Inductive and
Deductive Reasoning

How do you think your
way through a problem? How do you know
what you know about the world?

You observe and you
reason. You observe to obtain
information. Using information, you
reason your way to conclusions. How do
you observe? How do you reason?

You might use numbers
to quantify things and make the observations more precise and the conclusions
more exact. But whether or not numbers
are involved, there are different methods of reasoning and they produce different
types of conclusions of differing certainty.

An understanding of
the various ways of observing and reasoning is essential in dealing with the
myriad sources of information and misinformation, which confront all of us
daily.

We all make casual
observations everyday and arrive at intuitive conclusions based upon them.

Consider your fellow
students here at Anne Arundel Community College. How do they get to the campus?
They drive, right? Why are there
so many big parking lots? Because
everyone drives, right?

Maybe … and maybe
not.

Maybe some students
live so close that they walk or bicycle?
Still, what if it rains? They’d
drive then, right? And they must have
driver’s license, right? Everyone at
AACC has a driver’s license, right?

That’s a conclusion –
a **conjecture** – arrived at intuitively.
What’s a conjecture? A
conjecture is a generalization that you think might be always true – a rule
that is never broken. Forming a
conjecture often involves recognizing a pattern from past experience and
forming a generalization from it.

Your conjecture is:
All AACC students have a driver’s license.

How might you test
whether your conjecture is true?

You might interview
many of your fellow students. “Do you
have a driver’s license?” You ask
dozens of your fellow students and they all respond, “Yes”. Your conjecture is thus confirmed – so
far. This is **inductive reasoning**. You’ve purposefully examined many specific
cases, and in each case your conjecture is confirmed.

But is it really
always true? Suppose – and this is not
a fact – but suppose you did a little research and discovered that AACC has a
rule that to register for a class, you MUST present a valid driver’s license
with photo ID. They did ask you for a
driver’s license when you registered, right?
In that case, it MUST be true that all AACC students have a driver’s
license, because, if they did not, they could not register for a class and would
hence not be students.

This is an example of
**deductive reasoning**. You begin with an
assumption – you must present a valid driver’s license to register for a course
- and deduce a conclusion – all students have a driver’s license. Of course, your conclusion is only as
certain as your assumption – which, in this case, is false – there is no such
rule at AACC. AACC requires proof of
identity, but it need not be a driver’s license.

Is your conjecture
really true? Do all AACC students have
a driver’s license? You keep
interviewing your fellow students. And
finally, you meet, say, a 15 year old student who, due to her young age, does
not have a driver’s license, of course.
Your conjecture is falsified.
The 15 year-old is a **counter-example**, an instance which shows
that the conjecture is not always true..

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__Inductive Reasoning__

Let’s consider
another example of inductive reasoning.

Have you noticed the
white swans so common on the waters of the Chesapeake Bay? They typically swim in pairs and they are
always white. They are mute swans. They are a non-native species, imported from
Europe. They are causing problems to
the ecology of the Bay – their feeding habits destroy the Bay grasses - and
there is currently an effort underway to reduce their numbers by sterilizing
their eggs.

If you travel to the
more remote areas of the Bay during the winter, you might notice another type
of swan swimming together in large flocks.
These are whistling or tundra swans.
They winter on the Bay, but mate and breed in Canada during the
summers. They are a native North
American species – their feeding habits do not destroy the Bay grasses - and
they, too, are always white.

You form the
conjecture: All swans are white.

You embark on a
world-wide tour to investigate your conjecture.

You travel to Canada
and encounter more tundra swans and the bigger trumpeter swans. They are always white.

You travel to South
America and encounter Cascaroba swans.
They are also always white.

Your conjecture is
thus confirmed: All swans are white.

You travel to Europe
and find more mute swans, and they all are white. You travel to Africa and find more mute swans. Your conjecture -
all swans are white - is now confirmed by evidence from four continents.

You travel to Asia
and encounter whooper swans and Bewick swans (very closely related to the
American trumpeter and tundra swans, respectively) – and they, too, are always
white.

Your conjecture is
now all but proven – or is it? Are all
swans white?

