In 'The Myth of Instrumental Rationality', pp.20-1, Raz addresses the grounds of our lingering hostility to contradictions:
When we learn that there is a contradiction among our beliefs we learn (1) that some of our beliefs are false, and (2) that we hold some beliefs that if used together as premises in an argument may lead us astray in a special way [i.e. logical 'explosion']. Big deal! We hope that we all know that some of our beliefs are false anyway. And the risk that we will actually be led astray not by the logical implications of our false beliefs, but by their contradictory features, is, for all practical purposes, negligible...
To conclude: There is nothing wrong with holding contradictory beliefs as such, and the fact that one does is no reason to change one's beliefs. At most we could say that we should abandon our false beliefs. But that is so not because of the contradiction. Knowing that a set of propositions is contradictory has epistemic relevance: It tells us that the contradictory set contains a falsehood. It may be part of a case for believing that one particular proposition is false. But it is no such case by itself. Without such a case we have no reason to abandon any of them. For all we know, we may then abandon a true belief and remain with false ones. Nor do we have reason to suspend belief in all the propositions in the contradictory set. The cost, epistemic and otherwise, of doing so may be too great. That is why the logical paradoxes are rightly not generally taken as a reason to suspend our acceptance of the principles that generate them.
What do you think?