[FOM] Slater's position
Randall Holmes
holmes at diamond.boisestate.edu
Tue Sep 30 10:33:16 EDT 2003
Dear FOM colleagues,
Slater says (and I appreciate the clarification):
"Using standard predicate logic we
can construct a string of quantificational expressions which are
normally abbreviated as involving 'numerical quantifiers'. Thus we
can formulate, for instance,
(Ex)(Ey)(-(x=y).(z)(Fz iff z=x v z=y)),
and shorten this to
(2x)Fx,
which is normally read 'There are exactly 2 Fs'. The grammar of such
remarks requires the place of '2' to be filled by numerals, or
numerical variables, and quantifying over that place then enables us
to say, for instance,
(En)((nx)Fx.(m)((mx)Fx -> m=n).n=2),
which is equivalent to
iota-n(nx)Fx = 2,
and so we get, specifically
iota-n(nx)(x isin {{},{{}}})=2.
That would allow one to say, somewhat circuitously, if there are
exactly two Fs,
(iota-n(nx)(x isin {{},{{}}})y)Fy,
but if Holmes wants to put set abstracts in the place of numerals,
and such iota term descriptions, he will have to interpret instead,
for instance,
({{},{{}}}x)Fx.
A canonical set with two members, however, such as {{},{{}}}, merely
might have the same number of members as the set of Fs, {x|Fx}, and
so is not the number of members in either. The number of even
numbers, for instance, is not {n| n>0}."
Out of this I pick in particular:
"but if Holmes wants to put set abstracts in the place of numerals,
and such iota term descriptions, he will have to interpret instead,
for instance,
({{},{{}}}x)Fx."
In fact, this is not standard predicate logic. This is standard
predicate logic augmented with quantifiers over numerical quantifiers.
I've seen this kind of thing before and it is quite amusing. But I'm
not convinced that it is the correct way to treat numbers.
Query: is ((2+3)x)(Px) a legitimate way to say "there are five P's?"
(this bears on your use of substitution in this kind of context).
Second Query: How about (((\iota n)(nx)(Px))y)(Qy) as a way of
saying "there are the same (finite) number of P's as Q's"?
My direct reply to this is that if you interpret (Qx)(Px) (where Q is
a "quantifier") in such a way that you allow quantification over the
quantifier Q, the only way I can make sense of this is to read
(Qx)(Px) as {x | Px} \in Q: you have developed a new notation for
asserting that the extension of the predicate P belongs to a set (or
possibly a class, or even a superclass) Q. Otherwise, I would answer
that _standard_ syntax does not allow quantification over quantifiers.
But if quantifiers are to be taken as first class objects, I insist on
interpreting them as classes of extensions.
If I read your notation this way, then your sentence (nx)(Px) expands
to {x | Px} \in n. So the numeral n is a set (or class, or
superclass) whose elements are exactly the extensions with n elements.
In other words, your numerical quantifier n is exactly the same thing
as my Frege natural number n, on the only reading I allow of your
(_far_ more logically nonstandard) quantification over quantifiers :-)
By the way, I now can interpret
({{},{{}}}x)Fx:
it means {x | Fx} \in {{},{{}}} (which indeed does not mean
that there are two objects such that F).
In Godel-Bernays set theory (ZFC + classes), one would need
superclasses (classes of proper classes) to interpret "quantifiers as
first-class objects". (for example, the universal quantifier would be
the singleton of the universal class, which is not even a class). The
numerical quantifiers are actually proper classes (as long as one only
quantifies over sets). In NFU, all the usual quantifiers would
actually correspond to sets, so this doesn't make me _too_
uncomfortable :-)
Since Slater is arguing from a logic which is not generally accepted,
he doesn't have a knockdown argument for anything. What he has is yet
another program for interpreting the natural numbers. His program is
much more questionable on the face of things: fiddling with logic is
more dangerous than fiddling with the axioms of set theory. I can
report, though, that his system admits a sensible interpretation in
NFU, which I have briefly indicated. In the only way I can make sense
of his language, he agrees with me about what the numbers are :-)
And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
the gates of Cantor's paradise, that the | Boise State U. (disavows all)
slow-witted and the deliberately obtuse might | holmes at math.boisestate.edu
not glimpse the wonders therein. | http://math.boisestate.edu/~holmes
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