On 2008 May 6, at 8:37, PR Stanley wrote:
Thus, in traditional logic, if you induce "all apples are red",
simple
observation of a single non-red apple quickly reduces your result to
"at least one apple is not red on one side, all others may be red",
i.e, you can't deduce "all apples are red" with your samples anymore.
Paul: surely, you wouldn't come up with an incorrect premise
like "all apples are red" in the first place.
You could come up with it as a hypothesis, if you've never seen a
green or golden apple. This is all that's needed; induction starts
with "*if* P".
However, the real world is a really lousy way to understand inductive
logic: you can come up with hypotheses (base cases), but figuring out
*what* the inductive step is is difficult at best --- never mind the
impossibility of *proving* such. Here's what we're trying to assert:
IF... you have a red apple
AND YOU CAN PROVE... that another related apple is also red
THE YOU CAN CONCLUDE... that all such related apples are red
From a mathematical perspective this is impossible; we haven't
defined "apple", much less "related apple". In other words, we can't
assert either a hypothesis or an inductive case! So much for the real
world.
This only provides a way to construct if-thens, btw; it's easy to
construct such that are false. In mathematics you can sometimes
resolve this by constructing a new set for which the hypothesis does
hold: for example, if you start with a proposition `P(n) : n is a
natural number' and use the inductive case `P(n-1) : n-1 is a natural
number', you run into trouble with n=0. If you add the concept of
negative numbers, you come up with a new proposition: `P(n): n is an
integer'. This is more or less how the mathematical notion of integer
came about, as naturals came from whole numbers (add 0) and complex
numbers came from reals (add sqrt(-1)).
--
brandon s. allbery [solaris,freebsd,perl,pugs,haskell] [EMAIL PROTECTED]
system administrator [openafs,heimdal,too many hats] [EMAIL PROTECTED]
electrical and computer engineering, carnegie mellon university KF8NH
_______________________________________________
Haskell-Cafe mailing list
Haskell-Cafe@haskell.org
http://www.haskell.org/mailman/listinfo/haskell-cafe
