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OBJECTIVES:

•Students will be able to arrive at a conclusion from premises that are given and accepted.

•Students will be able to put several related conditional statements into logical order.

•Students will be able to critique deductive arguments for validity.

LESSON 25: Deductive Reasoning

Vocabulary:


A conditional statement is a statement in the if-then form where the if-clause = the hypothesis and the then-clause = the conclusion. (see pg. 203)

A conclusion asserts that the general statement, which applies to the group, also applies to the individual case. (see pg. 202)

Deductive reasoning is the process of arriving at a conclusion from premises that are given and accepted. (see pg. 202)

A major premise is a general statement about an entire group. (see pg. 202)

A minor premise is a specific statement indicating that an individual case belongs to the group. (see pg. 202)

What is a conditional statement? See page 202

What is a conclusion? See page 202

What is deductive reasoning? See page 202

What is a major premise? See page 202

What is a minor premise? See page 202

Vocabulary:


A syllogism [sil-uh-jiz-uhm] is a simple form of deductive reasoning which combines a major premise and a minor premise to reach a conclusion. (see pg. 202)

Valid reasoning arrives at conclusions that, based on the given premises and conditions, are absolutely true. (see pg. 206)

Invalid reasoning arrives at conclusions that, based on the given premises and conditions, are false or not necessarily true. (see pg. 206)

Critiquing is the process of evaluating deductive arguments for their validity. (see pg. 206)

What is a syllogism? See page 202

What is valid reasoning? See page 206

What is invalid reasoning? See page 206

What is critiquing? See page 206


Syllogism involve combining a major premise and a minor premise to reach a conclusion:

All cars have 4 wheels

Major Premise: General statement about an entire groupMinor Premise: Specific statement indicating an individual case belongs to a groupExample:

This is an example of a major premise because it is a statement about an entire group all cars.

Is this a major or minor premise?

My Nissan Sentra is a car

This is an example of a Minor premise because it is a statement about my Nissan Sentra

Is this a major or minor premise?

Therefore, a Nissan Sentra has four wheelsThis conclusion can be made by combining the major and minor premise.

LESSON 25: Conditional StatementsA conditional statement is written in the form if p, then q. If a given condition is met (if p), then another condition is true or an event will happen (then q).

The if-clause is the hypothesis; the then-clause is the conclusion. Conditional statements can be linked with minor premises to make conclusions.

Example

If tomorrow is Friday, then today is Thursday

What is your hypothesis? See page 203What is your conclusion? See page 203

Tomorrow is Friday, - Minor PremiseTherefore, today is Thursday - Conclusion

LESSON 25: Conditional StatementsA conditional statement is written in the form if p, then q. If a given condition is met (if p), then another condition is true or an event will happen (then q).

The if-clause is the hypothesis; the then-clause is the conclusion. Conditional statements can be linked with minor premises to make conclusions.

Example 2)

If a Summit student is in class then it is a weekday

What is your hypothesis? See page 203What is your conclusion? See page 203

A Summit student is in class, - Minor PremiseTherefore, it is a weekday - Conclusion

Can someone think of another conditional statement?

LESSON 25: Logical Order of Conditional StatementsWhen given several related conditional statements, you must put them into logical order.

Example: Put the following statement in proper logical order. – Follow along with example in your textbook

Look at the hypotheses and conclusions: The conclusion of one statement will flow into the hypothesis of the next.

What do you look at in order to put conditional statements in logical order? See page 204

A. If Grant can play songs, then he can play in a band.B. If Grant takes guitar lessons, then he can play chords.C. If Grant can play in a band, then he can play a concert.D. If Grant can play chords, then he can play songs.E. If Grant buys a guitar, then he can take guitar lessons.* You are looking for the conclusion (blue) of one statement

flowing into the hypothesis (red) of another statement.

Which conclusion does not flow into a hypothesis?If Grant can play in a band, then he can play a concert. The conclusion does not flow into a hypothesis so we know this is the last statement in logical order.

LESSON 25: Logical Order of Conditional StatementsA. If Grant can play songs, then he can play in a band.B. If Grant takes guitar lessons, then he can play chords.C. If Grant can play in a band, then he can play a concert.D. If Grant can play chords, then he can play songs.E. If Grant buys a guitar, then he can take guitar lessons.

