Using logic: Truth preservation, probability, and the lottery paradox
I defined a valid argument as one where the conclusion must be true if the premises are. Another way of putting this would be to say that in a valid argument it is impossible for the premises to be true and the conclusion false.
This is the very notion of possibility that we used in talking about possible worlds. So in terms of possible-world semantics we could say that a valid form of argument is one where a sentence of the form of the conclusion is true in every possible world where sentences of the forms of the premises are true. One shorthand way of saying this is to say that valid arguments are truthpreserving: if you've got true premises and you use a valid form of argument, you'll get a true conclusion.I mentioned earlier, when we were looking at Descartes' cogito in 2.3, that Descartes wanted an argument that transmitted not just truth but certainty from premises to conclusion. And, in fact, if you defined certainty as having a 100 percent probability of being true, then it turns out that arguments that preserve truth also preserve certainty. So Descartes was right to think that if he could find an argument that was valid—as the cogito certainly is—and its premise was certain, then the conclusion would be certain too. It also turns out, however, that a valid argument whose premises are merely probable can have a conclusion that's much less probable than any of its premises. So when you're using logically valid arguments, you need to keep track of probability as well as truth.
This fact is important in contexts where we are making an argument that has many premises. To see this, let's think about the so- called lottery paradox. Consider, for example, Mary Jo, who is thinking about lottery tickets in a ten-thousand-ticket lottery where each ticket has the same chance of winning.
As each ticket comes into her hand, she thinks, “That one won't win,” because it is indeed highly improbable that any particular ticket will win. Suppose she sits there for days, going through all the thousands of tickets, and in the end she has said to herself about each of them, “That one won't win.” That's certainly a perfectly reasonable thing to think about each of them given that the probability that any of them will win is only one in ten thousand. She also concludes, at the end, of her survey, “Well, those are all the tickets.” But from the premises:1: Ticket 1 won't win
2: Ticket 2 won't win
3: Ticket 3 won't win
up to
10,000: Ticket 10,000 won't win
she can conclude
Conclusion: Tickets 1 to 10,000 won't win.
From this, of course, given the further premise,
10,001: Tickets 1 to 10,000 are all the tickets
it follows that
None of the tickets will win!
And we certainly don't want her concluding that.
This paradox—that, in considering a lottery, it can be reasonable to believe that each ticket won't win, but not reasonable to think that they all won't win—is less worrying once you realize that, because each of the premises is less than certain, there is no logical guarantee that the conclusion will be as probable as each premise is. The argument preserves truth but not probability. (The rule, in fact, is that if you have n premises and the least likely premise has a probability of (1-e), then the conclusion can have a probability as low as (1—ne). Since, in this case, e is around 0.0001 and n is 10,000, (1— ne) is 0—so the probability of the conclusion can be as low as 0!)
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