What is the difference between a vicious and a benign infinite regress? We ought to look at a number of examples. Here is one. An entailment of a proposition p is any proposition that is a logical consequence of p. Now consider
1. Every proposition has entailments.
2. To know a proposition one must know its entailments.
(1) gives rise to infinite series. The entailments of a proposition are themselves propositions, so that if every proposition has entailments, then for every proposition there is an infinite series of propositions. For example, p entails ~~P, which entails ~~~~P, and so on ad infinitum. There is nothing problematic here.
(2), however, engenders a vicious infinite regress. For if to know a given proposition I must know its entailments, then to know a given proposition I must know infinitely many propositions. But I cannot know infinitely many propositions. So (2) implies that I cannot know any proposition.
What makes the regress vicious in the second case? What does viciousness consist in? It has to do with (2)'s being explanatory. (2) proposes a philosophical explanation: one knows a proposition by knowing its entailments. (2) proposes a theory as to what knowing a proposition consists in. But the explanation is faulty. Suppose p entails q which entails r which entails s, and so on. The theory proposes that in order to know p, I must know q. But to know q I must know r, and so on. This implies the impossibility of my knowing p. Viciousness, then, is the property of being explanatorily unsuccessful.
Perhaps we can hazard the following general formulation. A vicious infinite regress is an infinite regress that arises in the context of an attempted philosophical explanation when the explanation given permits the question that was to be answered to arise at successively higher levels ad infinitum. In the above example, to know that p one must know p's entailments, but to know them, one must know their entailments, and so on endlessly.
Now consider this pair:
3. Every event has a cause.
4. To explain an event one must explain its causes.
(3) engenders an infinite series: if every event has a cause, and causes are events, then there is an infinite regress of events. But the regress is benign. (4), however, is the answer to a philosophical question about the nature of explanation: What is it to explain an event? (4) proposes a philosophical explanation of explanation, namely, that to explain an event one must explain its causes. But this theory leads to a vicious infinite regress. Suppose z has y as a cause. The theory implies that to explain z one must explain y. But y is an event, so to explain it one must explain its cause x, and so on infinitely. The regress is vicious because it sets an impossible standard of explanation: if to explain an event one must explain every event in its causal ancestry, then no event can be explained. So (4) is false.