Edit, 9/8/2024:
Since I posted this 12 years ago, my thought on this matter has changed somewhat. Today I view the God's sovereignty/human volition conumdrum as quite irresolvable, both true, but a paradox that we simply have to live with. See also:
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The answer to that question depends upon whom you ask. There are Calvinistic Anglicans who, pointing to a term commonly used to describe Anglicanism, "Reformed Catholicism", place the emphasis on "Reformed" and will correctly note the role the Swiss Reformation played in the thinking of Anglicanism's original divines, such as Thomas Cranmer. These Anglicans tend to be found in "low church" denominations such as the Reformed Episcopal Church and in the Evangelical wing of the Church of England. Anglo-Catholics on the other hand play down the Reformation, play up the late Caroline, Oxford and Ritualist Movements and stress the Anglican tradition's medieval Catholic roots. The "Old High Churchman" curses both camps and celebrates high Catholic ritual and Catholic church order while at the same time owning the Protestant roots of the Church of England and her daughters throughout the world. Old High Churchmen tend to be Arminian (though not in the exact way the Remonstrants were Arminian), but there have been Calvinists among them, such as Archbishop of Canterbury John Whitgift.
Certain Anglican writers have mentioned the "Pelagian" tendency of the British people (Pelagius himself was from the British Isles), and it shows up in Anglicanism, rather paradoxically, in both the conservative High Church/Anglo-Catholic types on the one hand and liberal Anglican Protestants on the other. Anglican theologian C.B. Moss notes, for instance:
(Pelagianism) is very attractive to the ordinary man of independent will and common-sense religion and morals; and particularly to the Englishman. Probably 90 per cent of the English laity (that is, practising members of Christian congregations) are unconscious Pelagians. (The Christian Faith: An Introduction to Dogmatic Theology, p.155)
Theologian W. Taylor Stevenson agrees:
While it may be of very limited historical import, it is certainly appropriate that the only heresy associated traditionally with Britain is that of the British (or Irish) monk Pelagius (active 410-418) who argued that the individual, apart from divine grace, makes the initial and fundamental steps toward salvation. That is a practical, a 'sensible' idea. It is more significant historically, and quite in keeping with the English ethos, that English piety and theology has had a an earnest Pelagian flavour extending from the Puritanism of the seventeenth century through the Evangelical movements of the eighteenth and nineteenth centuries. ("Lex Orandi - Lex Credendi", The Study of Anglicanism, p. 192)
One often hears the argument from Old High Churchmen and Anglo-Catholics that since "God gave us free will" and desires all men to be saved, Christians become Christians by an act of their will as they respond positively to the Gospel, and likewise persevere as Christians by an exercise of holy volition (or not persevere, as the case may be, by an exercise of unholy volition). C.S. Lewis can be cited as one notable individual who held to this view.
The rub, of course, is that to the extent that such a belief is Pelagian, it is heretical. Pelagianism was condemned not just at the local Council of Carthage in 418, but at the ecumenical Council of Ephesus in 431 as well. Anti-predestinarianism did not suddenly evaporate, however, and a view later arose that avoided certain fundamental teachings of Pelagius while maintaining the free will doctrine. This view is called Semipelagianism, and while it was condemned at the Second Council of Orange in 529, it was never condemned at an ecumenical council, as Pelagianism was. Moreover, anti-Augustinianism (of which Semipelagianism is one variety) remained prevalent in the East and in some places in the West, so the free will doctrine survived intact as a fundamental Christian theologoumenon in much of the church.
But even the anti-predestinarian Old High Churchman Moss sees a continuing danger here. Quoting N.P. Williams, he writes (parenthetical comments mine):
Pelagianism is "fundamentally irreligious, so far as it tends to destroy in the heart of man the feeling of childlike dependence on his Maker" (presumably a reference to man's salvation). We have not got the unlimited power of free will asserted by Pelagius; man is weaker and more vicious than that sheltered monk knew (to which the Augustinian and the Calvinist say, "Amen"). The discoveries of Freud (and others), even though we accept them with great qualifications, at least show that there are vast depths of evil in the subconscious mind of man, of which he is usually quite unaware (to which the Augustinian and Calvinist again say, "Amen"). (The Christian Faith, p.155)
Does it follow from any of this, however, that the Calvinist system must be accepted? Only Calvinistic Anglicans would say that it does. Most traditional Anglicans would not say so, however. The only things they must say are that 1) Pelagianism is heretical and 2) the notion that the human will is situated neutrally between good and evil and therefore is always perfectly and arbitrarily free to choose one or the other is both empirically suspect and lacks clear scriptural support.
