categories: Subobject classifier and non-classical logic (intuistionistic)

[Galchin Vasili <vngalchin@yahoo.com>]

Hello Cat Community,

   I have been reading Goldblatt's book on topoi and
also a paper by Peter Johnstone on subobject
classifiers. Please "school" me .... In both places it
says that in general a subobject classifier's
"elements" are used as logic "values'. E.g. in the
topos Set, I can see that the s.c. (subobject
classifier) {0, 1} is a set of logic values that
defines a Boolean algebra. In the topos of Graph 
we also can come up with a set of logical values.
These examples of s.c. all clearly are sets or
structured sets .. hence have elements. I don't see
how in a totally general case of a topos we can say
that "elements" of it's s.c. define a set of logic
values .... after all the guiding principle of
category theory is that objects are opaque.

Thanks, Bill Halchin