From: David Madore
Newsgroups: sci.math
Subject: Re: Twin Primes
Date: Wed, 06 Nov 1996 11:32:58 +0100
Hauke Reddmann wrote:
>
> Maybe we should try to formalize a mathematical theory
> of "knowing"? E.g, what is the relation that connects
>
> 1. X knows A
> 2. X believes A
> 3. True(A)
> 4. Proof(A)
>
> 2/\4<=>3,maybe? I think that "know" isn't reducible on
> truth and provability.
> --
> Hauke Reddmann <:-EX8
> fc3a501@math.uni-hamburg.de PRIVATE EMAIL
> fc3a501@rzaixsrv1.rrz.uni-hamburg.de BACKUP
> reddmann@chemie.uni-hamburg.de SCIENCE ONLY
Such a theory exists. It is called some very strange
name (epistemic logic perhaps), and it is a subtheory of a
wider theory called "modal logic" (if you are interested
in that, you can read Sally Popkorn's "first steps in
modal logic").
You have to be careful in your choice of axioms, though.
One might expect the following to be reasonable (K
stands for "knows", B for "believes", ! for "not",
and -> for "implies")
a KA -> A (``if you know it it's true'')
b KA -> BA (``if you know it, you believe it'')
c KA -> KKA (``you know what you know'')
d !KA -> K!KA (``you know your ignorance'')
e BA -> KBA (``you know what you believe'')
f !BA -> K!BA (``you know what you don't believe'')
g BA -> BKA (``you believe to know what you believe'')
h BA -> !B!A (``you can't believe something and its converse too'')
Unfortunately, from them you can deduce
BA -> A
Which IS unpleasant. Proof:
1 BA -> BKA (g)
2 BKA -> !B!KA (h)
3 !B!KA -> !K!KA (contrapositive of b)
4 !K!KA -> !!KA (contrapositive of d)
5 !!KA -> KA (propositional calculus)
6 KA -> A (a)
7 BA -> A (from the above, by modus barbara)
David A. Madore
(david.madore@ens.fr,
http://www.eleves.ens.fr:8080/home/madore/index.html.en)