tech:logic matters

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

&
Okay. So {ro da broda} entails {da broda}. But all other uses of {ro}
don't entail existence - {ro da poi kea broda cu brode} and
{ro broda cu brode} entail neither {da broda} nor {da brode}.
One ought to be encouraged, I think, to not use poi clauses with da,
and to instead use logical connectives.
pc:
Well, in fact, _ro da poi broda_ was introduced ages ago exactly to
carry the implication that there are brodas.  The other expressions,
_ro broda_ and _ro lo broda_ are, I think, up for grabs, but Cowan
sems to have appropriated all of these distinct expressions for some
other set of distinctions.  And I, of course, think that _ro_ should be
treated as uniformly as possible, which would mean requiring the
_ganai _gi_ construction to get the "modern" interpretation.  (In fact,
the current theory treats the standard quantifiers of first order logic
as degenerate cases of restricted quantifiers -- restricted to
everything -- as they historically are, and the considers the full range
of possible quantifiers (= second order relations satisfying certain
restrictions) and goes for a more general notion than usual, beyond
the four traditional quantifiers (AEIO in Scholastic terms) and
absorbing all the interpretations of those types as well as cases like
the "modern" forms but less satisfying systematically: "many,"
"most," "denumerably many," and so on.  Lojban actually looks
pretty good in this respect -- no text I've looked at even talks about
"enough" (a three-place relation so a two place quantifier as a
determiner).  The Lojban treatment of these all needs some tidying
up, however.)  I applaud &'s last suggestion but think it is
unreasonable to expect, for -- as Carter shows pretty clearly -- many
people are terribly fond of the "modern" universal (though they, like
Carter, greatly overestimate how often they need to use it).

Speaking of Carter:
Now suppose that sentence is transformed with a prenex and a
restrictive clause: "for all x which is a unicorn, x drives a
Chevrolet".
 pc:
Note that this is not a meaning-preserving transform, but rather a
different sentence with a different underlying structure (no internal
reference to the whole universe of discourse, for example).

Carter:
Here's why it is true if there are no unicorns -- I
imagine I'm preaching to the converts on this.  If a few unicorns did
exist, the sentence would be interpreted as "(C(U1) and C(U2) and
C(U3)..."  Conjoin the sentence with some unlike conjunction such as
"E(F1) and E(F2)..." so that you have a single list of simple terms,
all conjunction-ed.  The conjunction of the whole list should equal the
conjunction of the two abstract sentences.  But now suppose zero
unicorns existed and do the same conjunction; the union list equals the
term list of the other sentence, and so the conjunction of the abstract
sentences equals (in truth value) the other sentence -- proving that
the unicorn sentence is true.  QED.
pc:
Sure, sure! IF a 0-member extended conjunction either disappears or
evaluates to T.  Now there are a variety of reasons for doing each of
these but also arguments against them, so the case is not
demonstrated yet.  We can, in fact do it either way, getting slightly
different quantifiers each time.  And IF we identify universals with
conjunctions over the relevant sets.  The tradition of logic has been
to require domains of variables to be non-empty (and largely to
ignore the conjunction argument, which gets confusing, esepcially
when compared with disjunctions).

Carter:
On the other hand, the logical formula "Au:C(u)" is (I guess) to be
interpreted like this:  "For this sentence the universe of discourse
consists of all unicorns.  For all u in that universe, u drives a
Chevrolet."
pc:
No quite; only that the variable u is to range only over unicorns.
There may be other things in the universe of discourse.

Carter:
A few people allege that when someone utters that sentence,
the mere appearance of "all unicorns" identified as a universe of
discourse is a commitment by the speaker that the set is not empty.
pc:
Again, just the range of the variables (that is the universe of
discourse only for the familiar quantifiers of standard logic).

Carter:
I don't get that interpretation.  The sub-universe of discourse (all
unicorns) has to be computed from some containing universe of discourse
(e.g. all residents of middle earth) as a set.
pc:
Yes indeed, except for the terminology.

Carter:
Sets are frequently
empty, and in reasoning by contradiction, a speaker might very well
utter an "Au:C(u)" sentence where he knows in advance and expects to
prove to the listener that the sub-universe of discourse is empty.
pc:
Point of this?  If he says "AuC(u)" intending to prove that there are
no unicorns, then what is trying to do is prove his assumed
sentence is false because part of what it asserts, that there are
unicorns, leads to a contradiction.  That is, this is an example of
"AuC(u)" requiring that there be unicorns.

Carter:
In my opinion, 99.999% of all sentences involving universal
quantification fit naturally into the 1858 interpretation (with an
imbedded implication or a restrictive clause), and I would not shed a
single tear if prolix circumlocutions were needed to express in Lojban
the "Au:C(u)" type of sentence.
pc:
When you get to the point in McCawley where he talks about this (Capter 4
in the '81 version), you will find that your statistical guess is off by a
pretty healthy margin. Does your confidence come up to the point of being
willing to use the 1858 locution as well (as the "logical" bit seems to
require)?  For my part, I am content (not enthusiatic or even happy,
maybe) to save _ro da poi broda_ (fairly prolix) for the traditional
interpretation (and get all the other traditonal ones in as well) and
allow _ro broda_ or _ro lo broda_ (or both, with other differentiations)
to take the modern version.  It makes Lojban odd in the family of
languages, but we are used to considering that a virtue.
pc>|83