Now to assess the proof; it seems clear that the premises are different than the conclusions, it is also clear that the conclusions follows from his premises, but however is the premise "hands are external things" actually true???... Moore certainly thinks so, however he never even tried proving that he has hands, he thinks proving this is absurd, in order to do so he would have to prove that he is not dreaming, which he knows he cant, he isn't even going to try.
He tries to counter attack skeptical philosophers who argued that how can one be sure if you are not currently dreaming in which you have hands. Moore simply replies by saying that he knows hes standing up, so he knows hes not dreaming, furthermore he thinks if that if you don't know that you are dreaming, than you really don't know that you are standing up, BUT Moore knows he is standing up, thus he is sure that he isn't dreaming. He further attacks the skeptic by saying that how can they be helping themselves to the idea of dreaming, when they themselves said that they cant be certain whether they are dreaming or not in the first place: if this was actually all a dream and there was no external world, the skeptic wouldn't really even have the slightest idea that he is dreaming to begin with...
Thats what Moore had to say on the external world.. good proof or not left aside.. certainly easier to understand than some others out there.
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