Let E (for 'election') be the proposition which says de re of each person who will in fact be saved that he or she will be saved. That is, E is the longest conjunction of the form 'John will be saved, and Mary will be saved, and Lois will be saved...' which is true. Let R (for 'reprobation') be the proposition which says de re of each person who will in fact be damned that he or she will be damned.
The doctrine of predestination is the doctrine that God, from eternity, has issued an efficacious decree of election - that is, God, from eternity, effectively chose that E should be true. The doctrine of double predestination states that in addition to the decree of election, God also issued a decree of reprobation - that is, in addition to effectively choosing that E should be true, God effectively chose that R should be true.
Double predestination is much more contentious among Christians than predestination (although predestination is not entirely uncontroversial - for instance, open theists will have to deny it). Many Christians would rather have single predestination, holding that all people are, on their own, bound for hell, and God intervenes to save those he wishes to save, and just leaves the rest alone.
In his Philosophical Theology (1969), James F. Ross proposes the following analysis of omnipotence:
S is omnipotent if and only if for every logically contingent state of affairs, p, whether p or ~p is the case is logically equivalent to the effective choice, by S, that p or that ~p (respectively). (p. 211)
This analysis appears to have the consequence that, if God is omnipotent, then double predestination is true. Both E and R are true contingent propositions, so if God is omnipotent then God effectively chooses that the corresponding states of affairs should be the case.
What could someone who accepted Ross's theory say about this consequence? Well, first, of course, one could just accept the consequence. There certainly are theologians who advocate double predestination on independent grounds, so maybe this isn't so bad. A second response would go like this: the doctrine of double predestination is supposed to be stronger than the doctrine of predestination, and this strengthening comes about by means of the 'in addition to' clause. But does it even make sense to speak of God's willing p in addition to q? There are reasons for doubting that it does. Leibniz argues at length in the Theodicy that God has a single simple act of will. Leibniz's argument, as I understand it, is an argument from divine rationality. However, the argument could also be made from immutability, atemporality, or simplicity. Furthermore, the argument could be made directly from Ross's theory of omnipotence: God decrees that E should be true, and that no one not listed in E should be saved, and that such and such collection of people should exist. These jointly entail R and so, together with Ross's theory of omnipotence, entail that God effectively chooses that R. In other words, given Ross's theory of omnipotence, the proponent of double predestination is faced with a dilemma: either the view is not genuinely stronger than mere predestination, or it is incoherent. (This response is parallel to the response Ross gave to worries about the consequences of his view for human freedom in his 1980 article "Creation".) A third option would be to distinguish between two ways in which God might choose things, and put E and R in opposite categories. For instance, one might distinguish between God's 'antecedent' and 'consequent' wills, as Leibniz does, and claim that God wills E antecedently and R consequently. Double predestination could then be defined as the view that God wills both E and R antecedently, and would thus be a genuine strengthening of predestination.
(cross-posted at blog.kennypearce.net)