Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Path: utzoo!utgpu!water!watmath!clyde!cbosgd!ihnp4!alberta!jiml
From: jiml@alberta.UUCP
Newsgroups: sci.philosophy.tech
Subject: Unbelievable but true...
Message-ID: <1150@cavell.UUCP>
Date: Thu, 28-May-87 05:14:19 EDT
Article-I.D.: cavell.1150
Posted: Thu May 28 05:14:19 1987
Date-Received: Sat, 30-May-87 00:38:10 EDT
Reply-To: jiml@cavell.UUCP (Jim Laycock)
Distribution: world
Organization: U. of Alberta, Edmonton, AB
Lines: 31
Keywords: epistemic logic

I hope this is the correct group in which to post an article on a
curiosity in epistemic logic.  It seems that there are a class of
sentences which, while true, may not be believed.  I'll give first
an example of an entirely reasonable statement, followed by a close
cousin which is unbelievable but true.

	let p = "it is raining"

	1.	Bel(Smith, p & ~Bel(Jones,p))

Smith believes that it is raining but that Jones doesn't believe it.

	2.	~Bel(Jones, p & ~Bel(Jones,p))

Note that in this case, p & ~Bel(Jones,p) may well be true, but that
Jones is incapable of believing it.
It's been a while since I've seen the derivation, but I recall that
the following is a theorem of epistemic logic:

	3.	(x)(p)~Bel(x, p & ~Bel(x,p))

provided that

	4.	Bel(x,p) > Bel(Bel(x,p))

is an axiom.

Good grief, my memory is getting rusty.  Is it S4 that is typically
used to model epistemic logics?

Anyway, can anyone out there add to this curious class of sentences?