We know that A leads into CB leads into DD leads into AE leads into B

C is the last one

C. If Grant can play in a band, then he can play a concert.

Which statement should go directly before C?

A. If Grant can play songs, then he can play in a band.

Which statement should go directly before A?

D. If Grant can play chords, then he can play songs.

Which statement should go directly before D?

B. If Grant can take guitar lessons, then he can play chords.

Which statement should go directly before B?

D. If Grant buys a guitar, then he can take guitar lessons.

LESSON 25: Logical Order of Conditional Statement

The statements in proper logical order:

E. If Grant buys a guitar, then he can take guitar lessons. conclusion

B. If Grant takes guitar lessons, then he can play chords.hypothesis conclusion

D. If Grant can play chords, then he can play songs.

hypothesis conclusion

A. If Grant can play songs, then he can play in a band.hypothesis conclusion

C. If Grant can play in a band, then he can play a concert. hypothesis

LESSON 25: Critiquing for Validity

Remember during our vocabulary review that Deductive reasoning can either be valid or invalid.

We will now learn how to critique deductive arguments to determine if they are valid or invalid.Example: This is an example of valid reasoning:

If a 4-sided polygon has 4 right angles, then it is a rectangle. Polygon ABCD has four right anglesTherefore, polygon ABCD is a rectangle.

Is the first statement true, assumed to be true or not necessarily true?

The first statement is true by definition of a rectangle

TRUE

Is the second statement true, assumed to be true or not necessarily true?

The second statement is assumed true because it is given

ASSUMED TRUE

Is the third statement true, assumed to be true or not necessarily true?

The third statement is true because the first statement which is true tells us that a polygon with 4 right angles is a rectangle and Polygon ABCD has four right angles.

TRUE

LESSON 25: Critiquing for ValidityExample: This is an example of invalid reasoning: Follow along on page 206

A person who lives in Arizona also lives in the United States. Mike lives in the United States. Therefore, Mike lives in Arizona.

Is the first statement true, assumed to be true or not necessarily true?

The first statement is true. Everybody who lives in Arizona also lives in the United States.

TRUE

Is the second statement true, assumed to be true or not necessarily true?

The second statement is assumed to be true because Mike can live in the United States

ASSUMED TRUE

Is the third statement true, assumed to be true or not necessarily true?

The third statement is not necessarily true because Mike can live in Florida. Florida is still in the United States

NOT NECESSARILY TRUE

LESSON 25: Converse, Inverse, Contrapositive

Recall that the form of a conditional statement if p then q.You can make new conditional statements by switching p and q, negating both p and q or by switching and negating at the same time

The converse f a conditional statement is formed by switching the places of the hypothesis and the conclusion. The sentence if p, then q becomes if q, then p

What is a Converse or a conditional statement? See page 208

Example:If the phone rings then someone is calling you

ConverseIf someone is calling you then the phone is ringing

If a number is even then it is divisible by 2

Who can tell me the converse of this statement?If a number is divisible by 2 then it is even


The converse of a true statement does not always produce a true statement or the converse of a false statement does not always produce a false statement.

If the phone rings then someone is calling you

If someone is calling you then the phone is ringing


If a number is divisible by 2 then it is even

Is the converse of the statement true or false?The converse is false because your phone can be on silent.

Is the converse of the statement true or false?The converse is true by definition of an even number.


The inverse of a conditional statement is formed by negating the hypothesis and the conclusion. The sentence if p, then q becomes if not p then not q.

The inverse is written as if ~ p then ~ q


How would you write the inverse of this statement? See page 210

If someone is not calling you then the phone is not ringing


If a number is not divisible by 2 then it is not even

How would you write the inverse of this statement? See page 210

Just like the converse, a true statement can produce a false inverse and a false statement can produce a true inverse


The contrapositive of a conditional statement is formed two steps.

Step 1: Form the converse


How would you write the contrapositive of this statement? See page 211

If the phone is ringing then someone is calling youStep 1: Form the Converse

Step 2: Form the inverse of the converse

If the phone is not ringing then someone is not calling you

Step 2: Form the Inverse of the Converse

The contrapositive is different then the converse and inverse. A true statement will always produce a true contrapositive and a false statement will always produce a false contrapositive

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