1. The Scriptures clearly teach that God "desires all to be saved and come to the knowledge of the truth" (1 Tim. 2:10) and God wills the salvation of only some of humankind, actively damning the rest (all, of course, to the praise of His glory -- no texts were used to argue this position, but I'm sure Romans 9 and Ephesians 1 were in the background).
2. These seem contradictory, for how can God will two incompatible things?
3. The reason they seem in contradiction is because of the fallen nature of human rationality, the "noetic effects of sin."
Here's the problem (or at least one of them) in this syllogism: if the noetic effects of sin are so profound, as the Reformed often argue, then the doctrine of perspicuity, that Scripture is clear and understandable in regards to salvation, is moot. Under the corruptive effects of sin and death, there is no accessible reality that we can call the "clear teaching of the Bible." We are fallen and so must necessarily always read the Bible through those lenses. No amount of historical-critical or redemptive-historical or grammatical-historical interpretation can reveal the "clear" meaning of the Scriptures, as these are rational, and therefore necessarily fallen, methods of inquiry. Since we are always interpreting texts (we never can access them in any so-called objective manner, as argued so cogently by Reformed philosopher James K. A. Smith in his The Fall of Interpretation), we can never come to the "clear" or "pure" meaning of the texts. (Add to this the problem of the non-existent inspired autographa and you've got a massive interpretive dilemma.)
Maybe, though, I'm overstating things. It certainly sounds like I'm saying that there is no hope for us to understand (and therefore live by) the Scriptures (I'm not, but explanation will have to wait awhile). Shouldn't we consider the clarifying work of generations of scholars to function sort of like Zeno's Paradox? That is, while we will never overcome the noetic effects of sin, our rationality still does function somewhat according to its creational design, so we can get approximately close to the intended meaning? Maybe, but the history of interpretation will destroy any lasting confidence in such a move: Unitarians, cultists, heretics, and so on cleave to a very similar principle. It depends on an untenable belief in progress: the more we study the text, the closer we must come. If that is the case, then there is no need to go back to the exegesis, say, of Calvin, as we have progressed from him. In order to save ourselves, we cut off the branch on which we reside.
All of this to say that the proposed syllogism collapses. The Bible may have a teaching on the relationship between God's desire for the salvation of all and predestination (I think it does, but it is radically different than the Reformed tradition has led us to believe, about which I hope to write more soon -- it has become a summer book project for me), but it is anything but "clear." The final premise neutralizes the first.
Is there no hope? Indeed, there is, but it requires a radically different approach to the Scriptures.
The author of the Scriptures is, at the ultimate level, the Holy Spirit. "Who spake by the prophets" as the Creed puts it. Or as St Peter describes, "For prophecy never had its origin in the will of man, but men spoke from God as they were carried along by the Holy Spirit" (2 Pt. 1:21). Modern interpretive methods rely on the human element of the text: the historical context, the literary genre, the cultural background, etc. These are good things, but they partake in the fallenness of the world (an important point brought up by Pete Enns, although I think he goes too far). These methods are essentially apophatic: they tell us what the Scriptures cannot mean, but not what they do mean. It is only as the fallen creational condition of the Scriptures is purified by the Holy Spirit (only thinks here of the role of the Spirit in the conception of the human nature of Jesus) that progress to the meaning and application of the Scriptures can be made. There is no correct interpretation of the Bible apart from the Holy Spirit.
If we are to interpret the Scriptures aright, therefore, we need to acquire (or better, be acquired by) the Spirit of God and Christ. Where does the Spirit reside? The Church. The problems with Sola Scriptura, many of which are already under debate in Reformed circles, always point us back to the "pillar and ground of the truth" (1 Tim. 3:15), of which Christ promised the Spirit would lead us (John 16:13). Of course, this thrusts us into more complicated (and more important) debates as to how we know the true Church. The Fathers, for their part, always argue for the necessity of holiness, that is, living in the Spirit, for proper interpretation and application of the Scriptures: one must engage in the life of the Church, the eternal Life of Jesus Christ, to be a theological authority.