Deductive Reasoning

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

What is deductive reasoning?  How do you use it?

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.) 

I hope this helps. Write back if you have more questions, about this 
or anything else. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/