What are the logical parts?

This question doesn't really make any sense. It's like saying, "What are the logical parts that are necessary to make a building useful?". Can you re-phrase it maybe?

I guess what I'm trying to say is I need help explaining  the logical parts that are necessary to make an iterative control structure useful.

To add to evilrix... For iteration you need not only a predicate but also something that enables it to be finite -- some functionality that leads to the next step. It can be increment or decrement of an integer variable, moving the iterator, whatever.

I personally consider the iteration control structure useful when it nicely reflect our mental picture of solving the problem. I mean, not our mental picture how the procesor should do it. The matematical approach is a good example: "for each element of the..." set, for example.

From that point of view I like the Python generalisation of the for loop:

    for element in "iterable":
        process the element

Then whatever is iterable, its processing can be prescribed via that simple syntax.

To summarize, the "necessary" or "useful" can be classified from low-level point of view or from high-level abstractions point of view.

Did you read the tutorial I posted a few times already in your other questions ?

        http://www.cplusplus.com/doc/tutorial/control.html

Yes

>> Yes

Then you should have already known this ;) That tutorial explains all parts of all loop types in detail :) You might want to refer to it when you have further problems ...