Everything follows from a contradiction, including, of course, a contradiction. However, consider the following scenario. Bill has 50,000 beliefs, only two of which are contradictory (i.e., he has no more than one belief which can’t be squared away with the rest of his belief set). Bart has 50,000 beliefs, and has up to 400 contradictory beliefs. Now, since everything follows from a contradiction, won’t it turn out that after Bart has contradicted himself once, all the rest of his beliefs follow from what he already believes? If this is the case, then aren’t Bart and Bill equally consistent? That can’t be right, but what’s wrong with it? We all, I suspect, intuitively want to say that Bill is clearly the one with the higher level of consistency (or whatever), but since contradictions are consistent with contradictions (given that they follow from each other), it seems not to be so.
Perhaps the solution is to introduce a distinction between consistency and coherence. Somebody either is or is not consistent (that is, their belief set either is or is not consistent), but there is no way to measure consistency on a scale. However, coherence can be measured. Suppose that Suzy has a consistent belief set, and that Sally does as well. Perhaps Sally’s belief set is more coherent than Suzy’s. For instance, perhaps Suzy believes a conjunction P&Q, but the probability of P on Q is 0.6, whereas Sally believes the conjunction N&Q, where the probability of N on Q is 0.9. Thus, because P(P|Q)<P(N|Q), the conjunction N&Q has greater coherence than P&Q. Perhaps, in this way, we can say that Sally’s belief set is more coherent than Suzy’s even if both of them are equally consistent, and similarly that Bill’s belief set is more coherent than Bart’s by reason of the same. The trouble here is that it implies that we can measure the relative coherence of inconsistent belief sets, and this seems a strange conclusion to draw. Maybe we just have to bite the bullet somewhere, either by saying that neither Bart nor Bill are in a worse situation than the other (both being equi-inconsistent), or by saying that two inconsistent belief sets are not necessarily equally (in)coherent.
Another problem is that, suppose we add another imaginary person, Alex, to the mix. Alex has one belief, which is not self-referentially incoherent (or, in case it isn’t possible to have only one belief, given holistic-heuristic considerations, then has as few beliefs as it is possible to have, none of which are inconsistent with any other). Wouldn’t Alex have a maximally coherent set of beliefs? The probability of P on P is 1. But doesn’t it stand to reason that Bill may have a preferable belief set, all things being equal, to Alex’s? Imagine that, save for at least one inconsistent belief, all of Bill’s beliefs are true, and none of Alex’s are true (or as few of Alex’s are true as is possible). Suppose Bill had 500,000 beliefs instead of 50,000; doesn’t it seem odd that his belief set would be less coherent than Alex’s?
[Edit: Here’s an objection somebody may give: what is the probability of a contradiction on a contradiction? If the probability is 1, then it seems that Bart will be more ‘coherent’ than Bill. If the probability is 0, how can we say that contradictions logically entail contradictions?]