Categorical Logic Part I

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Categorical Logic Part I. Terms. What is logic? “Logic is the study of the methods and principles used to distinguish correct from incorrect reasoning.” ( Copi , Introduction to Logic ). What is inference?

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### Categorical Logic Part I

Terms

What is logic?

“Logic is the study of the methods and principles used to distinguish correct from incorrect reasoning.”

• (Copi, Introduction to Logic)

What is inference?

“A process of linking propositions by affirming one proposition on the basis of one or more other propositions.”

• (Copi, Introduction to Logic)
Clarification of our Purpose

What is the value of reason and logic?

What, if anything, can logic tell us?

What, if anything, can logic and reason tell us about the nature of the world “out there?”

What, if anything, can logic and reason tell us about truth?

Categorical Logic

Concerned with relations of inclusion and exclusion among classes (or categories).

Categorical Logic vs. Propositional Logic

Propositional logic (or truth-functional logic) considers a statement the smallest indivisible unit and studies ways of joining & modifying statements to form other statements.

If Simon is a man (S), then he’s going to die (M).

Or “If S, then M.” Or “S → M.”

Standard Form Categorical Claims
• A: All_____ are _______.
• E: No _____ are _______.
• I: Some _____ are _____.
• O: Some ____are not ___.

Subject Term

Predicate Term

Affirmative Claim

Negative Claim

Equivalent Claims

Let’s review Exercise 9-2.

Exercise 9-2
• Every salamander is a lizard.
• Not every lizard is a salamander.
• Only reptiles can be lizards.
• Snakes are the only members of the suborder Ophidia.
• The only members of the suborder Ophidia are snakes.
• All salamanders are lizards.
• Some lizards are not salamanders.
• All lizards are reptiles.
• All members of the suborder Ophidia are snakes.
• All members of the suborder Ophidia are snakes.

6. None of the burrowing snakes is poisonous.

7. Anything that’s an alligator is a reptile.

8. Anything that qualifies as a frog qualifies as an amphibian.

9. There are frogs wherever there are snakes.

10. Wherever there are snakes, there are frogs.

11. Whenever the frog population decreases, the snake population decreases.

12. Nobody arrived except the cheerleaders.

6. No burrowing snakes are poisonous snakes.

7. All alligators are reptiles.

8. All frogs are amphibians.

9. All places there are snakes are places there are frogs.

10. All places there are snakes are places there are frogs.

11. All times the frog population decreases are times the snake population decreases.

12. All people who arrived are cheerleaders.

13. Except for vice-presidents, nobody got raises.

14. Unless people arrived early, they couldn’t get seats.

15. Most home movies are as boring as dirt.

16. Socrates is a Greek.

17. The bank robber is not Jane’s fiance.

18. If an automobile was built before 1950, it’s an antique.

19. Salt is a meat preservative.

20. Most corn does not make good popcorn.

13. All people who got raises are vice presidents.

14. All people who got seats are people who arrived early.

15. Some home movies are things that are as boring as dirt.

16. All people identical with Socrates are Greeks.

17. No people identical to the bank robber are people identical to Jane’s fiancé.

18. All automobiles built before 1950 are antiques.

19. All examples of salt are things that preserve meat.

20. Some examples of corn are not things that make good popcorn.

Exercise 9-5

Complementary Class: All things not included in another class.

Class: Students

Complementary Class: Non-Students

Contraposition: Switch S & P term positions and replace with complementary terms.

But…while all A- and O-claims are equivalent to their contrapositives, no E- and I-claims are equivalent to their contrapositives.

• No Sunnis are Christians.

6. Some Indians are not Hindus.

8. All Catholics are Christians.

Class: Students

Complementary Class: Non-Students

Obversion: Change claim from affirmative to negative (of vice-versa) and replace the predicate term with its complementary term.

All A-, E-, I-, O- claims are equivalent to their obverses.

2. Some Arabs are Christians.

3. All Sunnis are Muslims.

10. No Muslims are Christians.

Conversion: Switch positions of S & P terms.

But….while all E- and I-claims are equivalent to their converse claims, no A- and O-claims are equivalent to their converses.

4. Some Kurds are not Christians.

5. No Hindus are Muslims.

7. All Shiites are Muslims.

9. All Protestants are Christians.

Square of Opposition

A & E claims can both be false but cannot both be true.

I & O claims can both be true but cannot both be false.

A & O and E & I claims can never have the same truth value.

There’s not such thing as a harmless drug. (translate to standard form?)

No drugs are drugs that are completely harmless. (True)

• A-claim is________;
• I-claim is ________;
• O-claim is _______.

A or E is true, then we can infer the remaining truth values.

I or O is false, then we can infer the remaining truth values.

Logic exercises are easy. (translate to standard form)

All logic exercises are exercises that are easy. (false)

E-claim is________;

I-claim is ________;

O-claim is _______.

Exercise 9-11
• Whenever the battery is dead, the screen goes blank; that means, of course, that whenever the screen goes blank, the battery is dead.

Valid?

How was the 2nd claim derived from the first: conversion, obversion, or contraposition?

Requires the conversion of an A-claim: invalid.

2. For a while there, some students were desperate for good grades, which meant some weren’t right?

Valid?

What kind of claim is the first claim?

What does a “true” I-claim allow us to infer about A-, E-, and O-claims?

All other claims are indeterminate.

3. Some players in the last election weren’t members of the Reform Party. Obviously, therefore, some members of the Reform Party weren’t players in the last election.

Valid?

What kind of claim is the first claim?

How was the 2nd claim derived from the first: conversion, obversion, or contraposition?

