[FOM] A question about the history of a notation

[Vladimir N.Krupski]

> I would also like to get a sense, also, of the terminological preferences
> of members of this list. Which do you prefer for the uniqueness
> quantifier:
>
>       E! x F(x) ;
> or
>       E_1 x F(x) ?

I feel the following difference:

1) (E! x F(x) ) means that x s.t. F(X) does exist and is well formed, i.e.

      (E x) F(x)) and (A x)(A y) (x=y -> (F(x) <-> F(y) ) ).

2) (E_1 x) F(x)  -- "there exists a unique x s.t. F(x) "

This may be not the same because of the formal treatment of "=".
For example, in the FO language it is impossible to axiomatize
"=" as the real equality (a=a and (not a=b) when a, b are different).
So E_1 may be the semantical uniqueness quantifier and E! -- its
formal "approximation".

As for the syntax, the Russell's notation
>       E! iota x F(x)
or, more uniform,
   <opname> { bind{ x , F }
             }
is much more adequate then the usual form  of quantification.
Here "bind" is a special ground opname representing nothing but
the placeholder construction. (May be the Russell's "iota" is not
just the same as "bind" but it is very near to it.)
With the ordinary  notation for quantifiers we really
have many quantifiers (E x), (E y), ... instead of the unique (E).



Vladimir N. Krupski
Associate Professor
Lomonosov Moscow State University, Russia

email: krupski at lpcs.math.msu.ru
http://lpcs.math.msu.ru/~krupski/