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AIList Digest Volume 8 Issue 084
AIList Digest Thursday, 15 Sep 1988 Volume 8 : Issue 84
Mathematics and Logic:
The Ignorant assumption (leftover Religion) (5 messages)
Rules .vs. axioms
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Date: 9 Sep 88 03:40:56 GMT
From: s.cc.purdue.edu!afo@h.cc.purdue.edu (Neil Rhodes)
Subject: Re: The Ignorant assumption
In a previous article, Greg Lee writes:
>From article <1383@garth.UUCP>, by smryan@garth.UUCP (Steven Ryan):
>" That's it. Any formal system requires such assumptions.
>
>Well, I would say that some natural deduction systems of logic have
>no assumptions -- only rules of derivation. But you can probably
>find a definition of 'assumption' that makes what you say true.
>
I have a problem with Mr. Lee's reasoning in the above statement, and it
seems to be the foundation of most of his recent arguments.
If a formal system were to contain "only rules of derivation," what would
these rules act upon to form statements (theorems) about the system?
Rules alone in a formal system give you nothing. For this reason, you
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.
Since many scientific statements are derived within formal systems, to
believe these statements you must also believe other statements which
cannot be proved.
If Mr. Lee still believes that science asks us to take nothing on
"faith," then I am curious to know what flaws he finds in *my*
reasoning.
--
Neil Rhodes
afo@s.cc.purdue.edu
------------------------------
Date: 9 Sep 88 14:27:25 GMT
From: uhccux!lee@humu.nosc.mil (Greg Lee)
Subject: Re: The Ignorant assumption
>From article <3546@s.cc.purdue.edu>, by afo@s.cc.purdue.edu (Neil Rhodes):
" ...
" I have a problem with Mr. Lee's reasoning in the above statement, and it
" seems to be the foundation of most of his recent arguments.
"
" If a formal system were to contain "only rules of derivation," what would
" these rules act upon to form statements (theorems) about the system?
" 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)
" Since many scientific statements are derived within formal systems, to
" believe these statements you must also believe other statements which
" cannot be proved.
Perhaps that's so. My example does not concern "scientific statements".
I was reacting to a statement that "formal systems" require assumptions.
They don't -- maybe formalized scientific systems do, in a sense,
but even there assumptions can be treated as provisional rather than
as axioms. This is not to disagree with what Neil Rhodes said just
above.
As you will observe, a Gentzen system does involve assumptions, but
no specific assumption is given as part of the system. That is, there
are no axioms.
" If Mr. Lee still believes that science asks us to take nothing on
""faith," then I am curious to know what flaws he finds in *my*
" reasoning.
I find no flaws. If you are to have faith in scientific conclusions,
you must have faith in scientific assumptions. But why have faith in
anything? Why does "science ask us" to do this? If you have a need to
believe in things, other than tautologies, I think you ought not to lay
this at the door of science. It's a personal problem, which I think you
should try to get over.
Greg, lee@uhccux.uhcc.hawaii.edu
------------------------------
Date: 10 Sep 88 10:32:39 GMT
From: l.cc.purdue.edu!cik@k.cc.purdue.edu (Herman Rubin)
Subject: Re: The Ignorant assumption
In a previous article, 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)
------------------------------
Date: 10 Sep 88 15:17:45 GMT
From: glacier!jbn@labrea.stanford.edu (John B. Nagle)
Subject: Re: Rules vs axioms
In theorem proving, previously proved theorems are often used, correctly,
as rewrite rules. Free addition of new "axioms" to theorem proving systems
generally results in unsoundness. As Boyer and Moore once wrote, "It is
one thing to use axioms about a concept known to mathematics for
a century. It is quite another to write down axioms about an idea
invented yesterday." See Boyer and Moore's "A Computational Logic"
for the constructivist's way of avoiding this problem.
Attempts to use the theorem proving paradigm in less formal domains
were made in the late 70s and early 80s, but without notable success.
John Nagle
------------------------------
Date: 11 Sep 88 00:07:50 GMT
From: garth!smryan@unix.sri.com (Steven Ryan)
Subject: Re: The Ignorant assumption
>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)
Well, I see an assumption--it assumes the existence of a formal system.
------------------------------
Date: 11 Sep 88 00:26:44 GMT
From: garth!smryan@unix.sri.com (Steven Ryan)
Subject: Re: The Ignorant assumption
>" Isn't adopting provisional assumptions an act of faith?
>
>Not really. Consider the provisional assumption of a reductio ad
>absurdum argument.
>
>" ... I define faith as adopting assumptions without proof.
>
>It's an odd definition -- if we adopt it, we are led to the conclusion
>that all of us have faith and are therefore religious.
>
>" That's it. Any formal system requires such assumptions.
>
>Well, I would say that some natural deduction systems of logic have
>no assumptions -- only rules of derivation. But you can probably
>find a definition of 'assumption' that makes what you say true.
I really was hoping people would be content with an intentionally imprecise
and informal discussion. If we want to be rigourous, I think it is important
to define a process. I will propose:
A process P is an orderred triple (S,M,Q).
S is a undefined set (of states).
M is a set of pdfs m:S->[0,1].
Q is a relation on MxM called transistions, denoted m->n.
P is probablistic if for any m,s, 0<m(s)<1.
P is not probablistics if for all m,s, m(s)=0 or m(s)=1.
P is deterministic if Q is a function.
P is nondeterministic if Q is not a function.
P is a formal system if S is denumerable and Q is effectively computable.
I think science and religion and CT could be explained as different constraints
on S, M, and Q. If the consensus is to move the discussion into a cryptoformal
notations, that's fine by me, since my education was in math anyway rather than
philosophy.
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