Lewis:  “Subject S knows proposition P iff P holds in every possibility left uneliminated by S’s evidence” (p. 222-3)

“…possibility W is uneliminated iff the subject’s perceptual experience and memory in W exactly match his perceptual experience and memory in actuality.” (p. 224)

[Note: does “memory” mean “seeming memory” or actual memory?  To actually have the same memories, S must have the same experiences of the same things (leaving out, I suppose, the things that S forgets) in each world.  If S remembers touching a tree in each world, then there has to have been a tree in each world that S touched.  That would mean no demon worlds or BIV worlds.  That makes it—I don’t know whether to say “too strong” or “too weak”. It seems to get rid of the skeptical problem much too easily by making it too difficult for a world to count as an uneliminated possibility.  Then again, maybe that’s externalism.  But Lewis doesn’t sound like he thinks the problem has disappeared just like that.]

    Lewis does not seem to require that S be aware that P in order to know that P.  So maybe that makes it less implausible that if S knows that P, then S knows that S knows that P.  S needn’t ever realize that S knows that P, or that S knows that she knows that P.  However, since knowledge attributions require context, perhaps someone has to be aware that P or at least think about P.

    Taking the most difficult case, where we don’t leave out some possibilities (on the sotto voce proviso), S knows that P iff P in all worlds where S has matching perceptual experience and memory.  So that also means that if S knows that P in actuality, S also knows that P in all those matching worlds (the uneliminated possibilities).  Why?  Because in each of these worlds, it is true that S knows that P iff P in all uneliminated possibilities—which is the same group of worlds each time, including the actual world.  So if S knows it in one of these worlds (such as actuality), then P must be true in each of them, so S knows P in all of them.

    Does S know that S knows that P?  S knows that S knows that P iff S knows that P in all the uneliminated possibilities—in all the matching worlds.  So if S knows that P, then indeed S also knows that S knows that P.

     Is it true that if S doesn’t know that P, then S knows that S doesn’t know that P?  If S doesn’t know that P, then it must be that not-P in one of the uneliminated possibilities—one of the worlds that matches actuality in S’s perceptual experience and memory.  It will then be true in each of those worlds that S doesn’t know that P (because not-P in one of the uneliminated worlds).  And if S doesn’t know that P in each of those worlds, then S knows that S doesn’t know that P.  So, if S doesn’t know that P, then S knows that S doesn’t know that P.

    Note: it doesn’t follow that if S doesn’t know that P, then S knows that not-P—which is as it should be.

“S knows that P iff S’s evidence eliminates every possibility in which not-P—Psst!—except for those possibilities that we are properly ignoring.”(p. 225)

    However, if we don’t leave out any of the possibilities, we may not have any knowledge at all—or very little, at any rate.  That was the skeptical problem, and the reason for the sotto voce proviso.  What about contexts where some possibilities can be properly ignored?

    Does S know that it is a zebra?  We aren’t concerned with zoo fraud, suppose, or demon worlds.  In all the worlds where S has the same perceptual experience and memory (except those we are properly ignoring) it is a zebra that S sees.  So S knows it is a zebra.

    Does S know that S knows it is a zebra?  Is this meant to be the same context?  It probably wouldn’t be usually—more likely, someone has come up to S and said, “Is that a zebra?”  S confidently asserts that it is.  “But are you sure about that?”  Now S stops to ponder.  Could it be a painted mule after all?  Does she really know that it is a zebra?  S doesn’t know whether she knows it is a zebra after all—if it is a painted mule, it isn’t a zebra, and it has occurred to her that she can’t eliminate that possibility.  It is consistent with all her perceptual experience and memory.  So S doesn’t know that she knows it’s a zebra, even though she knew (before the question arose) that it is a zebra.

    Now suppose that S is aware that zoo fraud is rampant (I hope this is some other possible world—I like zoos!!).  She can’t properly ignore the possibility that that this is a painted mule.  So she doesn’t know that it is a zebra.  And she knows that she doesn’t know it is a zebra.

    But suppose S hasn’t been reading the papers, which have been talking about zoo fraud, and still trusts zookeepers with wonderful naivete.  So she believes it is a zebra, even though (perhaps) she ought to believe that zoo fraud is rampant and therefore can’t properly ignore that possibility.  So she doesn’t know that it is a zebra (though she believes that it is).  Does she, in that context, know that she doesn’t know it is a zebra?  Since it is the same set of possibilities, apparently she does (though she isn’t aware of it because she has been ignoring the headlines at the newstand she passes every day).  More likely the question would come up in a different context—someone comes up and says, “Gee, given all this zoo fraud, I no longer know whether this is a zebra.”  “What zoo fraud?” S asks, and gets the whole story.  Now she knows that she doesn’t know it is a zebra, since she doesn’t improperly ignore the possibility of zoo fraud.