Logic: Definitions

Date: 04/04/2000 at 18:05:31
From: jenn
Subject: Logic - definitions

What does deductive reasoning mean? What does inductive reasoning 
mean?

Date: 04/05/2000 at 18:36:44
From: Doctor Ian
Subject: Re: Logic - definitions

Hi Jenn,

Deductive Reasoning
-------------------
Deductive reasoning is when you move from things you know or assume to 
be true - called 'premises' - to conclusions that must follow from 
them.  The most famous example of deduction is

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

The first two statements are premises, and the third statement is a 
conclusion. By the rules of deduction, if the first two statements are 
true, the conclusion _must_ be true.  

Note that this is the case even if the premises appear to be nonsense:

  All ducks play golf.
  No one who plays golf is a dentist. 
  Therefore, no ducks are dentists. 

The conclusion follows deductively, or is 'deduced', from the 
premises. IF the premises are true, THEN the conclusion must be true.  
If the premises happen to be false, then of course all bets are off.  
The conclusion might still be true, or it might not.  

Lewis Carroll was fond of using examples like these in his books on 
symbolic logic, partly because they prevent you from applying your 
'common sense'. After all, the whole point of logic is to make up for 
errors in common sense!  

Inductive Reasoning
-------------------
Inductive reasoning is when you move from a set of examples to a 
theory that you think explains all the examples, as well as examples 
that will appear in the future. The simplest kind of induction looks 
like this:

  The sun came up this morning.
  The sun came up the day before that.
  The sun came up the day before that.
   .
   .
  Therefore, the sun comes up every day, and will come up tomorrow 
  too. 

Note that while a conclusion deduced by deduction _must_be true if the 
premises are true, the conclusions induced by induction _may_ be true, 
or they may not. For example, people who visit Seattle for short 
periods often find that it rains every day of the visit. They could 
induce (or infer, or draw the conclusion) that it rains every day in 
Seattle - but this wouldn't be true.  

In mathematics, inductive reasoning is often used to make a guess at a 
property, and then deductive reasoning is used to prove that the 
property must hold for all cases, or for some delimited set of cases. 

For example, by playing around, you might notice that every time you 
inscribe a triangle in a circle so that one leg lies along a diameter 
of the circle, it seems to be a right triangle. You might then guess 
that this is _always_ the case. That's induction. You would then set 
out to use the axioms of geometry to _prove_ that this must be the 
case. That's deduction. 

There are other kinds of reasoning, too. One of the most common is 
abductive reasoning, or 'reasoning to the best explanation'. This is 
when you try to generate the 'most likely' chain of events that could 
have led to some observation.  

For example, if you come to school one day and find that all the doors 
are locked, you might be able to think of a number of explanations:  
It's a holiday; it's a weekend; it's summer; the janitor got drunk 
last night and lost his key; some pranksters put glue in the locks; 
you've entered the Twilight zone; you're dreaming; everyone else is 
planning a surprise party for you and they don't want you to come in 
before it's ready; you're at the wrong school; and so on. 

There are an infinite number of possible explanations for any 
occurrence! But some are more likely than others, and when you choose 
the one that seems most likely given the other information that you 
have (e.g., "Oh, yeah, last night was the football game, so yesterday 
was Friday..."), you're using abductive reasoning.  

Note that Sherlock Holmes often called what he was doing deduction, 
when in fact it was usually abduction.  

It's interesting to note that of the three types of reasoning that 
I've mentioned, the one that is easiest for humans (abduction) happens 
to be hardest for computers, and the one that is easiest for computers 
(deduction) happens to be the hardest for humans. Do you suppose 
that's just a coincidence, or do you think it might have something to 
do with some fundamental differences between brains and computers?  

I hope this helps. 

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