You have not been to
Australia. You go there. And you discover the Australian black swan –
with its beautiful pink eyes. It is not
white. It is a counter-example. Your conjecture has been falsified.

This illustrates the
conundrum of inductive reasoning. No matter how many instances of confirmation
you have obtained, you cannot be certain of your conclusion. No conjecture can ever be proven beyond all
doubt by inductive reasoning. There is
always the possibility of a counter-example.

This is also the
conundrum of science. You form a
conjecture, a generalization about the world around you. In science, such a conjecture is called a
hypothesis. You embark upon a rigorous
program of testing if this conjecture is true.
In science, such testing involves **experimentation**, which is
observation in specifically controlled situations. BUT! No matter how many
times you successfully test your conjecture, no matter how many experiments
confirm the hypothesis, you cannot be absolutely certain of its
truthfulness. The next test might
falsify it. The next experiment might reveal a counter-example.

Inductive reasoning
differs from intuitive reasoning in that the conjecture is explicitly stated
and it is tested and confirmed by a planned program of observations. With intuitive reasoning the observations
are more casual – counter-examples might not even be noticed if you are not
thinking about the conjecture at the moment the counter-example is encountered.

You might intuitively
arrive at the conjecture that SUV drivers are tailgaters because whenever you
notice someone tailgating you, they seem to be driving an SUV. But this could be because SUVs are big and
you are more likely to notice a big SUV than a smaller car when someone is
tailgating you. A planned, quantified
program of always observing what is behind you would probably reveal that SUV
drivers are no more likely to tailgate than any other type of driver.

__Assumptions and
Deductive Reasoning__

Deductive reasoning
is not based upon observation: it is based upon assumptions and the laws of
logic.

If we assume that
“all birds have wings” and assume that “a penguin is a bird”, then it must be
true that “a penguin has wings”. This
is one example of deductive reasoning called a syllogism. It has the form: If
”some assumptions”, then “a conclusion”, where we say the conclusion is
logically implied by the assumptions. If the assumptions are true, then the
conclusion must also be true.

But is the statement
really true about the real world? That
is a different question. That is an
empirical question, a question about reality, and not a question of logic. Are the assumptions true about the real
world? Do all birds have wings? Do
penguins have wings? Is it fair to call
the penguin’s flippers wings? Maybe …
though maybe not. But that doesn’t
affect the logical justification of the implication.

The assumptions are
assumed to be true; and, if they are, the conclusion is deduced with
certainty. If the assumptions aren’t
true, then it doesn’t matter. The
statement is not really false as a logical deduction. If the conclusion is false about the real
world, then since the deductive reasoning is valid – it follows that one of the
assumptions must also be false about the real world. In this case, if we decide that a penguin’s flippers are not
wings, then either the assumption that “all birds have wings” is false or the
assumption that “a penguin is a bird” is false.

Our first example of
deductive reasoning about student driver’s licenses can also be phrased as a
syllogism. If “a person must present a
valid driver’s license to register for a course at AACC” and “a student at AAAC
must be registered for a course” then it must also be true that “all AACC students have a valid driver’s
license”. But even though the logic is
flawless, the conclusion is false because one of the assumptions is false.

In mathematics the
role of reasoning changes. The
assumptions become definitions or axioms that are “absolutely true”; and hence,
the deductions, the conclusions, are also true with absolute certainty.

“ **1 + 1 = 2** ”
is not just a conjecture, it is the definition of the number two.

“ **2 + 1 = 3** ”
is not just a conjecture, it is the definition of the number three.

“ **3 + 1 = 4** ”
is not just a conjecture, it is the definition of the number four.

“ **2 + 2 = 4** ”
is not just a conjecture, it is a logically provable truth. (By proving
addition is associative, we prove that:

“ **2 + 2 = (1 + 1) + (1 + 1) = 2 + (1 + 1) = (2 + 1) + 1 = 3 + 1 = 4** ”.

What is the next
number in the following sequence of numbers?

**1,
3, 5, 7, 9, 11, 13, ...**

Clearly it’s **15**
- these are the odd numbers.

Inductive thinking in
mathematics often involves recognizing a pattern. Let’s try another.

