FOM: Arbitrary Objects

[charles silver]

    I have started puzzling over what an "arbitrary object" is.   Suppose
you want to prove A is a subset of B.   So, you say, let x be an arbitrary
element of A.   Then, you prove x is also an element of B.   End of Proof,
because x was completely arbitrary.  Okay, do you think x was *really* an
object in this proof, or something else?  If it really *was* an object--an
"arbitrary" one--which one was it, and how is that object determined.   If
it really *wasn't* an object at all, what was it?  Was 'x' just a schematic
letter, used to range over all the elements of A?   If so, why do we say:
Let x be arbitrary?   Why not say that the letter 'x' will be used as a
standin for any element of A?   I know everyone reading this "knows" what an
arbitrary object is, but I don't--not anymore. Since reading Kit Fine's
fascinating book on the subject (_Reasoning With Arbitrary Objects_), plus
reading some other articles, I'm now puzzled what arbitrary objects *are*.
    In case one is tempted to think arbitrary objects pertain only to proofs
like the above, there are also proofs starting out with existential
sentences that seem to require arbitrary objects as well.  For example,
suppose you prove something of the form ExFx (something has F), and you want
to reason to a conclusion, say C.   After ExFx, you may say something, like,
"let t have F" and then reason about t, hoping to arrive at C, though,
formally speaking, C should not contain the letter 't' in it (which again,
suggests that we are speaking here about the letter, not the object the
letter stands for [??].   In this case, t is also arbitrary; it's an
arbitrary object having F, where all that's known about F is that it's
non-empty.  (And, actually, F *could* turn out to be empty, because we could
later arrive at a contradiction.)  Well, there's more to this, but I'd like
to know whether any of you have thought about this and come to any
conclusions.   Clearly Kit Fine (and a few others) have thought about it a
great deal, but normally I think it is just taken for granted what an
arbitrary object is, without it being subjected to any detailed scrutiny.

Charlie Silver
University of Memphis