Chap 9 - Categorical Logic

8/12/2019 Chap 9 - Categorical Logic

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REVIEW OF CHAPTER 3

Deductive arguments

Patterns of deductive arguments

Inductive arguments

Patterns of inductive arguments


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Chapter 9: Categorical Logic

Part 1: Categorical propositions

Part 2: Translating into standard categorical form

Part 3: Testing validity with Venn Diagrams


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Part 1: Categorical propositions

Categorical propositions make declarations about

entities belonging to, or not belonging to, categories or

classes. Each categorical proposition has 4 basic parts:

1.Quantifier: all, no or some2.Subject: (S)

3.Predicate: (P)

4.Corpula: Linking verb

Ex: AllIU studentsareCritical Thinking learners.

1 2 4 3


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Venn Diagrams for Categorical Propositions

Venn diagrams, invented by John Venn, is a

very useful method of diagramming the

informational content of categoricalpropositions.

A Venn diagram for a categorical proposition

consists of 2 overlapping circles.


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A Venn diagram for 2 classes, S and P


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Two simple rules governing Venn diagrams:

1. Shade a region to show that it is empty.

2. Place an X in a region to show that it is occupied

by some.

X

1 2


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A- All S are P


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E- No S are P


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I-Some S are P


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O- Some S are not P


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Categorical Claims

"All S are P" (A): "The class of S outside of P is empty."

"No S are P" (E): "The class of S inside P is empty."

"Some S are P" (I): "The class of S inside P has at least one

member."

"Some S are not P" (O): "The class of S outside of P has at least

one member.


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Common stylistic variants

A: All S are PEvery S is aP. Whoever is an S is aP.

Whatever is an S is aP. If anything is an S, then it is aP.

Any S is aP. If something is not aP, then it is not an S.

Each S is aP. S are allP.

S are alwaysP. OnlyP are S.

The only S areP. Only if something is aP is it an S.

Something is an S only if it is aP.


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SomeP are S.

A few S areP.

There are S that areP.

Several S areP.

Many S areP.

Most S areP.Nearly all S areP.

A: Some S are P


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Notall S areP.

Noteveryone who is an S is aP.

S are notalwaysP.

Some S are non-P.

There are S that are notP.

A few S are notP.Several S are notP.

Most S are notP.

Nearly all S are notP.

A: Some S are NOT P


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Practice: Translate the following only sentences

into standard categorical form.1. Only doctors are psychiatrists.

2. Only fools rush in.

3. Employees restroom only.

4. None except senior citizens are eligible for the discount.

5. Teachers alone may use the Teachers Lounge.


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Practice: Translate the following only sentences

into standard categorical form.1. Only doctors are psychiatrists.

All psychiatrists are doctors.

2. Only fools rush in.All people who rush in are fools.

3. Employees restroom only.

All people who use the restroom are employees.

4. None except senior citizens are eligible for the discount.All people who are eligible for the discount are senior citizens.

5. Teachers alone may use the Teachers Lounge.

All people who may use the Teachers Lounge are teachers.


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Practice: Translate the following sentences into

standard categorical form.1. There are birds that cannot sing. (Q.13, p236)

2. The grass is always greener on the other side. (Q.16, p236)

3. Polar bears live in Canada. (Q.19, p236)

4. If you dont learn this lesson, youre not here today.

5. Not all friendly teachers are easy-going.


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Practice: Translate the following sentences into

standard categorical form.1. There are birds that cannot sing. (Q.13, p236)

Some birds are animals that cannot sing.

2. The grass is always greener on the other side. (Q.16, p236)All places on the other side are places where grass is greener.

3. Polar bears live in Canada. (Q.19, p236)

Some polar bears are animals which live in Canada.

4. If you dont learn this lesson, youre not here today.All non-learners of this lessons are absentees today.

5. Not all friendly teachers are easy-going.

Some friendly teachers are not easy-going teachers.


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Tip 1:Rephrase all nonstandard subject and

predicate terms so that they refer to categories.

Some roses are white.

Some roses are white flowers.S P


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Tip 3: Fill in any unexpressed quantifiers.

Vietnamese people are friendly.

Some Vietnamese people are friendly people.

Q S C P


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Tip 4: Translate singular statements as all or no

statements.

Paris is the capital of France.

All places identical with Paris are places that are

Q S C Pthe capital of France.


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Tip 5: Translate stylistic variants into the

appropriate categorical form.

Every S is a P.

S are always P. All S are P.

Any S is a P.


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Part 3: Testing validity using Venn Diagrams


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Testing validity of categorical syllogism

A categorical deductive argument with two

premises and a conclusion.

Ex: All snakes (S) are reptiles (R).

All reptiles are cold-blooded animals (C).

So, all snakes are cold-blooded animals.


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All S are R.All R are C.

All S are C.

The two lower circles represent the

two categories in the conclusion.

Sample 1

All snakes (S) are reptiles (R).




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There are three steps in this process:

1. Draw premise one.

2. Draw premise two.

3. Check the validity.


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Step 1.- Shade the area where

snakes are not reptiles.

All S are R.

All R are C.

All S are C.





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All S are R.

All R are C.

All S are C.




Step 2.

