Re: kifattardi@ICSI.Berkeley.EDU (Giuseppe Attardi)
From: attardi@ICSI.Berkeley.EDU (Giuseppe Attardi)
Date: Thu, 16 Dec 93 20:13:46 PST
Cc: interlingua@ISI.EDU, email@example.com
In-reply-to: <9312152011.AA25350@Sunburn.Stanford.EDU> (firstname.lastname@example.org)
Subject: Re: kif
Date: Wed, 15 Dec 1993 12:10:31 -0800
From: email@example.com (Michael R. Genesereth)
X-Sender: firstname.lastname@example.org (Unverified)
(4) The problem of metalevel paradoxes is also mentioned in the
specification, though in this case the discussion is even briefer.
There are numerous ways to eliminate the metalevel paradoxes popularized by
Montague. Kripke (Journal of Philosophy 1975) gives an axiomatic approach
based on Kleene's strong three-valued logic.
This solution, as the eccellent book by Turner points out, has some drawbacks:
the principle of bivalence
T(A) V T(~A)
does not hold, nor the principle of logical truth:
If A is a logical truth then T(A)
With our semantics, both these principles are maintained, as well as the
law of excluded middle (A V ~A) and its metalevel version T(A V ~A).
The semantics is defined through a process of approximations which
leaves out the paradoxical sentences, to which we ultimately assign
false as truth value.
Technically this is not much different from the semantics for Truth
which I remember seeing in an earlier version of KIF.
Perlis's version is the one used in KIF.
Also in this approach the principle of bivalence does not hold for
all sentences, even though it holds for those non paradoxical.
An intuitive reason why in our approach we are able to preserve
the properties of classical logic is that we push negation in the
opposite direction than Perlis': he pushes negation inside T, while we
push it outside, or rather we push T as far inside as possible.
Paradoxical sentences are not grounded and so you keep pushing T