Can an O-claim be converted?

Invalid?

4. Since some of the students who failed the exam were students who didn’t attend to review session, it must be that some students who weren’t at the review session failed the exam.

Valid?

What kind of claim is the first claim?

How was the 2nd claim derived from the first: conversion, obversion, or contraposition?

Can an I-claim be converted?

Valid?

5. None of the people who arrived late were people who got good seats, so none of the good seats were occupied by late comers.

Valid?

What kind of claim is the first claim?

How was the 2nd claim derived from the first: conversion, obversion, or contraposition?

Can an E-claim be converted?

Valid?

6. Everybody who arrived on time was given a box lunch, so the people who did not get a box lunch were those who didn’t there on time.

Valid?

What kind of claim is the first claim?

How was the 2nd claim derived from the first: conversion, obversion, or contraposition?

Can an A-claim be contraposed?

Valid?

7. None of the people who gave blood are people who were tested, so everybody who gave blood must have been untested.

Valid?

What kind of claim is the first claim?

How was the 2nd claim derived from the first: conversion, obversion, or contraposition?

Can an E-claim be obverted?

Valid?

8. Some of the people who were not tested are people who were allowed to give blood, from which it follows that some of the people who were not allowed to give blood must have been people who were tested.

Valid?

What kind of claim is the first claim?

How was the 2nd claim derived from the first: conversion, obversion, or contraposition?

Can an I-claim be contraposed?

Invalid?

9. Everybody who was in uniform was able to play, so nobody who was out of uniform must have been able to play.

Valid?

What kind of claim is the first claim?

How was the 2nd claim derived from the first: conversion, obversion, or contraposition?

Can an A-claim be converted?

Invalid?

10. Not everybody in uniform was allowed to play, so some people who were not allowed to play must have not been people in uniform.

Valid?

What kind of claim is the first claim.

What kind of claim is the second claim?

Does a “true” value of an I- or O-claim imply the truth value of any other claims?

Indeterminate.

Invalid.

Translate to Standard Form

Students who wrote poor exams didn’t get admitted to the program.

No students who wrote poor exams are students who were admitted to the program.

• All students who wrote poor exams are students who were not admitted to the program. (by obversion)
Translate to Standard Form

If you live in the dorms, you can’t own a car.

No people who live in the dorms are people who own cars.

All people who live in the dorms are people who do not own cars. (by obversion)

All people who own cars are people who do not live in the dorms. (by contraposition)

Translate to Standard Form

There are a few left-handed students in this class.

Some students in this class are students who are left-handed.

Some students in this class are students who are not right-handed (or non left-handed). (by obversion)

• Some students who are left-handed are students who are in this class. (by conversion)
Translate to Standard Form

I’ve had days like this before.

Some days I’ve had are days like this day.

Some days I’ve had are not days unlike this day. (by obversion)

Some days like this day are days I’ve had. (by conversion)

Categorical Syllogism

A two-premise deductive argument whose every claim is standard form categorical claim.

Three terms occur exactly twice in exactly two of the claims.

Which of the following is a categorical syllogism?

All PHIL 1 students are honest.

Some PHIL 1 students are hard-working.

Therefore, some hard-working students are honest.

All PHIL 1 students are honest.

Many PHIL 1 students are hard-working.

Therefore, many PHIL 1 students are honest and hard-working.

Validity: not possible for premises to be true while the conclusion is false.

Major term is predicate term of conclusion.

Minor term is subject term of conclusion.

Middle term is term of premises but not conclusion.

Some CR students are men.

Some CR students are women.

Some men are women.

All CR students are men.

Some CR students are superheroes.

Some men are superheroes.

Some CR students are zombies.

What is the conclusion AT&T is hoping we’ll supply?

Translate premises to standard form.

Is the argument valid?

Why would AT&T leave the premise unstated?

How could we make this into a valid argument?

Invalid Forms

All A’s are X.

All B’s are X.

Therefore, all B’s are A’s.

All A’s are X.

No A’s are Y.

Therefore, no X’s are Y’s.

All X’s are Y’s.

Therefore, all Y’s are X’s.

Some X’s are not Y’s.

Therefore, some Y’s are not X’s.

Some X’s are Y’s.

Therefore, some X’s are not Y’s.

Some X’s are not Y’s.

Therefore, some X’s are Y’s.

What follows from the premises?

All business executives have accounting experience, and some business executives are not economists.

A. Some economists do not have accounting experience.

B. Some people with accounting experience are not economists.

C. All people with accounting experience are business executives.

D. More than one of these.

E. None of these.

What premise must be added for validity?

Coffee is a stimulant, since coffee contains caffeine.

A. All substances that contain caffeine are stimulants.

B. All stimulants are substances that contain caffeine.

C. Neither creates validity.

D. Both create validity.

What premise must be added to create validity?

All physicians own mutual funds, from which it follows that no professors are physicians.

Translate this argument into standard form and determine validity (hint: there are 2 syllogisms)

Pornography violates women’s rights. It carries a demeaning message about a woman’s worth and purpose and promotes genuine violence. This is, indeed, a violation of women’s civil rights and justifies the Minneapolis City Council in attempting to ban pornography.

All instances of pornography are things that demean women’s worth and promote violence.

[Unstated:] All things that demean women’s worth and promote violence are things that violate women’s rights.]

All instances of pornography are things that violate women’s rights.

All instances of pornography are things that violate women’s rights.

[Unstated:] All things that violate women’s rights are things that are justifiably banned by the Minneapolis City Council.

All instances of pornography are things that are justifiably banned by the Minneapolis City Council.