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From: jon@XN.LL.MIT.EDU (Jonathan Leivent)
Newsgroups: comp.ai.digest
Subject: Are all Reasoning Systems Inconsistent?
Message-ID: <19880724060132.9.NICK@HOWARD-JOHNSONS.LCS.MIT.EDU>
Date: 24 Jul 88 06:01:00 GMT
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Organization: The Internet
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Approved: ailist@ai.ai.mit.edu

Date: Tue, 19 Jul 88 10:16 EDT
From: Jonathan Leivent <jon@XN.LL.MIT.EDU>
Posted-Date: Tue, 19 Jul 88 10:16:06 EDT
To: AILIST@AI.AI.MIT.EDU
Subject: Are all Reasoning Systems Inconsistent?
cc: adams@XN.LL.MIT.EDU, jon@XN.LL.MIT.EDU



Within any (finite) reasoning system, it is possible to construct a sentence S
from any preposition A such that S = (S -> A) using Godel-like methods to
establish the recursion.  However, such a sentence leads to the inconsistent
conclusion that A is true - any A!

1. S = (S -> A)		; the definition of S, true by construction
2. S -> (S -> A)	; a weaker form of 1.
				[U = V, so U -> V]
3. S -> S		; an obvious tautology
4. S -> (S ^ (S -> A))	; from 2. and 3. by conjunction of the consequents
				[U -> V and U -> W, so U -> (V ^ W)]
5. (S ^ (S -> A)) -> A	; modus ponens
6. S -> A		; from 4. and 5. by transitivity of ->
				[U -> V and V -> W, so U -> W]
7. S			; from 1. and 6.
				[U = V and V, so U]
8. A			; from 6. and 7. by modus ponens

Am I doing something wrong, or did logic just fall down the rabbit hole?



-- Jon Leivent