What is the difference between “vicious circularity” and “virtuous circularity”?

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I cannot say it as eloquently as Van Til or Bahnsen, but I’ll give it a try. Virtuous circularity is simply the acknowledgment that all knowledge is ultimately circular. In other words, all knowledge requires an ultimate standard beyond or higher than which there can be no further appeal. For the Christian, this is God. Vicious circularity is absurd circularity, where circular appeals are made to things not capable of holding the title of ultimate standard. For the humanist, this is human reason.

I think I remember Bahnsen dealing with this distinction in his book Van Til’s Apologetic. I will see if I can find them later this week. (I’m 600 miles from home right now.) Others, I am sure, can chime in in the meantime.
 
The foundations of logic are circular. You have to use logic to prove logic. Vicious circularity is more of a second order issue. It could be something like x is so because x is so.
 
Van Til’s Apologetic, edited by Bahnsen, may well be the most important book I have read. Below are some quotations on circular reasoning I made for myself from it. I made a graphic for myself, too. The presuppositional approach is transcendental because for truth to be objective, it must come from outside ourselves and ultimately outside the universe. Man is limited by his position in the universe. Since he is inside and is subject to the effects of what he may observe, he is a subjective viewer and not objective. The result is that all human reasoning is ultimately circular. We begin and end with ourselves as the viewer.

The only solution to that is the transcendent revelation of God. The revelation transcends the limitations of our minds and our subjectivity. Even so, our thinking remains circular—or rather “spiral.” As Christians we consult the revelation and apply its implications to reality at hand. Nevertheless, we are still limited by our grasp of the revelation and our inability to transcend existence as God can. It is the best we can do.

Our thinking may always be circular, but informed and guided by the transcendent revelation, we can trust what God tells us. Thus despite the inherent circularity of all human reasoning, our trust in God’s revelation is virtuous.

“We hold it to be true that CIRCULAR REASONING is the only reasoning that is possible to FINITE MAN” (p 518).

“The method of implication is circular reasoning. Or we may call it SPIRAL REASONING” (see note 122; p 518).

Note 122: “The ‘circularity’ of a transcendental argument is NOT AT ALL THE SAME as the fallacious ‘circularity’ of AN ARGUMENT IN WHICH THE CONCLUSION IS A RESTATEMENT (in one form or another) OF ONE OF ITS PREMISES [= circular reasoning]. Rather, it is the circularity involved in a coherent theory (where all parts are consistent with or assume each other) and which is required when one reasons about a precondition for reasoning. Because autonomous philosophy does not provide the preconditions for rationality or reasoning, its “circles” are destructive of human thought—i.e., ‘vicious’ and human endeavors. (Because there is more than one kind of ‘circularity,’ Van Til sometimes repudiated and sometimes tolerated the notion that his apologetics was circular—which has undoubtedly been confusing to his students)” (i.e., apparent “circularity” is not at all the same as circular reasoning; n122, p 518).

I also saved more quotations from the book and keep them handy for myself.

Note that Van Til’s and Bahnsen’s explanations here are different from currently popular versions. Here, Van Til explains that circularity is a problem of human limitations. We are part of the reality that we observe. The solution to our being finite is the divine revelation, because it transcends everything, especially circular reasoning. Consulting the revelation may be circular or spiral, but the revelation is not circular reasoning since it comes from outside us.

Currently popular explanations usually skip transcendence to say that all knowledge is circular reasoning, that circular reasoning from God is virtuous, but circular reasoning apart from God is vicious. I think this misses the point because transcendence is not circular at all. Rather our own human reasoning that consults the revelation is circular. The revelation is the solution to circular reasoning because it transcends everything.
 

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