Re: quantifiers

[Jim Carter <jimc@MATH.UCLA.EDU>]

Responding to pc's recent discussion about quantifiers.
Is it true that all unicorns are blue?  Reasons why
"Ax member(x, empty-set) F(x)" should be considered true:

Through negation:
        "not (Ax member(x, S) -> F(x))" =
        "not (Ax (not member(x, S)) v F(x))" =   #definition of implication
        "Ex member(x, S) & not F(x)"             #de Morgan's rules

Since S is an empty set, "Ex member(x, S)" is false, so the whole
proposition is false, so true is its negation, "Ax member(x, S) -> F(x)".
"If you doubt that all unicorns are blue, show me one that isn't."

Mechanistically: Define universal quantification over a set recursively
like this:
        "Ax member(x, S) -> F(x)"  is said by jimc to mean the same as

        if {S is finite but not "small"} {
                Ex member(x, S) & F(x) & (Ay member(y, S-{x}) -> F(y))
                (actually true for all members but we only consider one
                 at at time to avoid a circular definition)
        } elseif {S is "small"} {
                starting proposition discussed below
        } else {
                punt (infinite sets, irrelevant to this discussion)
        }

Now if "small" means "cardinality(S,1)" then the starting proposition
would be just F(y) where y is the unique member of S.  On the other
hand, if "small" means "cardinality(S,0)" then the starting proposition
is literal "true", and it seems to me to be perverse to say either that
"Ax member(x, S) -> F(x)" is false or undefined on an empty set, or that
such a recursive definition could not be applied in this case.

The invalid objection has probably been raised that you can't have a
statement and its negation both true, and "all unicorns are blue" and
"all unicorns are not blue" are both true in this analysis (correct)
and are mutually negations (wrong).  "Ax member(x,S) -> F(x)" is not
the negation of "Ax member(x,S) -> not F(x)" ("Ex member(x,S) & not F(x)"
is the negation; equivalently but more confusing, "Ex F(x) -> member(x, S)".)

pc says: "Even if there are no unicorns, what compels us to claim that
_ro pavyseljirna cu blanu_ is true?  It is a universal claim, so the
minimum truth value of its instances.  It has no instances, so,
presumably, it has no truth value."  jimc says: I thought we were going
to do this by logic, i.e. by and, or, etc., not by minimums.

pc continuing: "Or, since I suppose _su'o pavyslejirna cu blanu_ is
false and it is the maximum value of its instances (since it is a
particular claim) and the minimum is never greater than the maximum, it
follows that the universal claim is false too."  jimc continuing in a
similar vein:  If you're going to map the field of cardinality 2 (Boolean
values) into the field of integers and then use numeric comparisons,
there isn't any Boolean value for "has no truth value" and so you can't
compare the pc-preferred truth value of "ro pavyseljirna cu blanu" with
anything else.

I agree that it's odd to assert "ro pavyseljirna cu blanu" when the count
of unicorns is zero, but by trying to set up special cases to block out
such a statement, you throw sand (or sabots) into the gears of logic.

I agree with your pragmatic statement that most speakers using a
universal quantifier expect in a non-logical way that terms of the
logical expression that have lesser emphasis (restrictive terms) will
be true for at least some referents.  That is, the guy says "Ax
member(x,S) -> F(x)" and expects that S is nonempty.  In fact, most
people think "G(x) -> F(x)" includes assertion of "G(x)".  However, I
think we should do the same as when using English to express logic: if
pragmatic considerations conflict with logic, logic prevails.

Further on, pc says "In a similar way, the empty set is a perfectly
normal set in some sense, but its member(s?) are (almost?) never the
topic of conversation nor the sort of thing one wants to make claims
about."  But in math, very frequently you want to prove that something
F(x) is impossible, so you prove various propositions of the form Ax:
F(x) -> whatever" and finish by showing that the whatevers are mutually
exclusive.  We need to keep well oiled the machinery that works on the
various members of the empty set.

pc says: "...Lojban has identified three originally very distinct
notions, _ro da poi broda_, _ro broda_ and _ro lo broda_.  Since the
first of these was created exactly to have a universal quantifier with
existential import, the rest fell into that pattern as well..."  Maybe
"ro da poi broda" may have been alleged to have existential import, and
maybe speakers not trained in logic may assume that it has existential
import, but on the face of it (says jimc) it means exactly "Ax B(x) ->
(main bridi(x))" and there's no reason to screw up the logic by saying
otherwise.  I think pc regrets the interpretation, which he was not a
party to.  He points out that the existential import is stated in the
draft textbook -- but, jimc says, this is an error that ought to be
corrected.

James F. Carter        Voice 310 825 2897       FAX 310 206 6673
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