Hello Hal,
---The basis of Leibniz's charge is twofold; miracles need to remain special and sacred and predictions about the world have to be possible.--- This is okay - you use the semi-colon to seperate two independent clauses, but it's actually a great time to use the colon, because what follows is an elaboration or explanation of what comes before.
The portion after the semicolon contains two sentences combined with a conjuction. I know this is usually wrong, No, its' not. That's exactly what you use a conjunction for.
Any feed back on semicolons would be appreciated. At least one of the teachers here will tell you to use them as sparingly as possible if at all. I find it a useful mark to join what would otherwise be two sentences that are very closely related.
---These are: Descartes' interactionism, Malebranche's occasionalism, and Leibniz's pre-established harmony.--- Don't use a colon here. What follows simply completes the sentence. In running text, use a colon after a complete thought and before the thing that describes, elaborates, or gives examples. There are three theories to support this: Descarte's...
---The two events compliment each other through their relation in time, and this leads to correlation. This is what we mean by harmony: inter-substantially and intra-substantially. This is a tough case because I have no idea what it's try to say, but it seems to be a good use of the colon - those two elements that follow are what is meant by harmony?