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.

* 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)

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

* 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

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:

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

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.

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

* 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

* 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

* 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

* 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))

* 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

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

* 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

* 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

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

* 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

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

* 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 (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

* 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

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