FOM: Re: Arbitrary Objects

[Kanovei]

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