# FOM: Re: Arbitrary Objects

Kanovei kanovei at wmwap1.math.uni-wuppertal.de
Wed Feb 6 11:18:46 EST 2002

Clearly the Arbitrary Objects discussion has deviated
into very interesting domains of Latin morphology, "Dynamic Predicate
Logic", formal proof theory (to mention some of them).
Still I would suggest to stick to the original
(in fact challenging) question:

QUESTION: the expression
"let x be an arbitrary element of (a given collection) X",
does it have any semantical meaning in the form of a
choice of some object x (be it really an element of X or
something more complicated, but the requirement, to avoid
trivialities, is that the choice is made ONCE).

The answer that the expression is illegal because it
is a left bracket which has to be matched by a right
bracket to form a semantically meaningful statement,
somehow negates the problem.

The suggestions to replace "arbitrary" to "any" and the
like, perhaps, interesting with grammatical point of view,
solve nothing, as does the suggestion to reformulate the
sentence as follows: "reset register x to an element of X".

I proposed a plausible semantical solution of the question,
a few days ago, in terms of a discussion between two
opponents (after all, any science develops as a discussion
between opponents), which, to repeat, is the following:

A: I claim that any x\in X satisfies F
B: Prove!
A: give me any x\in X

Now, if B quits then A wins by default.
If B gives an element, then if A demonstrates F then A wins
and if A fails then B wins.

Note: semantically, there is ONLY ONE choice of an arbitrary
x here.

V.Kanovei