Do we have a duty to resolve contradictions within our own thoughts and opinions? For example, does a person who thinks killing animals is very wrong, but who has no qualms eating meat, need to revise one opinion or the other? What about someone who doesn't really believe in a god, yet insists on worshipping one and arguing for its existence? Or is it our choice to live with contradictions as we choose?

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That's a very interesting question,thanks for asking. There seems to be a difference between your twoexamples that is worth thinking about. The first example clearly anddirectly involves a moral choice. There we have a person who lives acontradiction in that they believe that X is wrong in a specificallymoral sense of 'wrong', and yet are complicit in X. In the secondexample, though, there doesn't appear to be anything moral at stake(there may in fact be, but for the sake of argument here let usassume that there is not). So, we have a person who thinks that X iswrong in the sense of false, but still behaves as if X.

If there is a duty to removecontradictions in our beliefs and behaviours, it seems more urgent inthe first case. The contradiction there involves some moral wrong, orsome failure in the consistency of moral character. Consistency is afeature valued in most moral systems. See this question and answer:http://www.askphilosophers.org/question/715

In the second case, there may beself-delusion, some form of deception, or just plain stupidity, but(provided it doesn't do any harm) is there a moral duty? To take amore banal example: I'm pretty sure that my daughter knows there isno Santa Claus, but also knows that if she plays along, Christmaswill be a bit more fun, more colourful, there'll be stockings, andprobably more presents. This deception seems innocent enough. Only amoral theory that gave intrinsic value to consistency of beliefs (aswell as moral consistency) would posit such a duty ofnon-contradiction. (It has to be intrinsic, i.e. something good initself, since we have already assumed that the contradiction does noharm.)

The question now, is: can we think ofreasons to value such consistency? Although Plato, for example, wouldcount such a weeding out of contradictions in our beliefs as a goodthing, arguably he does so not because it is good in itself, separatefrom other goods, but rather because it is an integral part of livingthe philosophical life. For that reason, such contradictions wouldinvolve harm to the soul. Plato, then, would be arguing that lack ofconsistency in our beliefs is never innocent, but always directly orindirectly does harm (to the person holding the beliefs, if no oneelse). For example, it might be innocent now, but make medispositionally less likely to notice contradictions when they reallycount.

I would like to add a couple of further considerations to Douglas Burnham's response.

I there is too much inconsistency between what a person professes to believe and what that person does, we have reason to doubt that she really believes what she says she does. If a friend claims to believe that killing animals is seriously wrong, yet continues to eat meat with no qualms whatsoever, I will doubt the sincerity of her belief; I will suspect that her professed 'belief' is merely a popular thing to say in certain contexts, or that it is merely an expression of the repulsion that she feels when she imagines certain scenes. Likewise, if a person professes to be an atheist but regularly and earnestly worships god, I will doubt his claim to be an atheist.

If there is too much inconsistency between your beliefs one day and your beliefs the next, or your beliefs in one context and your beliefs in another, people will have a hard time relating to you as a single person and you will have a hard time recognizing yourself across time; in extreme cases, you may be diagnoses as suffering from multiple personality disorder. This is not a moral defect so much as a failure to meet the conditions for being a responsible moral agent since the part of you that believes and acts in one way will not have much control over the part of you that believes and acts in some other way. If we simply accept the contradiction between a person's objection to eating meat today and acceptance of eating meat tomorrow, we cease to hold them responsible for what they do and thus cease to treat them as a moral agent at all.

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