Problem Solving and Critical
“Inductive reasoning” is the process of
arriving at a general conclusion based
on observations of specific examples.
You purchased textbooks for 4 classes.
Each book cost more than $50.00.
Conclusion: College textbooks cost more
There is no guarantee that the conclusions
reached by “inductive reasoning” are correct
with no exceptions.
Even though all your textbooks this term
cost more than $50.00 it is still possible (and
even probable) that a textbook for some
course will cost less than $50.00.
The conclusion is really a conjecture or
hypothesis and a case where the conclusion
is not true is a counterexample.
It takes one counterexample to show a
hypothesis is false.
In mathematics “inductive reasoning” is often
used to find patterns.
Find the pattern in the following sequence of 6
numbers and use that pattern to decide what the
next number should be.
Each number is obtained from the previous one
by multiplying by 4. The next number is
“Deductive reasoning” is the process of
proving a specific conclusion from one or
more general statements. A conclusion that is
proved true by deductive reasoning is called
a theorem. For example:
The catalog states that all entering freshmen
must take a mathematics placement test.
You are an entering freshman.
Conclusion: You will have to take a
mathematics placement test.
Examples from Mathematics
Suppose 3x = 12. We conclude x = 4.
The length of a rectangle is 6 and its width
is 5. We conclude its area is 30.
Select a number. Multiply it times 10.
Subtract 6 from the result. Divide what
you have by 2. Add 3 to this.
Repeat a few times with other numbers and
make a conclusion [hypothesis].
What type of reasoning did you use.
Show why this works. Relate deduction to
logic and proof.
Can you think of examples of inductive and
deductive reasoning in your areas of
interest? In social sciences, humanities,
arts, and sciences?