Chapter 1

Mathematical Reasoning

Section 1.2

Deductive Reasoning

Deductive Reasoning is one of the most important tools that is used to establish mathematical facts and results. Previously we said that deductive reasoning uses a collection of general principles (called a hypothesis or premise) to generate a conclusion about something.

In the examples we will look at the hypothesis will be a series of statements. In order to help us reason I want to show you how we can draw a “picture” or diagram of certain types of statements that relate categories of things.

Statement: Statement:All professors are millionaires. All millionaires are professors.

millionaires

professors millionaires

professors

This is called a Venn Diagram for a statement. In the diagram the outside box or rectangle is used to represent everything. Circles are used to represent general collections of things. Dots are used to represent specific items in a collection. There are 3 basic ways that categories of things can fit together.

women

democrats

women democrats

women democrats

All democrats are women. Some democrats are women. No democrats are women.

One circle inside another. Circles overlap. Circles do not touch.

democrats

Laura Bush

women

George Bush

Laura Bush is a democrat. George Bush is not a woman.

What statement do each of the following Venn Diagrams depict?

Some of the statements depicted are true and some are false. The point here is draw what each statement is logically saying. We want to learn how to reason regardless of the truth of the statements we are using to reason. There are certain key words we can look for in a statement to determine which of the diagrams we are going to use.

Key wordsAllEveryEveryone

Key wordsSomeA fewThere are

Key wordsNone, NoEveryNobody

Putting more than one statement in a diagram. (A previous example)

Hypothesis:

Dr. Daquila voted in the last election.

Only people over 18 years old vote.

Conclusion:

Dr. Daquila is over 18 years old.

Dr. Daquila

voters

people over 18

IF__THEN__ Statement Construction

Many categorical statements can be made using an if then sentence construction. For example if we have the statement: All tigers are cats.

cats

tigers

This can be written as an If_then_ statement in the following way:

If it is a tiger then it is a cat.

This is consistent with how we have been thinking of logical statements. In fact the parts of this statement even have the same names:

If it is a tiger then it is a cat.

The phrase “it is a tiger” is called the hypothesis. The phrase “it is a cat” is called the conclusion.

One method that can be used to determine if a statement can be deduced from a collection of statements that form a hypothesis we draw the Venn Diagram in all possible ways that the hypothesis would allow. If any of the ways we have drawn is inconsistent with the conclusion the statement can not be deduced.

Example

Hypothesis:

All football players are talented people.

Pittsburg Steelers are talented people.

Conclusion:

Pittsburg Steelers are football players. (CAN NOT BE DEDUCED!)

talented people

football players

Pittsburg Steelers

talented people

football players

Pittsburg Steelers

talented people

football players

Pittsburg Steelers

Even though one of the ways is consistent with the conclusion there is at least one that is not so this statement can not be deduced.

Can the following statement be deduced?

Hypothesis:

If you are cool then you sit in the back.

If you sit in the back then you can’t see.

Conclusion

If you are cool then you can’t see.

There is only one way this can be drawn! So it can be deduced!

people who can’t see

people who sit in back

cool people