Moss affirms, correctly in my view, that "antinomy" marks much of what is affirmed theologically in the Bible. An antinomy is an apparent paradox, "two sides, which cannot be fully reconciled by reason." (The Christian Faith, p. 49.) While I personally lean Augustinian as do many Anglicans, I do so because I think Augustine's theology more accurately reflects the soteriology of the New Testament, and especially Pauline soteriology. In a nutshell, that soteriology is this: we do not become Christians, or stay Christians, due to our own power. God must graciously make us Christians, which is to say that faith is a gift, not something we exercise naturally. Our coming to faith in Christ is quite supernatural, actually. Some representative texts from St. Paul (bolded emphases mine):
And you hath he quickened, who were dead in trespasses and sins; herein in time past ye walked according to the course of this world, according to the prince of the power of the air, the spirit that now worketh in the children of disobedience: Among whom also we all had our conversation in times past in the lusts of our flesh, fulfilling the desires of the flesh and of the mind; and were by nature the children of wrath, even as others. But God, who is rich in mercy, for his great love wherewith he loved us, even when we were dead in sins, hath quickened us together with Christ, (by grace ye are saved;) and hath raised us up together, and made us sit together in heavenly places in Christ Jesus: That in the ages to come he might shew the exceeding riches of his grace in his kindness toward us through Christ Jesus. For by grace are ye saved through faith; and that not of yourselves: it is the gift of God: Not of works, lest any man should boast. For we are his workmanship, created in Christ Jesus unto good works, which God hath before ordained that we should walk in them. (Eph. 2:1-10, KJV)
Wherefore, my beloved, as ye have always obeyed, not as in my presence only, but now much more in my absence, work out your own salvation with fear and trembling. For it is God which worketh in you both to will and to do of his good pleasure. (Philippians 2:12-13, KJV)
And St. Augustine's favorite text:
. . . and what hast thou that thou didst not receive? now if thou didst receive it, why dost thou glory, as if thou hadst not received it? (I Cor. 4:7, KJV)
However, I am unwilling to draw certain of the same conclusions Augustine and some of his followers (especially the Calvinists) did. It is quite clear from the New Testament, against the Calvinists and against certain things that even Augustine seemed to say concerning the limited scope of the atonement, that Christ died for all men and that God therefore wills the salvation of all men. But that does not commit me to a Pelagian, Semipelagian or Semipelagian-like conclusion that God has given all men free will (defined as an uncaused cause of human behavior) and looks nervously down from heaven to see who among these "all" will take Him up on his offer and who will not. There very well may be other theological options.
However, it's a difficult theological area to be sure. Ignoring the phenomenon of antinomy in the Bible and accordingly trying to boil down rationally the predestinarian or the voluntarist theology tends to land us in trouble. But that doesn't mean we don't have a clear choice to make. Augustine scholar Gerald Bonner summarizes the argument of the Anglo-Catholic theologian J.B. Mozley (again, bolded emphasis mine):
In a study of Augustinian predestination first published in 1855, J.B Mozley, brother-in-law of John Henry Newman and later Canon of Christ Church, Oxford, and Regius Professor of Divinity, theologically orthodox but fair-minded and aware of the limitations of the human intellect, noted the ideas of Divine Power and human free will, while sufficiently clear for the purposes of practical religion, are, in this world, truths from which we cannot derive definite and absolute systems. "All that we build upon either of them must partake of the imperfect nature of the premise which supports it, and be held under a reserve of consistency with a counter conclusion from the opposite truth." The Pelagian and Augustinian systems both arise upon partial and exclusive bases. Mozley held that while both systems were at fault, the Augustinian offends in carrying certain religious ideas to an excess, whereas the Pelagian offends against the first principles of religion: "Pelagianism . . . offends against the first principles of piety, and opposes the great religious instincts and ideas of mankind. It. . . tampers with the sense of sin. . . . (Augustine's) doctrine of the Fall, the doctrine of Grace, and the doctrine of the Atonement are grounded in the instincts of mankind." (Freedom and Necessity: St. Augustine's Teaching on Divine Power and Human Freedom)
In other words, to paraphrase Mozley, it is better to err in the direction of St. Augustine than Pelagius. Augustine is a saint and Doctor of the Church, and widely regarded to be the greatest of all the Church Fathers; Pelagius is a heretic and the Semipelagians have been more or less judged by the Church to be the bearers of errant doctrine. Augustine points us to the hope of grace, Pelagius to the hopelessness of free will and works salvation. Augustine, following St. Paul, taught that the will was not free without grace, and even then needed the constant assistance of God's grace to choose daily to live the Christian life. Salvation is not by works, but grace. If anything is clear from the New Testament, it is that. But we must be careful about what conclusions we draw from all this. Most Anglicans are thus careful, and so when they are asked whether traditional Anglicanism is Calvinist or Arminian, they will answer with a deliberate and clear "yes."
That's our story, and we're sticking to it.
Ergo: the potential Evangelical convert to traditional Anglicanism who happens to be on the Reformed side of things should not be put off by Anglican anti-predestiniarianism. The Augustinian (and in some churches, the Calvinist) doctrines of grace are held by a goodly number of Anglicans. Whether the anti-predestinarian Anglicans want to admit it or not, Augustinian soteriology is an acceptable theologoumenon in the Continuing Anglican Church, just as it is in the Roman Catholic Church. And the Arminian Evangelical who joins Anglicanism will be welcomed with many open arms, including those of old St. Clive himself.