1. 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 agreements for renting an apartment
completing catalog requirements for a major, etc.
Understanding instructions on medicine
Understand graduation requirements
Etc.

2. * Standard Categorical Claims 06
(S)ubject: noun or noun phrase*. Example: Methodists (Class members)
(P)redicate: noun or noun phrase. Example: Christians (College Students)
(S)ubject
(P)redicate_
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)

3. All shitzus are dogs.
Some dogs are not animals.
No men are teachers.
Some teachers are parents.

4. * Venn Diagrams of 4 Standard Claims
Circles-classes/categories
Shaded-empty A E I O
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

5. 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:

6. 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

7. 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 Greek All people identical with Socrates are Greeks.

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

9. * Determining Truth Values for Corresponding Claims 1
All Aluminum cans are recyclable
No Aluminum cans are recyclable T
thus F
Known
Some Aluminum cans are recyclable
Some Aluminum cans are not recyclable
thus T
thus F

10. * Determining Truth Values for Corresponding Claims 2
All Muslims are Christians
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

11. * 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

12. * 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))

13. * 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

14. Obversion Claims 3 T thus F thus T thus F Known
No Aluminum cans are (recyclable)
No Aluminum cans are non-(recyclable)
All aluminum cans are (recyclable)
All Aluminum cans are non-(recyclable) 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

15. * Two Syllogisms All animals have X Man has X
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

16. * 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

17. Relationship of Terms Consumers (Collectivists) Americans (Socialists)
Democrats
(Republicans)

18. * 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

19. * Venn Diagram Validity Test-1
Minor
Major
Middle
No Republicans are Collectivists

20. * 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

21. * Venn Diagram test of Validity
(p267…)
(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

22. * 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

23. Distributed: claim says something about every member of the class
* Distributed: claim says something about every member of the class. Memorize this to apply rules method.
A-claim all S are P
E-claim No S are P
I- claim Some S are P
O-claim Some S are not P
The circled terms are distributed