kube@cogsci.berkeley.edu (Paul Kube) (07/09/87)
In article <4865@milano.UUCP> wex@milano.UUCP writes: >My favorite example is the one >of the proofreader. He has just finished proofreading a 350-page book >and seen all the typos corrected. If we ask him "Do you believe there >is a typo on page <n> of this book?" for all 350 possible values of ><n>, he will say "no" each time. > >However, if we ask "Do you believe there is a typo somewhere in the >350 pages of this book?" he will answer "yes." Inconsistent? Yes. Yes, indeed, inconsistency--not just omega inconsistency, since what's at issue is the relation between a finite conjunction and its conjuncts, not a universally quanitfied sentence and its substitution instances. >So why does he hold this set of beliefs? > >The best answer I could give him was that his beliefs were not a >matter of simple truth/falsity, but were a matter of degree... >... [His counter-claim was that my answer >was not an explanation, simply a way to rationalize a set of beliefs >that he, the belief-holder, considered inconsistent.] Depending on what gets to count as an answer to the question "why does he hold this set of beliefs", the proofreader may well be right. If you're looking for a justificatory account, lottery-paradox examples of this sort can be handled in a framework of probability theory as you suggest. However, probability theory is, like "logic", also systematically violated by standard doxastic practice (e.g. in "cognitive illusion" phenomena of the sort documented by D. Kahneman and A. Tversky). If it's standard doxastic practice that you want an explanation of---well, I suspect that the true psychological story of the fixation and revision of belief and its role in action is going to be very messy and not very flattering. On the topic, a nifty discussion of a shortcoming of logical inference as an account of reasoning can be found in Vann McGee's paper "A Counterexample to Modus Ponens", Journal of Philosophy, sometime fall 1986. The counterexample: It's October 1980. You hold the following plausible beliefs: 1. If it's a Republican that will win the election, then if Reagan doesn't win, Anderson will. 2. It's a Republican that will win the election. However, you don't believe what follows from these by modus ponens, viz. that if Reagan doesn't win, Anderson will (everyone believed that if Reagan didn't win, Carter would). --Paul kube@berkeley.edu, ...!ucbvax!kube
andrews@ubc-red.uucp (Jamie Andrews) (07/10/87)
In article <19647@ucbvax.BERKELEY.EDU> kube@cogsci.berkeley.edu.UUCP (Paul Kube) writes: > It's October 1980. You hold the following plausible beliefs: > 1. If it's a Republican that will win the election, then if > Reagan doesn't win, Anderson will. > 2. It's a Republican that will win the election. >However, you don't believe what follows from these by modus ponens, >viz. that if Reagan doesn't win, Anderson will (everyone believed that >if Reagan didn't win, Carter would). Let's see if I remember the resolution of this paradox. Let R be the proposition that Reagan will win, similarly A and C. One approach is to consider the statements probabilistically, and to note that modus ponens does not hold in this case for probabilistic logic. This approach seems the most sensible. The classical-logic approach is to forget about probabilities and beliefs, in which case the statements become 1. ((R or A) -> (~R -> A)) 2. (R or A) ...but the implication is that (R or A) is true because R is true. Thus (~R -> A) is a perfectly reasonable conclusion because we know that (~R) is always false, and by classical logic the implication is always true. Seen this way, the paradox becomes a challenge to the validity of classical implication and an argument for relevance logic. Another way to look at the classical-logic approach is that (1) is a tautology (since (~R -> A) equiv. (~~R or A)), and so is meaningless as a premise. --Jamie. ...!seismo!ubc-vision!ubc-cs!andrews "They hold the sky on the other side of border lines"