Logical consistency is important. If my positions fail to be consistent, this is clear proof that at least one of them is mistaken.
Nonetheless, logical consistency is not the supreme value. It would not be the supreme value even for someone who cared about truth alone. The reason for this is that someone may see that his position is inconsistent, and know that it follows that his position is partially mistaken, without knowing any concrete way to modify his position which will improve it. His best estimate of any particular modification may be that his position would end up containing a greater amount of error. In this case, caring about the truth will entail that he preserve his position without modification, despite recognizing that his position contains some falsity.
This happens to all of us, as we can see from one particular case. I would be foolish and arrogant if I believed, “All of my beliefs are true.” In order for that to be the case, I would have to be essentially infallible. So it is far more reasonable for me to believe that I am mistaken about some things. But then if we take all of my beliefs together, including the belief that “some of my beliefs are mistaken,” we can derive a contradiction.
And it is clear, in the general case, that I have no way to improve my position to eliminate the inconsistency. Believing that all of my beliefs are true would not improve it. It would add some consistency, but only by means of adding a definite falsity. Likewise, changing some other belief would not help, since even if some of my beliefs are mistaken, I do not know which ones those are. Changing a belief at random is likely to worsen my position overall. And if I examine my beliefs and come to the conclusion that I should modify one belief or another, this will not and should not change my conviction that some of my beliefs are mistaken.