Is it proper to say "Bob wants Joe and I to stop speaking," OR "Bob wants Joe and me to stop speaking"?
I know the answer has to do with a gap in my subject/object knowledge, but please direct me to the rule.
The suggested questions do not address an example like the one above, wherein the object of the sentence is performing an action.
wants in your sentence is a transitive verb and transitive verbs take objects after them. Sometimes they take only one object after them (called Direct Object) and sometimes they can take two objects after them (Direct Object and Indirect Object). In the latter case, they are called ditransitive verbs.
So the bottom line is this— Transitive Verbs take Nouns and Pronouns in the Objective/Accusative Case after them. This role of verbs is also common to prepositions— which also take Nouns and Pronouns in the Objective/Accusative Case after them. Thus, This is a book for Joe and me (not Joe and I).
Another way of looking at it is this— Would you say Bob wants us (that's Joe and the other person) to stop speaking or Bob wants we (that's Joe and the other person) to stop speaking? If us hit you even before having to think about it, you might have guessed that verbs take objective case of pronouns after them. Likewise, it should be me and not I. As far as Joe is concerned, this problem doesn't arise there because, well, Joe is Joe only in both Subjective as well as Objective cases.
