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From: kube@cogsci.berkeley.edu (Paul Kube)
Newsgroups: sci.philosophy.tech
Subject: The nature of belief
Message-ID: <19647@ucbvax.BERKELEY.EDU>
Date: Wed, 8-Jul-87 20:52:14 EDT
Article-I.D.: ucbvax.19647
Posted: Wed Jul  8 20:52:14 1987
Date-Received: Sat, 11-Jul-87 18:27:50 EDT
References: <3587e521.44e6@apollo.uucp> <680@gargoyle.UChicago.EDU> <121@cavell.UUCP> <4865@milano.UUCP>
Sender: usenet@ucbvax.BERKELEY.EDU
Reply-To: kube@cogsci.berkeley.edu.UUCP (Paul Kube)
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Keywords: logic probability theory belief truth consistency

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