In the case of false beliefs due to ignorance and not irrationality "that clock is working" is better than "that broken clock is working" because that clock is working can be either true or false, while "that broken clock is working" is contradictory so always false i.e. it can not adequately express a case of ignorance. — neomac
Sure it can adequately express a case of ignorance. I mean, it is impossible to knowingly believe that broken clock is working(treat this as if we are pointing at the clock). So, when one believes that broken clock is working, it is most certainly a case of ignorance. Jack's is precisely such a case.
While it is impossible to knowingly believe a falsehood, it is not impossible to believe one. Jack's belief is false. We know this. It cannot be either true or false. Jack's belief can only be false. Therefore, if we say that he believes a falsehood(as compared/contrasted to just being mistaken), then our report of Jack's belief ought produce a candidate that is impossible to knowingly believe. Falsehoods are language constructs on par with false assertions. So...
The candidate "that clock is working" is not impossible to knowingly believe. Jack's belief is false, and all false belief is impossible to knowingly believe. Therefore, "that clock is working" cannot be what Jack believes.
If we say that Jack holds false or mistaken belief, then our report ought produce a candidate that is impossible to knowingly believe.
If, at time t1, we're pointing at the clock and saying Jack believes that clock is working, we could be saying that the content of his belief is not equivalent to a proposition because clocks are not, and his belief is about that particular clock. All of this would be true for he does indeed believe that particular clock to be working. So, our report would seem fine. However, if we're aiming at the most accurate or precise description of the content of Jack's belief, we cannot leave out the fact that that particular clock, which has become meaningful to Jack as a result of his belief formation, is a broken one.
It is as a result of Jack's belief not being propositional in content, that our saying that he believes that that broken clock is working does not attribute a contradictory belief to Jack, because it is only the words that are contradictory. Jack's belief does not consist of those words. Jack's belief that that broken clock is working consists of correlations drawn between the broken clock and his wondering what time it was. Jack's belief that that broken clock is working does not have propositional content. It's not belief about the words I just used to report upon it. It's not belief about assertions, sentences, utterances, or any other language use. It's belief about a broken clock. Broken clocks are not propositions. Thus, Jack's belief that that particular broken clock is working does not have propositional content.
Jack's belief has that broken clock as it's content, in very large part at least. The broken clock is not all there is to Jack's belief(for it is a complex one indeed with other necessary prerequisites), but without that particular broken clock to look at, Jack's belief would not have been formed at all.
Jack's belief would pose no problem at all if it were not broken. Perhaps that's why there is such a fervor to leave that much out.