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From: ap1i+@andrew.cmu.edu (Andrew C. Plotkin)
Newsgroups: comp.ai
Subject: Re: Fun with the semantics of paradox
Message-ID: <0Xrvixy00V4D81KlB4@andrew.cmu.edu>
Date: 27 Jan 89 00:45:49 GMT
References: <1883@buengc.BU.EDU> <2996@uhccux.uhcc.hawaii.edu> <905@ubu.warwick.UUCP> <479@aipna.ed.ac.uk> <1036@hudson.acc.virginia.edu>,
	<3715@uklirb.UUCP>
Organization: Carnegie Mellon, Pittsburgh, PA
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In-Reply-To: <3715@uklirb.UUCP>

/>>       "The following sentence is true"
/>>       "The preceeding sentence is false"    ?
/
/ In order to avoid paradoxies Russell introduces a strict hierarchy of types.
/ The first sentence of the above example is of type "sentence about sentence".
/ Then the second must be of type "sentence". On the other hand in order to make
/ a statement about the first, the second must be of type "sentence about
sentence
/ about sentence", both is impossible. So such a self-reference, direct or
/ indirect, is excluded.

I'm not positive about this, but it was my understanding that that produces a
weaker system. Godel showed that a mathematical system *can* talk about the
truth (at least, about provability within itself) without self-contradiction.
That is, the above sentences can be allowed, but are given the status
"unprovable".
   Now we humans think we can do better than that; we keep saying that we can
define truth without that sort of cop-out, which is why we get so tangled by
these paradoxes...

--Z