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From: kube@cogsci.berkeley.edu (Paul Kube)
Newsgroups: sci.philosophy.tech
Subject: Re: Unbelievable but true...
Message-ID: <19097@ucbvax.BERKELEY.EDU>
Date: Thu, 28-May-87 23:21:39 EDT
Article-I.D.: ucbvax.19097
Posted: Thu May 28 23:21:39 1987
Date-Received: Sat, 30-May-87 09:48:04 EDT
References: <1150@cavell.UUCP>
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Reply-To: kube@cogsci.berkeley.edu.UUCP (Paul Kube)
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Keywords: epistemic logic

In article <1150@cavell.UUCP> jiml@cavell.UUCP (Jim Laycock) writes:
>
>Good grief, my memory is getting rusty.  Is it S4 that is typically
>used to model epistemic logics?

Which set of modal axioms you use for your epistemic logic of course
depends on what intuitions you want to capture, but also on whether
you're after a logic of belief or of knowledge.  I've seen T, S4, and
S5 used as axiom sets for logics of belief, but the axiom schema 
LP -> P in each of them makes better sense interpreting L as `knows' 
than as `believes'.

>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.

-L(P & -LP) (i.e., (x)(p)~Bel(x, p & ~Bel(x,p)) ) is a theorem in each of
these systems.  You don't need LP -> LLP (which is missing from T),
only LP -> P :

1.      L(P & -LP)      (assume for contradiction)
2.      P & -LP	        (from 1. by LP -> P)
3.      P             	(from 2. by conjunction elimination)
4.      LP              (from 3. by necessitation)
5.      -LP             (from 2. by conjunction elimination)
6.      LP & -LP        (from 4., 5.)

For anyone who wants to get into this stuff, Hintikka's book
_Knowledge and Belief_ is the place to start; and there's been a bunch
of recent stuff in the AI knowledge representation literature (e.g.
see D. McDermott, "Nonmonotonic Logic II: Nonmonotonic Modal
Theories", JACM, January 1982).

--Paul kube@berkeley.edu, ...!ucbvax!kube