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From: cik@l.cc.purdue.edu (Herman Rubin)
Newsgroups: comp.ai
Subject: Re: The Ignorant assumption
Summary: Rules are stronger than axioms
Message-ID: <918@l.cc.purdue.edu>
Date: 10 Sep 88 10:32:39 GMT
References: <3546@s.cc.purdue.edu> <2365@uhccux.uhcc.hawaii.edu>
Organization: Purdue University Statistics Department
Lines: 35

In article <2365@uhccux.uhcc.hawaii.edu>, lee@uhccux.uhcc.hawaii.edu (Greg Lee) writes:
> From article <3546@s.cc.purdue.edu>, by afo@s.cc.purdue.edu (Neil Rhodes):
			....................
< " Rules alone in a formal system give you nothing.  For this reason, you

> They give you nothing but tautologies, at least.

< " need a given set of statements (axioms) from which these rules can derive
< " other statements (theorems).  Since these axioms are not derived and are
< " necessary to the formal system, then you must "believe" them to be true
< " while working within the system.
 
> There are formalizations of logic that require axioms, but not all
> do.  Gerhard Gentzen created systems that have no axioms.  For
> instance:
> 	Suppose p (one can introduce provisional assumptions freely)
> 	Conclude p (one can repeat an assumption as a conclusion)
>   So, p implies p (since p was concluded on the basis of the provisional
> 	  assumption p, one can derive the implication)

In a treatment of natural deduction mentioned above, one shows that 

	The customary axioms and axiom schemes are derivable.

	The customary rules of derivation are valid.

	Any theorem provable by natural deduction can be proved by using
the customary axioms, axiom schemes, and rules of derivation.

However, starting with a set of axioms and no rules, nothing more can be
derived.  Thus we see that rules are stronger than axioms.  
-- 
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907
Phone: (317)494-6054
hrubin@l.cc.purdue.edu (Internet, bitnet, UUCP)