Today’s Lesson is on

In Chapter 1 you learned that inductive reasoning is based on observing what has happened and then making a conjecture about what will happen. In this lesson, youwill study deductive reasoning. Deductive reasoning (or logical reasoning) is the process of reasoning logicallyfrom given statements to a conclusion. If the given statements are true, deductive reasoning produces a true conclusion. Many people use deductive reasoning in their jobs. A physician diagnosing a patient’s illness uses deductive reasoning. A carpenter uses deductive reasoning to determine what materials are needed at a work site.

Law of Detachment• Auto Maintenance An auto mechanic knows

that if a car has a dead battery, the car will not start. A mechanic begins work on a car and finds the battery is dead.

• What conclusion can she make?• Exactly: The mechanic can conclude that the

car will not start.• Critical Thinking Suppose that a mechanic

begins work on a car and finds that• the car will not start. Can the mechanic

conclude that the car has a dead battery?• Explain.

The mechanic is using a law of deductive reasoning called

antecedent consequent

P QP_______Q

Examples

• If a baseball player is a pitcher, then that player can not pitch a complete game two days in a row. Vladimir Nuñez is a pitcher. On Monday, he pitches a complete game.What can you conclude?

• For the given true statements, what can you conclude?

• Given: If M is the midpoint of a segment, then it divides the segment into two congruent segments.

• M is the midpoint of segment AB.

Does the following argument illustrate the Law of Detachment?

• Given: If it is snowing, then the temperature is less than or equal to 32°F.

• The temperature is 20°F.• You conclude: It must

be snowing.

• You are given that a conditional and its conclusion are true.

• You cannot apply the Law of Detachment and conclude that the hypothesis is true.

• You cannot come to any conclusion about whether it is snowing from the

• information given.

The error illustrated is sometimescalled the fallacy of the converse.A fallacy is an error in logical thinking. Why is this error a “fallacy of the converse”?The fallacy is concluding that theconverse is true because theconditional is true.

PQ

Q

Can you assume that P is true based on this law of detachment?

P

Qx

x

If possible, use the Law of Detachment to draw a conclusion. If it is not possible to

use this law, explain why.• Given: If a road is icy, then driving

conditions are hazardous.

• Driving conditions are hazardous.

Another law of deductive reasoning is

• The Law of Syllogism allows you to state a conclusion from two true conditional statements when the consequent of one statement is the antecedent of the other statement.

Algebra Use the Law of Syllogism to draw a conclusion from the following

true statements.

• If a number is prime, then it does not have repeated factors.

• If a number does not have repeated factors, then it is not a perfect square.

• What conclusion can you draw from this information?

If possible, state a conclusion using the Law of Syllogism. If it is not possible to use

this law, explain why.

• If a number ends in 0, then it is divisible by 10.

• If a number is divisible by 10, then it is divisible by 5.

• If a number ends in 6, then it is divisible by 2.

• If a number ends in 4, then it is divisible by 2

Try this: and explain your conclusions by stating

which law(s) were used • If a quadrilateral is a

square, then it contains four right angles.

• If a quadrilateral contains four right angles, then it is a rectangle.

• PQ, QR, Then• PR

Try this: and explain your conclusions by stating

which law(s) were used • If the circus is in

town, then there are tents at the fairground.

• If there are tents at the fairground, then Paul is working as a night watchman.

• The circus is in town.

If the circus is in town, then Paul is working as a night watchman.

Look at the symbolic representation • CT• TP• This means that using

the Law of Syllogism you can get that

• CP• C• Using the Law of

Detachment, you can get P

• Therefore, Paul is working as a night watchman.

• If the circus is in town, then there are tents at the fairground.

• If there are tents at the fairground, then Paul is working as a night watchman.

• If the circus is in town, then Paul is working as a night watchman.

• The circus is in town.

Deductive reasoning is a processof reasoning logically from givenfacts to a conclusion. The Law ofDetachment is found in almostevery line of two-column proofs,where q is the “conclusion” andp q is the “justification” for q.It is p, the given, that is sometimeslost when thinking about each line.

The use of the word Givenbefore the two statements is an important concept that I just figured you would understand. However, in mathematics, the statements following Given are always considered true. What do you call statements that areassumed to be true without proof?