB.BITNET>
From:         "John E. Clifford" <pcliffje@CRL.COM>
Subject:      quantifiers
To:           Veijo Vilva <veion@XIRON.PC.HELSINKI.FI>
Content-Length: 8744
Lines: 144

        Djer's Quine quote points out -- as does xorxes -- that we do not
need set talk to do the work here. As I have said, I am using it as a
convenience, since the purely predicate forms tend to get unwieldy (cf.
the use of "3x" rather than the full form with only particular quantifiers
and identities and all). It also allows me to to stick closer to the
Lojban, which has two sets of locutions (really 3), lo broda and da poi
broda which each needs explanation, even if they ultimately turn out to be
equival ent.  But I mentioned Quine for another reason: at least one of
his systems of logic operates totally without referring expressions,
having only bound variable and, defined in terms of them and some rather
unlikely predicates, definite descriptions whic h (of familiar sound!)
look like referring expressions but are, in fact, only disguises for very
complex quantified ones.

        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.  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;  and that latter is one of the few things that
just about everyone in the business agrees about (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).  Of course, I have made no secret of my
willingness to split these up every which way to get some referring
expressions in, but the point of this reconstruction was to account for
the relations which xorxes insists are in Lojban.

        As for why we have the restricted as well as the unrestricted
quantifier, the correct question from the linguistic point of view is, why
do we have the UNrestricted quantifier at all?  When we try to reproduce
it in a natural language, we almost always end up using a restricted one
with some very general restriction:"everyTHING."  (Indeed that is how
unrestricted quantifiers are defined in restricted quantifier theory -- or
with a tautological restriction.) As for existential import, we have that
because that is what natural languages do have (it is not a convention):
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).  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").  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.
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 would have almost no use other
than simplifying negation shifting, but that is already more use than I
have seen of some of these critters, so might be worth doing.  (I should
admit here that this assumes one side in an ongoing controversy -- since
the 11th century, so probably a philosophic controversy -- about reading
categoric propo sitions.  This side is C.L. Dodgson's and who can argue
with a mathematician with a sense of humor?)

        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.

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

        Xorxes has dealt admirably with most of djer's 7/30 posting.  Let
me add only a few comments, including some on xorxes' comments.

        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
)) The xu'u just covers all the equating of new variables with old ones.
It has nothing to do, so far as I can remember, with the dogs and men
issue (which I think is all taken care of -- but that's just me).

        The numerical quantifiers (and the fuller forms with just
existential quantifiers and identities) ARE a throwback to Principia
Mathematica, to numerosity without numbers.  We say how many things there
are -- or that we are interested in -- without referr ing to numbers
(sorry about the k({broda})=n notation, it is inaccurate but soooo handy,
better would be nx x broda, which also gets rid of the set notation.  I
meant the same thing by either system).  In PM numbers come much latter as
sets of equinumerou s sets, so at heart cardinals.  To get ordinals, you
have to have an ordering and outside the natural numbers that gets tricky
(and the unity of the two notions breaks down since there may be many
different ordinals that are not different cardinal :omega+ 1 is a
different ordinal from omega (the next one, pretty much) but not a
different cardinal).

        I am not sure what the question is about
>1. ro lo ci nanmu ku goida ci lo gerku ku goi de zo'u tu'e da pencu de
>2. ro lo ci nanmu ku goi da ci lo so gerku ku goi de zo'u tu'e da pencu
>de
>3. ro lo ci nanmu ku goi da ci lo ci gerku ku goi de zo'u tu'e da pencu
>de
But, besides not
knowing what tu'e does, I am uncomfortable about goi-ing to bound
variables, the more so when what is being identified as the referent of
the variable is a plural set -- variable tend to stand for individuals.  I
feel more comfortable wit h da and de replaced with, say, ny and gy
respectively.  The other problem I have is the claims that there are only
three men altogether (in reality, not just the universe of discourse,
since the internal quantifiers are in lo's) and nine dogs or three dogs.
Djer's earlier remark that
>there is virtually no difference between "ci lo broda" and "lo ci
>broda", except perhaps the convention that "lo ci broda" claims only ci
>broda exist.
seems to make light of an important
distinction between the size of the set (or the number of things that have
a certain property, to get away from set talk) and the number of those
things we are talking about at the moment.  I suppose that all three of t
he sentences listed above are false simply because they have the wrong
number of men or dogs or both in the universe.

        Xorxes exposition of re prenu cu pencu re gerku seems essentially
correct; the right reading is 3a or something more or less equivalent to
it (using sets, a la an earlier And, simplifies it somewhat, I think).
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).

        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."

        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.  The relevant
comprosmise seems needed (it also simplifies the formulaic versions by
allowing the universal part of , say, re prenu to work only with prenu,
not with the whole sentence -- called "relevant prenu" in some earlier
version).