**2, 5, 8, 11, 14, 17, 20, ...**

Did you say **23**? One way to find a pattern is to find the
differences between successive terms.
Here it is the constant **3**, each term is **3** more than the previous.

Let’s try another:

**0, 1, 4, 9, 16, 25, 36, ...**

Hmmmm, the
differences between the terms are:

** 1, 3, 5, 7, 9, 11, **

These are the odd
numbers again; so the next number must
be **36 + 13 = 49**.

But think about the
original sequences again. It can also
be written as:

**0 ^{2}, 1^{2}, 2^{2},
4^{2}, 5^{2}, 6^{2}, ...**

** **

Are the differences
between any consecutive perfect squares the consecutive odd numbers? That’s a mathematical conjecture arrived at
by recognizing two related patterns.
But it’s just a conjecture. In
mathematics we sometimes are able to **prove** that conjectures MUST be
true. How could we prove this one? We could use a little algebra.

If **n ^{2}**
is a perfect square, the next perfect square is

** (n+1) ^{2} **

And notice that **2n
+ 1** are exactly the consecutive odd numbers, each being one more than the
consecutive even numbers ( **2n** = multipiles of **2
**)..

Let’s do a more
interesting example of proving a mathematical truth to be true with absolute
certainty.

A prime number is a
whole number that can only be divided, without remainder, by itself and **1**.

**2** is a prime number;
as is **3**.

The first few prime
numbers are: **2, 3, 5, 7, 11, 13, 17,
19, …**.

Looking at the list,
we might conjecture that **2** is the only even prime. (Can you prove it? It’s
true.)

Any whole number,
which is not a prime, is divisible by at least one prime: **4** is divisible by **2**;
**6** by **2** and **3**, **8** by
**2**, **9**
by **3**, **10** by **2** and **5**, etc. (Can you prove it?
It’s true.)

We also might
conjecture that any odd prime differs by **2** from some other odd prime. It seems to be true.

So let’s extend the
list: **2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, …**

OOPS! **23** falsifies that conjecture, as does
**37**. So let’s extend the list some more:

**2, 3, 5, 7, 11, 13,
17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97,
101, …**

We keep finding
bigger and bigger primes. Can we find
one over **1000**. A little playing with a
calculator shows that **1009** is a prime.

So we form the
conjecture that there is no largest prime.
Can we deductively prove it?

Yes! Let’s suppose there is a largest prime –
call it **P**. Now, do the following
multiplication:

**2*3*****5*7*****11*13*****17*19*****23*29*****31*37*****41*43*****47*53*****59*61*****67*71*****73*79*****83*89*****97*101***** … **
*P

It might take a long
time to compute! But call this product
**N**. Notice that each of the primes does
divide **N** evenly. But how about
**N+1**? None of the primes divides it
evenly since they divide **N**. So
**N+1**
must be a new prime. The assumption
that **P** is the largest prime is contradicted and must be false. There is no largest prime. There are infinitely many prime numbers.

Let’s look at another
curious property of prime numbers.
Consider the even numbers greater than **2**:

**4 = 2 + 2**

**6 = 3 + 3**

**8 = 5 + 3**

**10 = 5 + 5 = 7 + 3**

**12 = 7 + 5**

**14 = 7 + 7 = 11 + 3**

**16 = 11 + 5 = 13 + 3**

**18 = 11 + 7 = 13 + 5**

**20 = 13 + 7 = 17 + 3**

**22 = 11 + 11 = 17 + 5
= 19 + 3**

Every even number
greater than **2** appears to be the sum of two prime numbers (sometimes in more than one
way). That this is always true is
known as Goldbach’s Conjecture. No
counter-example is known – computers have checked all numbers up to the trillions. But no one has been able to prove it is
always true. No one knows if it is
always true.

This will be one of the lessons of this course. We will look at lots of examples. We will form conjectures. Sometimes the conjectures will be proven to be true and sometimes counter-examples will be found. And sometimes we will be left in limbo – no counter-examples and no proof. There are many unsolved problems in mathematics. Not every question has an answer.

©James S. Freeman, 2006