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Chapter 9 Categorical Logic w07. A system of logic developed to clarify and evaluate deductive arguments. The study of categorical logic dates back to Aristotle. Based on the relations of:. Inclusion Exclusion. Relevance: Understand car purchase , loans, etc.

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Chapter 9 Categorical Logic w07

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Chapter 9 categorical logic w07 l.jpg

Chapter 9 Categorical Logic w07

A system of logic developed to clarify and evaluate deductive arguments. The study of categorical logic dates back to Aristotle. Based on the relations of:

  • Inclusion

  • Exclusion

  • Relevance:

  • Understand car purchase, loans, etc.

  • Understand contractual agreementsfor renting an apartment

  • completing catalog requirements for a major, etc.

  • Understanding instructions on medicine

  • Understand graduation requirements

  • Etc.


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* Standard CategoricalClaims 06

(S)ubject: noun or noun phrase*. Example: Methodists (Class members)

(P)redicate: noun or noun phrase. Example: Christians (College Students)

(P)redicate_

(S)ubject

A: All _______ are _________(affirmative)

E: No________are__________(negative)

I: Some_______are__________(affirmative)

O: Some______are not _______(negative)

*Only noun or noun phrases are allowed--Not All fire trucks are red (adj)


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All shitzus are dogs.

Some dogs are not animals.

No men are teachers.

Some teachers are parents.


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A

E

I

O

Circles-classes/categories

Shaded-empty

* Venn Diagrams of 4 Standard Claims

Methodists Christians

Buddists Christians

All Methodists are Christians

No Buddhists are Christians

Christians Methodists

Christians Methodist

Some Christians are Methodists

Some Christians are not Methodists

Blank-no mention

X-some, at least one


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Translation of claims into standard form: “equivalent claims”06

  • Purpose is to translate an ordinary claim into an equivalent standard form p261

  • Easy translations e.g “Every A is a B --> All A’s are B’s [A: Claim]

    “Minors are not eligible --> No minors are eligible [E: Claim]

    3.Past to present: “There were….” To “Some …”p261

    4.Only; Only adults are admitted to see Napoleon Dynamite

    All admitted to Napoleon Dynamite are adults

    5.The only; The only people allowed to drink beer are over 21

    All people allowed to drink beer are over 21

    6.Times, occasions, places (whenever, wherever); She makes friends wherever she goes All places she goes are places she makes friends

    7,Claims about an individual (object, occasion or place);

    Hitler was a psychopath All people identical with Hitler are psychopaths

    8.Mass nouns; Daisy Dukes are too out of style to get one now All Daisy Dukes are too out of style to have now

    Etc. An introduction, not possible to cover all possibilities.

Introduces

predicate of A:

Introduces

subject of A:

A: or E:

All…:

Treat as

A: are E: claim:

Treat as

A: claim:


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Translation Practice

  • Every salamander is a lizard

  • Snakes are the only members of the suborder Ophidia

  • Anything that’s an alligator is a reptile

  • Socrates is a Greek


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Translation Practice Answers

  • Every salamander is a lizard.

    All salamanders are lizards.

  • Snakes are the only members of the suborder Ophidia.All members of the suborder Ophidia are snakes.

  • Anything that’s an alligator is a reptile.All alligators are reptiles.

  • Socrates is a GreekAll people identical with Socrates are Greeks.


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* The Square of Opposition: Correspondence (same S and P)


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All Aluminum cans are recyclable

* Determining Truth Values for Corresponding Claims 1

No Aluminum cans are recyclable

T

thus F

Known

Some Aluminum cans are recyclable

Some Aluminum cans are not recyclable

thus T

thus F


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All Muslims are Christians

* Determining Truth Values for Corresponding Claims 2

No Muslims are Christians

F

?

Known

Limits

If T at top all known

If F at bottom all known

If F at top or T at bottom only contradictory known

Some Muslims are Christians

Some Muslims are not Christians

?

thus T


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* Limits on determining Truth value

  • If we have one truth value, it is often possible to determine other Truth values.

