FOM: Re: Arbitrary Objects
Mitchell Spector
spector at seattleu.edu
Wed Jan 30 13:12:02 EST 2002
Charlie,
Here's one way of formulating a notion of "arbitrary object". Use a
many-worlds interpretation, as is sometimes done in quantum mechanics.
Every time you say something like "Let x be an arbitrary object", you're
duplicating the universe of discourse into many (usually infinitely
many) different universes, each one with a different distinguished
constant symbolized by x. (Here, x is a new constant symbol, of course.)
I hesitate to get into this, because I tend to agree, in fact, with
the prevailing sentiment that "arbitrary" isn't an adjective in this
case, that the phrase "arbitrary object" is just a convenient way of
speaking when carrying out certain kinds of logical arguments.
Nevertheless, a many-worlds approach would provide a way of formulating
arbitrary objects as virtual objects, or real objects in virtual
universes, depending how you think of it.
Mitchell Spector
Seattle University
On Friday, January 25, 2002, at 01:39 PM, charles silver wrote:
> 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
>
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