- Shade the area where

reptiles are not cold-blooded

animals.


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All S are R.

All R are C.

All S are C.




Step 3.

- The conclusion tells us that all snakes

are cold-blooded animals. The area ofthe Snake circle that does not overlap

the Cold-blooded circle must be shaded.

- The drawing clearly shows that the

conclusion is necessarily true.This is a valid syllogism.


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All educated people respect books.

Some bookstore personnel are not truly educated.Some bookstore personnel dont respect books.

E R

B

All E are RSome B are not E

Some B are not R

Translated intostandard form

Be clear that:E = Educated people

R = People who respect books

B = Bookstore personnel

Sample 2


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All E are R

Some B are not E

Some B are not R

Draw the first premise. All

E are inside R, so we know

that the rest of E is empty.

We represent this empty

area by shading it.

E

B R


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All E are R

Some B are not E

Some B are not R

Now the second premise.

We read some as at least

one and represent it with an

X. So we want to put an X

inside the B circle but outside

of the E circle.

X

E

B R

X


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All E are R

Some B are not E

Some B are not R

So, is the conclusion necessarily

true? Is it true that some B are

not R?

No, this is an invalid argument.

The X shows that there maybesome B that are not R, but not necessarily.

E

B R

X


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No islands are part of the mainland. Hawaii is an

island. Therefore, Hawaii is not on the mainland.

Translated into

standard form

No I are M

All H are I

No H are M

I M H

I

MH

Be clear that:

I = Islands

M = Mainland placesH = (Places identical to) Hawaii

Sample 3


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No I are M

All H are I

No H are M

Step 2: Draw the second

premise.

Step 3: Check if the part in H not overlapping M is shaded?

Yes, this is a valid argument!

I

MH

Step 1: Draw the first

premise.

I

H M

S l 4


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Some modems are cable connections. Some cable

connections are digital. Thus, some modems are digital.

Some M are C

Some C are D

Some M are D

M C

D

Sample 4

M C


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Some modems are cable connections. Some cable

connections are digital. Thus, some modems are digital.

Translated into

standard form

Some M are C

Some C are D

Some M are D

M C

D

M D

C

X

M D

C

X X

Invalid


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If you apply the step by step approach to using Venn

Diagrams you will quickly become an expert. Keep

these things in mind:

1. Put your syllogism in standard form first.

2. Be consistent in how you draw your diagram: alwaysshade the premises with No and All before putting the X

for Some.

3. Test validity by looking for the necessity of the conclusion.


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PRACTICE

Translate the following into standard categorical form.

Then use Venn diagrams to test their validity.


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Translate the argument into standard form,

and test its validity

All my Critical Thinking students are bright guys and girls.

No bright guys and girls sleep in class.

Therefore, no Critical Thinking students sleep in class.

All CT students are bright people.

No bright people are class sleepers.

Therefore, no CT students are class sleepers.


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All men who dont give flowers to their sweethearts on March 8 are

not romantic.

Some unromantic men are not ideal partners.

Therefore, some men who dont give flowers to their sweetheartson March 8 are not ideal partners!

All no-flower men on March 8 are unromantic people.

Some unromantic people are not ideal partners.

Therefore, some no-flower men on March 8

are not ideal partners.


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Some students who register for Critical Thinking are frequentlyabsent.

All students who are frequently absent cannot take the tests.

Therefore, some students who register for Critical Thinking cannot

take the test.

Some CT students are frequent absentees.

All frequent absentees are not test takers.

Therefore, some CT students are not test takers.

Cl k


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Classwork

There are e-mail messages that are not spell-checked. There are interoffice

memos that are e-mail messages. Therefore, there are interoffice memos that

are not spell-checked.

If anything is a truck, then it is not a car. There are Mazdas that are trucks. It

follows that there are Mazdas that are not cars.

Every person who drinks and drives is an irresponsible person. Not every

person who talks on a car phone is an irresponsible person. Hence, not every

person who talks on a car phone is a person who drinks and drives.

Joey is in kindergarten. Only children in kindergarten fingerpaint in school.

So, Joey fingerpaints in school.


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FURTHER PRACTICE

Use Venn diagrams to test the validity of the following arguments.


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No fish are reptiles. All trout are fish. So, some

trout are not reptiles.

No fish are reptiles.

All trout are fish.

So, some trout are not reptiles.

Test the validity of the argument


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No one who is a Nobel Prize winner is a rock star. A number of

astrophysicists are Nobel Prize winners. Therefore, a number of

astrophysicists are not rock stars.

No Nobel Prize winners are rock stars.

Some astrophysicists are Nobel Prize winners .

Therefore, some astrophysicists are not rock stars.


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At least one lawyer is not a golfer. Only persons who have attended

law school are lawyers. So, at least one person who has attended

law school is not a golfer.

Some lawyers are not golfers.

All lawyers are persons who have attended law school.

So, some persons who have attended law school are not golfers.


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Every tax evader is a lawbreaker. Hence, because no one who is a

lawbreaker is a model citizen, no one who is a model citizen is a

tax evader.

All tax evaders are lawbreakers.

No lawbreakers are model citizens.

So, no model citizens are tax evaders.

Chap 9 - Categorical Logic

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