FOM: "Arbitrary Objects" - two more remarks

[Gordon Fisher]

Kanovei wrote:

> From: Arnon Avron <aa at post.tau.ac.il>
> Date: Mon, 11 Feb 2002 10:53:12 +0200 (IST)
>
> >Just think how much energy has been invested in
> sentences like "the present king of France is bald"
>
> This seems to be written in disapproval, but in fact the
> to understand that the sentence
>
> "the present king of France is bald"
>
> that is,
>
> \forall x (x is the present king of France --> x is bald)
>
> means to understand the predicate calculus in general.
> You may complain that >2000 years since Aristotle is too
> long a term to achieve a proper understanding here, but
> hardly that the efforts have led to something not worthwhile.
>
> V.Kanovei

Would I be getting closer to the idea of arbitrary objects if
I said the following:

According to the predicate calculus, I take it that
(Ax)(P(x) ->Q(x)) is true if there is no a such that P(a),
where a presumably is looked for (so to speak) in some set S.
Arbitrary objects have been introduced by Fine in order
to provide some sort of formalization where
we don't get this result upon universal quanitification,
insasmuch as one works with P(x) only if there is at
least one a in S such that P(a).  ???

As I expect everyone knows, there is a kind of
counter-intuitive ring to the statement
" 'every present king of France is bald' is true."
One wants to say this shouldn't be considered as
"true" (with some definition of truth) since there
is no present king of France.  Maybe "meaningless"
or "not having a value of either true or false" --
but not "true" (or not "true" in any possible world,
perhaps).

Gordon Fisher     gfisher at shentel.net