Formal Coherence Theory and Contradictory Beliefs

It seems plausible that most of us mere humans have some contradictory beliefs. However, according to most formal measures of coherence, if a set of beliefs contains a contradiction, its degree of coherence is zero. (The two measures discussed in chapter 6 of Olsson (2005) both contain the probability of the conjunction of the propositions in the numerator. I should note that I'm not all that familiar with the literature on coherence, so I may well have missed some proposals that avoid this problem.) Thus the beliefs of an actual human, when taken all together, has zero coherence. Is this a problem for such formal coherence theories? There are a few different ways one might respond.

First, we could say that formal coherence theory is concerned with idealized agents, and such agents don't have contradictory beliefs, so the problem never arises. Of course, if one actually holds a coherence theory of justification, this response will not be appealing, because it would follow that none of our beliefs are justified. One way around this might be to specify an idealization procedure, and claim that the degree of coherence of your actual beliefs is given by the degree of coherence of your idealized beliefs. This response leads us to the next possible solution.

Second, we could limit ourselves to a contradiction-free subset of an agent's beliefs. My main worry about this is that I don't know how to choose a subset in a non-arbitrary way. But perhaps this can be done. (Maybe something like AGM theory could help here?)

Third, we could introduce a new measure of coherence which allows that two distinct sets of beliefs which both contain contradictions may nevertheless have different degrees of coherence. I haven't tried to do this myself, so I've no idea if it could actually work.

Fourth, we could claim that, in general, humans actually don't have contradictory beliefs. This doesn't strike me as very promising, but perhaps one could muster an argument to this effect based on work on the individuation of beliefs.