Sorry for the lack of posts and for being bad about approving comments. I've been very busy this quarter as I'm taking three math classes, and this consumes most of my time. Here is an explanation (albeit somewhat oversimplified) of Suarez's idea of the 'modal distinction'. Suarez is an excellent, extremely clear late medieval philosopher. I wish more of his works were in English and I hope he will be canonized some day. The reference for Suarez's theory of distinctions is his Metaphysical Disputation VII.
Suarez makes three types of metaphysical distinctions. There are two which most philosophers of his era admit to exist: the real distinction and the distinction of reason. A real distinction holds between A and B just in case A can exist without B and B can exist without A. A prime example of a real distinction is between two substances, like me and my chair. A distinction of reason on the other hand holds between A and B just in case A and B are really the same and thus mutually inseparable, yet we conceive A and B using distinct and incomplete concepts. So, one of the primary examples Suarez appeals to is the divine attributes. Given the theory of divine simplicity (the idea that God has no proper parts at all) it seems to follow that God's justice is really the same as God's mercy. They are both mutually inseparable in the sense that neither can exist without the other. However, we use different and necessarily incomplete concepts to think about each of them. This explains why it is not obviously true that God's mercy is identical to God's justice, even though both phrases and their corresponding concepts refer to the same being.
Aside from these two Suarez admits a third distinction, called a modal distinction. The motivation for this distinction comes in large part from problems about the relationship between an accident and its inherence in a substance. The problem is raised by Bl. John Duns Scotus, taking as his starting point the case of the Eucharist (though his argument doesn't really depend on this example in particular). We know in the case of the Eucharist that the accidents of the bread such as the quantity remain even though the substance of the bread does not. Hence, the accidents no longer inhere in the substance. So suppose that the quantity of the bread were not really distinct from its relation of inhering in the substance of the bread. Then since the quantity of the bread exists after transubstantiation, so would its inhering in the substance of the bread. But clearly the quantity's inhering in the bread cannot exist unless it inheres in the bread! This is a contradiction, since in the case of the Eucharist the accidents no longer inhere in the bread. Hence, it seems to follow there is a real distinction between a quantity and its relation of inhering (similar arguments can be run for other relations of union or composition).
But this leads to problems of its own. Let I be the relation of inhering between an accident a and substance b. Suppose I is really distinct from a as Scotus's argument appears to show. Then since I is an accident of a (it's a feature of a connecting it to b), it follows there must be some relation I* between I and a in virtue of which I inheres in a. But then there must be some relation I** between I* and I in virtue of which I* inheres in I, and so on. This creates an infinite regress, which is problematic to say the least. So it seems whichever way we go we have an issue. If the relation of inhering is really distinct from the accident, we have an infinite regress. If it is not really distinct from the accident, we have a contradiction.
While nominalists like Ockham have their own solution to this quandary, Suarez is a metaphysical realist, and thus unlike the nominalists he thinks as a matter of semantic principle that there must correspond to the term 'inheres' some extramental being. It is with this concern in mind that Suarez develops his notion of a modal distinction. A modal distinction is a distinction which obtains between a being and its mode. Suarez is using the word 'being' here in a strong sense, to denote a real, particular individual, in scholastic terminology a 'res' (though this need not be a substance). A mode on the other hand is just a way in which a being or 'res' can exist. So Suarez wants to say that instead of letting an accident's inhering be a really distinct relation, we let it simply be a mode or way of being of the accident. Suarez considers this modal distinction to be a distinction intermediate between a real distinction and a distinction of reason, because the accident can exist separately from its mode (at least by the power of God), but the mode cannot exist separately from the accident. Since this is the only intermediate distinction Suarez admits, he takes A and B to be modally distinct just in case A can exist without B but B cannot exist without A, or vice versa (but not both).
Clearly then Scotus's problem was with his first argument; Scotus assumed that if the quantity of the bread and its inhering are not really distinct then it follows they are both mutually inseparable. Suarez on the other hand allows that it is possible for a being to be separable from its mode even though they are not really distinct; after all, a being's mode is not in itself a 'being' in the strong sense, so it is not some separate 'being', and thus it is not really distinct. With his modal distinction laid out Suarez has the tools to give a very nice solution to the problem of the Eucharist. The accidents of the bread and wine can continue to exist even without the substance because it is not part of their essence that they inhere in something; inhering is only one of their modes. While it is obviously not a natural occurrence that an accident exist without its mode of inhering, it is not metaphysically impossible, and thus God can bring it about by his power.
While Suarez's account seems secure there is one objection. Someone like St. Thomas Aquinas might object that the essence of an accident includes mention of the substance of which it is a part. However, at least in the case of the Eucharist, even Aquinas makes exceptions, and thus this account must obviously be modified. And Suarez can easily modify it by saying that accidents are beings whose definition include that it is metaphysically possible that they inhere in a subject. Substances by contrast are beings for which it is metaphysically impossible that they possess the mode of inhering. So it seems Suarez's modal distinction provides a coherent solution to all the puzzles set out above.