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.

## Intuitive Reasoning

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.

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..

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.

# Inductive and Deductive Reasoning in Mathematics

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:

02, 12, 22, 42, 52, 62, ...

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 n2 is a perfect square, the next perfect square is (n+1)2.  Their difference is:

(n+1)2 - n2  = (n2 + 2n +1) - n2 = n2 - n2 +2n + 1 = 2n + 1.

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.