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### Deductive Reasoning

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Date: 09/21/2000 at 02:03:08
From: Toi
Subject: Deductive Reasoning

What is deductive reasoning?  How do you use it?
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Date: 09/24/2000 at 02:00:52
From: Doctor Ian
Subject: Re: Deductive Reasoning

Hi Toi,

Deductive reasoning is when you start from things you assume to be
true, and draw conclusions that must be true if your assumptions are
true.

The classic example is

All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.

Lewis Carroll, in his classic works on logic, tended to use more
whimsical examples:

All ducks who are dentists play golf on Tuesdays.
Anyone who plays golf can't be trusted.
Therefore, ducks who are dentists can't be trusted.

It's important to realize that when you use deductive reasoning to
arrive at a conclusion, your conclusion can be false, even when your
reasoning is valid. For example,

Every member of Congress is a woman.
Mel Gibson is a member of Congress.
Therefore, Mel Gibson is a woman.

The conclusion is false, because the premises are false, but the
reasoning is valid, because if the premises WERE true, then the
conclusion WOULD BE true.

It gets weirder. The truth of your conclusion doesn't guarantee the
validity of your argument.  For example,

Everyone born in Connecticut likes square dancing.
My cousin Cathy likes square dancing.
Therefore, my cousin Cathy was born in Connecticut.

In fact, my cousin Cathy _was_ born in Connecticut; but the argument
is not valid, for reasons you can probably appreciate, if not explain.

(The rules of deduction say that from 'If A then B' and 'A' you can
conclude 'B'; but from 'If A then B' and 'B', you can't conclude 'A'.
That doesn't keep a lot of people from trying, though, especially in
political debates.)

Deductive proofs are essential to mathematics. One very important kind
of deductive proof is 'reduction to absurdity', or proof by
contradiction. In this kind of proof, you assume the opposite of the
thing you want to prove, and show that it leads to a contradiction.
Anything that leads to a contraduction must be false, so that means
the thing you wanted to prove must be true.

It sounds weirder than it is. You can see this technique in action in
the following item from the Dr. Math archive:

Proof that Sqrt(2) is Irrational
http://mathforum.org/dr.math/problems/tim.8.29.96.html

Proof by contradiction has gained importance in recent years, because
it's easy to get computers to do it. (All the other kinds of deduction
are much harder.) You load up your premises, and the opposite of the
thing you want to prove, and let the computer start deriving
conclusions, until it finds two conclusions that contradict each
other.

Note that Sherlock Holmes often spoke of using 'deduction', but in
fact he usually relied on abduction, which is an entirely different
kind of reasoning. ('Abduction' is a style of reasoning where you
think up several possible explanations for something, and then pick
the one that you think is most likely. For example, if I show up at
work and no one is there, I can think of any number of explanations,
including (1) my co-workers were all abducted by aliens, (2) my co-
workers have all become invisible, (3) the company went out of
business, but no one told me, and (4) I've forgotten that it's a
weekend. I have no way of _proving_ that (1-3) aren't true, but
I'm probably going to go with (4) on the basis of likelihood.)

or anything else.

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Logic

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