  • True claim, top of square, we can determine all others

  • If we know A is false all we can infer is corresponding O (not E or I)

  • False claim at the bottom (I or O) we can infer other 3

  • If false at top all can infer is value of contradictory


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* Three Categorical Operations

  • Conversion: (E and I claims not A and O) switch S and P [All E an I claims are equivalent]

  • Obversion: (A ↔ E, I ↔ O) horizontal change affirmative to negative (vice versa) and replace predicate with its complementary term* [All 4 A, E, I, O are equivalent]

  • Contraposition: (A and O not E and I) switch S and P and replace both with complementary terms.[All A and O claims are equivalent]

  • *Universe of discourse-context that limits scope of terms (“everyone passes” [in class not world])

  • Complementary class-everything in the universe not in first category (everyone not in the class, simplest to put “non” in front of class p273)

  • complementary term-the names of complementary classes (students vs non students (p273))


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* Three Categorical Operations--Practice by making change and determine whether it is equivalent to starting claim

  • Converse: “All Shiites are Muslims”

    All Muslims are Shiites. (not equivalent)

  • Obversion: “No Muslims are Christians”

    All Muslims are non-Christians. (equivalent)

  • Contrapositive: “No Sunnis are Christians”

    No non-Christians are non-Sunnis. (not equivalent)

equivalency


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No Aluminum cans are (recyclable)

No Aluminum cans are non-(recyclable)

All aluminum cans are (recyclable)

All Aluminum cans are non-(recyclable)

Obversion Claims 3

T

thus F

Known

Some Aluminum cans are (recyclable)

Some Aluminum cans are not-(not recyclable)

Some Aluminum cans are (not recyclable)

Some Aluminum cans are not non-recyclable

thus T

thus F


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* Two Syllogisms

Two common Nature vs Nurture arguments

  • All animals have X

  • Man has X

  • Therefore man is an animal

  • Man is an animal

  • Animals have Y

  • Therefore man has Y

Conclusion used as Premise for another argument

* We would have to convert these to standard form for analysis


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* Categorical Syllogisms

  • Standard form, two premise deductive argument, whose every claim is a standard form categorical claim in which three terms occur exactly twice in exactly two of the claims

  • Example:

    All CSUB students are college students

    Some college students are not dorm residents

    Therefore some CSUB students are not dorm residents

  • Terms:

    P Major (predicate of conclusion) -- dorm residents

    S Minor (subject of conclusion) -- CSUB students

    M Middle (both premises but not in conclusion) -- college students


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Consumers

(Collectivists)

Relationship of Terms

Americans

(Socialists)

Democrats

(Republicans)


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* Venn Diagram Validity Test-0

(p267 and Categorical Logic)

No Republicans are collectivists

All socialists are collectivists

Therefore, no socialists are Republicans

Minor

Major

Middle


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* Venn Diagram Validity Test-1

Minor

Major

Middle

No Republicans are Collectivists


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* Venn Diagram Validity Test-2

Minor

Major

Since result (green) is an overlap of shaded area, thus empty, we have a correct diagram of the conclusion, a valid syllogism

No Rs are collectivists

Middle

All Socialists are Collectivists


Venn diagram test of validity l.jpg

(p267…)

* Venn Diagram test of Validity

  • (1) Some syllogisms are problematic

    -I or O as one premise, where to place the X

    If one premise A or E and other premise is I or O diagram A or E first (p287) and there is no longer a choice of where to place the X

  • (2) Some syllogisms still have a problem-an X could go either of two places. Place the X on the line

    If the the X falls entirely within the appropriate area the argument is valid. If the X fails to entirely fall within the area the argument is invalid (p289)

  • (3) When both premises of a syllogism are A or E (shading) and the conclusion is an I or O (an X), a diagram cannot possibly yield a diagram of the conclusion

    • If any area has only one area unshaded place the X there and then the conclusion can possibly be read—valid, if not the conclusion is invalid


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* Rules Method for Test of Validity p294

  • (1) # Negative claims premises = # negative claims conclusion

  • (2) One premise must distribute * the middle term

  • (3) Any term distributed* in conclusion must be distributed in premise

* Distributed: see next slide


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* Distributed: claim says something about every member of the class. Memorize this to apply rules method.

  • A-claimall S are P

  • E-claimNo S are P

  • I- claimSome S are P

  • O-claimSome S are not P

The circled terms are distributed


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