Path: utzoo!attcan!uunet!lll-winken!ncis.llnl.gov!helios.ee.lbl.gov!pasteur!ucbvax!decwrl!purdue!bu-cs!buengc!bph
From: bph@buengc.BU.EDU (Blair P. Houghton)
Newsgroups: comp.ai
Subject: Re: Fun with the semantics of paradox
Message-ID: <1984@buengc.BU.EDU>
Date: 26 Jan 89 17:35:07 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>
Reply-To: bph@buengc.bu.edu (Blair P. Houghton)
Followup-To: comp.ai
Organization: Boston Univ. Col. of Eng.
Lines: 34

In article <3715@uklirb.UUCP> kerber@uklirb.UUCP (Manfred Kerber) writes:
>Heiko Hecht writes:
>>> But what if choices 1. to 3. don't seem to work, does anyone have suggestions
>>> as to how to resolve the following paradox:
>>>    
>>>       "The following sentence is true"
>>>       "The preceeding sentence is false"    ?
>
>This can be excluded by Russell's "Theory of Types" as described in "Principia
>Mathematica" or in the American Journal of Mathematics p.222 ff, Vol.XXX, 1908.
>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.

How bout a few symbols?

1.  S1 ==> S2
           __
2.  S2 ==> S1

note the precise notation.  "==>" says that if S1 is true, then S2 is
true.  It says nothing about what happens if S1 is false.  Thus,
since S2 claims S1 is false, then we know nothing of the validity of S2.

It's not a paradox, it's incomplete.

				--Blair
				  "Any ideas?  Any at all.  About
				   anything.  C'mon, just a _little_
				   thought.  Don't be such a
				   politician.  C'mon, you can do it..."