How so? I've heard this treatment and read Sproul's views of the so-called "mischaracterization," but...
From John Calvin, Institutes of the Christian Religion, translated by John Allen.
In conformity, therefore, to the clear doctrine of the Scripture, we assert, that by an eternal and immutable counsel, God has once for all determined, both whom he would admit to salvation, and whom he would condemn to destruction. We affirm that this counsel, as far as concerns the elect, is founded on his gratuitous mercy, totally irrespective of human merit; but that to those whom he devotes to condemnation, the gate of life is closed by a just and irreprehensible, but incomprehensible, judgment. In the elect, we consider calling as an evidence of election, and justification as another token of its manifestation, till they arrive in glory, which constitutes its completion. As God seals his elect by vocation and justification, so by excluding the reprobate from the knowledge of his name and the sanctification of his Spirit, he affords an indication of the judgement that awaits them.
Though not subject to simple cut and paste, his view is even more clearly spelled out here:
Even before Calvin, Gottschalk clearly taught this from Augustine onward:
GOTTSCHALK OF ORBAIS
SHORTER CONFESSION
© 2003 Victor Genke, translation
(PL 121, Col. 347–350)
I believe and confess that omnipotent and unchangeable God foreknew and predestined saint angels and elect men to eternal life gratis and that He equally predestined devil, head of all demons, with all of his apostates, and also reprobate men, namely his members, on account of their own most certainly foreknown evil merits, through the most right judgment to deserved eternal death; for thus says the Lord himself in His Gospel: “The prince of this world is already judged” (1). What blessed Augustine explained for people is therefore publicly expressed so: “That is irrevocably destined to eternal fire” (2). Likewise the Truth itself about the reprobate: “Whoever does not believe is judged already,” (3) that is to say is already damned. He (4) says: “The judgment has not yet appeared, but already it has taken place” (5). Likewise, commenting on these words of John the Baptist: “His testimony is not accepted by anybody,” (6) he says thus: “By anybody that is by a certain people, which is prepared for the wrath of God, to be damned with the devil” (7). Likewise about Jews: “They were angry, those dead and predestined to eternal death” (8). Likewise: “Why did the Lord say to Jews: ‘You do not believe, because you are no sheep of mine’ (9), unless He saw them predestined to everlasting destruction, not won to eternal life by the price of His own blood?” (10) Likewise commenting on these words of the Lord: “The sheep that belong to Me listen to My voice; I know them and they follow Me. I give them eternal life; they will never be lost and no one will ever steal them from My hand. The Father, for what He has given Me, is greater than anyone, and no one can steal anything from the Father’s hand” (11), he said so: “What can the wolf do? What can the thief and the robber? They destroy none but those predestined to destruction” (12). Likewise speaking about two worlds: “The whole world is the Church, and yet the whole world hates the Church. The world therefore hates the world, the hostile that which is reconciled, the condemned that which is saved, the polluted that which is cleansed” (13). Likewise: “For there is a world about which the apostle says: ‘That we might not be condemned with this world’ (14) For that world the Lord does not pray, for He does not ignore to what it is predestined” (15). Likewise: “Judas, the betrayer of Christ, was called the son of perdition, predestined to perdition” (16). Likewise in Enchiridion: (17) “To the damnation of those whom He justly predestined to punishment.” Likewise in his book On the Perfection of Human Righteousness (18) he says: “This good, which is required, nobody has done, not a single person. But so in that race of men, which is predestined for perdition. For God’s foreknowledge took notice of them and pronounced the sentence on them.” Likewise in his book On the City of God: (19) “Who has given such things even to those who He has predestined to death.” Likewise blessed pope Gregory: “This Leviathan with all of his members is deputed to eternal tortures” (20). Likewise Saint Fulgentius in the third book On the Truth of Predestination and Grace (21) says: “God certainly prepared punishment for those sinners whom He justly predestined to suffer torments.”
Blessed Fulgentius composed a whole book concerning this particular question, that is on the predestination of the reprobate to destruction, for his friend named Monimus.
On which basis also Saint Isidore says: “Predestination is twofold: either of the elect to rest or of the reprobate to death” (22). Therefore so I (23) believe and confess everything with these elect of God and catholic men, inasmuch as I am divinely inspired, animated, equipped. Amen.
A false witness indeed is he who corrupts anything either in meaning or form of that which has been stated.
[footnotes are mostly to Augustine] -- see
Though we could argue as to the "causative agent" of their reprobation, I think it's pretty clear that at least Augustine and Calvin had the common understanding of double predestination. After Calvin, the --lapsarians diverged a bit, but in some sense don't all mainstream Calvinists believe that God "made one vessel for dishonor" in a salvific sense with respect to certain individuals?
Created for the purpose of being reprobate and exercising His judgment on them is what we're talking about (I think) regardless of whether God used their known behavior to reprobate themselves or whether He actively caused them to refuse to believe.