Yesterday I wrote about the lottery paradox, the epistemological crux of some of my recent reading. I refer you back to yesterday's post if you come to this not knowing what the term means. I will proceed today presuming that you know.
Now, clearly, part of what makes the paradox ... paradoxical is the rule of conjunction. This is the idea that if A is known and B is known then it follows that A + B is known. Or, in a weaker alternative statement: if A is rationally believable and B is rationally believable then A + B is rationally believable.
After all, why can we not simply say that "it is true that my ticket is a loser and I can know that this is true by considering the laws of probability"? One reason I can't say that with a clean logical conscience is that if I can say it of my own ticket I can say it of any one of the thousand (or however many) others that were sold. And if I can do THAT, then by the principle of conjunction I can say that all the tickets are losers. That seems a paradoxical conclusion. Can I in good conscience believe both that SOME number will be drawn and that ALL the numbers are losers? If not: can I limit the principle of conjunction?
In a very great number of cases, the principle of conjunction is quite useful. Indeed, it is the principle that makes Venn diagrams valuable. Imagine a simple Venn diagram: two circles with a small space of overlap/intersection. One set is A. the other is B. The intersection is A + B. If I know that something is true of all Greeks and I know that it is true of all persons named Socrates, then I know that it is true of all Greek persons named Socrates. Still better, if I know that "all humans are mortal" and I know that "The one fellow known as Socrates is a human," I know that "all humans are mortal and Socrates is a human," putting them within a single thought and voila! "Socrates is mortal" becomes inevitable. Abandoning the rule of conjunction would be a big loss for logical inference.
But it seems to be the principle of conjunction that gets us into trouble if it is applied as iteratively as it must be applied to the lead paradoxical conclusion mentioned above. If I know of every lottery ticket that "it is a loser" then don't I know that the lottery has no winners? Can we limit this principle while keeping its value?
Perhaps the key to an answer comes in a contribution to the Douven book by Dana Kay Nelkin. The problem may be simply with the application of the principle of closure in conjunction to a statistical inference in particular. (Nelson, by the way, is a professor of philosophy at the University of California, San Diego.) My inference that I will be in Chicago next week, in the circumstance I discussed in yesterday's post, is NOT a statistical inference. My evidence that I will be in Chicago is not merely statistical. It includes "my awareness of my intentions and plans, my visual confirmation of a receipt for my plane ticket, and so on."
The argument that tells me my lottery ticket is a loser, though, is purely statistical. In Nelkin's words, the "evidential support for the belief that [my] ticket will lose is exhausted by statistical evidence."
Nelkin goes further than I think is necessary in hostility to statistical evidence. But I think he offers a piece of the puzzle. If we combine some suspicion of statistical reasoning of that sort WITH some suspicion of unlimited conjunctivitis, we've got our solution.
We have defanged the paradox if we can agree that, in any case where evidential support is solely statistical, but only in such cases, the rule of conjunction may be ignored. This may also defang the better-known Preface Paradox.
After all, why can we not simply say that "it is true that my ticket is a loser and I can know that this is true by considering the laws of probability"?
ReplyDeleteYou can't say that because it is false. Until the lottery is completed, your ticket (or any other ticket) is not a loser; it is probably a loser. That means that no paradox exists.
I am tempted to say that I know my ticket is a loser because I know that I think of lots of other things as known that are less certain than the failure of my million-dollar-lottery ticket. Most knowledge is probabilistic. Indeed, if we are willing to entertain various philosophical thought experiments about a dreamer, a brain in a vat, a malicious demon, then virtually nothing is certain. Even, "I have a thumb on each hand." If we are willing to entertain probabilistic knowledge as, beyond some point, "outright knowledge," not only about my thumbs but about my upcoming travels, then one might reasonably think we should allow ourselves KNOWLEDGE of "this ticket is a loser."
DeleteBut let us change the subject a little. Just a little. The preface paradox -- which I think goes back to G.E. Moore, works similarly. A scholar (a historian studying World War I, let us say) has written a 400 page book. Meticulously researched. Out of the same sense of meticulously, though, he cautions readers in his preface, "It is very likely that at least one statement in this book is false." He believes this on the basis of probability. After all, EVERY scholarly text beyond a certain quite modest size has errors. And yet if you went through the book with him, sentence by sentence, and asked him whether he stood by each, he might well say, "Yes! Check the footnote!" or even "Yes! I remember the day in the archives when I found that document!" or something analogous. And yet, it is a 400 page work and he has this overriding feeling based on his familiarity with scholarly work in general, that something is wrong.
I think this is quite analogous. Again the threat is that if we conjoin all the individual affirmations we get an affirmation of the whole that our knower neither knows nor believes, that "there is no winner:" in the one case and "there is no error" in the other. And again, I submit we have the same solution. We should distrust the conjunction of STATISTICAL evidence in particular with closure under conjunction
You can find both the lottery and the preface paradox in the Stanford Encyclopedia's list of epistemic paradoxes. https://plato.stanford.edu/entries/epistemic-paradoxes/
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ReplyDeleteChristopher, you write that a scholar cautions readers in his preface, "It is very likely that at least one statement in this book is false." Exactly. He doesn't say, "It is true that at least one statement in this book is false," as you would say, "it is true that my ticket is a loser." I suspect that you intended to have the scholar write, "It is true that at least one statement in this book is false," but you slipped and had him say what he does know to be true, rather than what he does not know to be true.
ReplyDeleteSpeaking of G.E. Moore, he purported to refute skepticism of the external world by holding out his hands during a lecture and saying “Here is one hand” and “here is another”, claiming to prove that there is an external world on that basis. In ON CERTAINTY, Wittgenstein claimed that Moore does not "know" that he has two hands as an empirical fact, but rather can say that he does because he knows how to speak English. Since one cannot doubt that he has two hands, one cannot know that he does. To say otherwise is to misuse "know."
I don't think that the preceding paragraph addresses your point, but it might.
If the issue is the word "knowledge" or "to know" then we can remove it and keep the paradox -- maybe even, from your P of V, create the paradox. Use the phrase "rationally believable," or something similar. It is not rationally believable that this is a winning ticket. That is true of any one of them, so it is true of all of them. So it is not believable that there IS a winning ticket. Which is absurd. Likewise: it is not rationally believable that I've gotten the first sentence in the book wrong. The same is true of any one of them, so it is true of all of them.
DeleteI'm afraid that changing "know" to "rationally believable" doesn't help. Of course it's not rationally believable that this is a winning ticket, because the winning ticket has not yet been drawn. If you mean it's not rationally believable that this will be the winning ticket, then you're mistaken. It has only an exceedingly small chance of being the winning ticket, but it might be. It might be a different story if you believe that the drawing will not be random, because God or fate or whatever has predetermined which ticket will be chosen, but I'm not going to take that road.
ReplyDeleteYou have helped me clarify a point here. The key difference between the lottery and preface paradoxes. The lottery paradox is specifically aimed at knowledge of the future. The preface paradox is aimed at knowledge of the recent past -- the scholar is imagined to be writing about a book that, otherwise, he considers a finished work. Of course in the former case the pertinent lottery is an ongoing fact -- the drawing is, let us say, tomorrow. As an indeterminist, though, I'm not going to press a deterministic account of drawings. (We could postulate a quantum-mechanical element in the drawing if we liked.) I still think you're missing the point, but I have to say the point isn't all-fired important enough for me to press it further.
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