Path: utzoo!utgpu!jarvis.csri.toronto.edu!rutgers!cs.utexas.edu!uunet!mitel!sce!cognos!rayt
From: rayt@cognos.UUCP (R.)
Newsgroups: comp.ai
Subject: Re: abduction vs. induction
Message-ID: <6204@cognos.UUCP>
Date: 28 May 89 00:31:31 GMT
References: <1480@crin.crin.fr <14820@paris.ics.uci.edu> <6144@cognos.UUCP> <6487@sdcsvax.UCSD.Edu>
Reply-To: rayt@cognos.UUCP (R.)
Organization: Cognos Inc., Ottawa, Canada
Lines: 39

In article <6487@sdcsvax.UCSD.Edu> Steve Bloch writes:
<>In article <6144@cognos.UUCP> I wrote:

<>>... I am surprised to hear that [mathematical induction] isn't
<>>[logically valid] since MANY mathematical proofs rest upon it. 

<>What exactly do you mean by "logically valid"?  The definition in my
<>math logic class gives a couple of really basic axioms about how
<>logical (truth-functional) connectives and quantifiers relate to one
<>another, and says something is "logically valid" iff it holds in all
<>models of those axioms.  Those axioms do not include mathematical
<>induction, and in fact there are lots of models of those minimal
<>axioms (indeed, there are lots of models of the more powerful
<>Robinson arithmetic, which looks as though it's talking about natural
<>numbers) in which induction does not hold.  Induction happens to be
<>true in the Platonic object we call the natural numbers, so you can
<>draw valid conclusions about the natural numbers using it, but it's
<>certainly not logically valid.

I had a much more simplistic concept of "Logically Valid" in mind;
to wit: Whenever all of the premises are true, the conclusion is
also true. Hence, in mathematical induction, IF the initial condition
AND the induction step are true, THEN the hypothesis is true, is a
logically valid deduction. To achieve valid conclusions for particular
instances, one must demonstrate that each of the propositions is true
for those objects in question (also that they are of the appropriate
type). Your's is undoubtedly the definitive 21st-Century word on this
issue, however.

						R.

P.S. I AM skeptical, though, that the SET of natural numbers could be
     considered to be a Platonic object (form) since I don't believe that
     the Greeks had a notion of the infinite.
-- 
Ray Tigg                          |  Cognos Incorporated
                                  |  P.O. Box 9707
(613) 738-1338 x5013              |  3755 Riverside Dr.
UUCP: rayt@cognos.uucp            |  Ottawa, Ontario CANADA K1G 3Z4