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Is "understanding" a proposition necessary, but not sufficient, for "believing" that same proposition? Further, where could one find arguments (discussion) for and/or against either position?

I confess I'm puzzled by Prof. Heck's reply. He defends the following three assumptions: (1) If you understand a proposition, then you also understand its negation. (2) It is necessary, if you are to believe a proposition, to understand it. (3) It's perfectly possible to believe a proposition and not understand its negation. I interpret those assumptions as follows: (1*) Understanding P entails understanding not-P. (2*) Believing P entails understanding P. (3*) Believing P doesn't entail understanding not-P. (1*)-(3*) imply a contradiction: Believing P does and doesn't entail understanding not-P. If so, then (1)-(3) imply everything (if I've interpreted them correctly). I also don't see how the falsity of (3) implies that we would always have to believe contradictions. If (3) is false, then believing P entails understanding not-P; I don't see how any unwelcome consequences follow from that. PLEASE NOTE : (3) above was taken from Professor Heck's original...

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