“Categories” of Stuff

1. All boogers are tasty objects.No tasty objects are unhealthy snacks.Therefore, no boogers are unhealthy snacks.

2. “Categories” of Stuff • We will be discussing categories of things:1.any general or comprehensive division; a class. 2.a classificatory division in any field of knowledge, as a phylum or any of its subdivisions in biology. • http://dictionary.reference.com/browse/category • We really mean “Grouping” : things that are “in” the “group” and things which are “out” of the “group”.

3. DEDUCTIVE ENTAILMENT: WHAT IS IT AGAIN? ANYBODY….?

4. IF IT IS LOGICALLY IMPOSSIBLE FOR THE CONCLUSION TO BE FALSE GIVEN THE TRUTH OF THE PREMISES…OR… • WHEN STATEMENT (1) IS TRUE, (2) BEING FALSE IS LOGICALLY IMPOSSIBLE.

5. Govier says, “That a person is a sister deductively entails that she is a female.” • DO YOU BUY THAT?

6. Returning to Boogers • All boogers are tasty objects.No tasty objects are unhealthy snacks.Therefore, no boogers are unhealthy snacks. • NOTICE THE FORM OF THE ARGUMENT: • ALL B are T • No T are U • Thus, no B are U • This is called a “CATEGORICAL FORM”

7. THE MAGIC OF CATEGORICAL FORMS • Categorical forms revolve around 4 kinds of statements: • “A” = ALL B are T = “Universal Affirmative” IF SOMEONE SAYS, “Politicians are corrupt,” and they fail to qualify it (i.e. “Most politicians” or “many politicians”) then we treat it as if they had said “ALL POLITICIANS” • “E” = NO B are T = “Universal Negative” • IF SOMEONE SAYS, “Boogers are not tasty” and they do not qualify it (i.e. “Most boogers” or “many boogers”) then we treat it as if they had said, “There are no boogers which are tasty.” 3) “I” = Some B are T = “Particular Affirmative” 4) “O” = Some B are not T = “Particular Negative”

8. SQUARE OF OPPOSITION • A E • I O A and E Statements are “CONTRARY” to each other: Both cannot be TRUE, but both can be FALSE. All B are TNo B are T I and O statements are “Sub-Contrary” to each other: Both can be TRUE but both cannot be FALSE. Some B are TSome B are not T

9. SQUARE OF OPPOSITION • A E • I O A and O statements CONTRADICT each other: If one is TRUE the other MUST be FALSE. All B are TSome B are not T Same for E and I Statements: they CONTRADICT, so if one is true, the other must be false. No B are TSome B are T

10. DISTRIBUTION • Any object which we talk about in a categorical argument is DISTRIBUTED when we know something about EVERY member of that category of thing.

11. Let’s look a little closer: • EXAMPLE: ALL CATS are stupid animals. • We know something about every member of the category “Cat”….which is…… THEY IS STOOPIT!

12. Let’s look a little closer: • EXAMPLE: ALL CATS are stupid animals. • We know something about every member of the category “Cat”….which is…… • SO “CAT” is a “DISTRIBUTED” category.

13. Let’s look a little closer: • EXAMPLE: ALL CATS are stupid animals. • Do we know ANYTHING about stupid animals?

14. Let’s look a little closer: • EXAMPLE: ALL CATS are stupid animals. • Do we know ANYTHING about stupid animals? • YES! We know that SOME stupid animals are cats… • BUT IS THE CATEGORY “STUPID ANIMALS” “DISTRIBUTED”?

15. NO! • COULD WE MAKE A PREMISE ABOUT STUPID ANIMALS WHICH DISTRIBUTES IT? • SURE! • “ALL STUPID ANIMALS GET RUN OVER.”

16. EVERYONE GETTING IT? • WHAT KIND OF STATEMENT IS THIS? • “No supervisors are brilliant.” • A • E • I • O • NONE OF THE ABOVE

17. LET’S TRY ANOTHER • WHAT KIND OF STATEMENT IS THIS? • “Some LaCross players are not smart.” • A • E • I • O • NONE OF THE ABOVE

18. LET’S TRY ANOTHER • WHAT KIND OF STATEMENT IS THIS? • “Korean cars are low quality.” • A • E • I • O • NONE OF THE ABOVE

19. LET’S TRY ANOTHER • WHAT KIND OF STATEMENT IS THIS? • “Lots of lizards are cute.” • A • E • I • O • NONE OF THE ABOVE

20. We can connect categorical statements together into “mini-arguments.” All dogs have fur. All animals with fur are smelly. Thus, all dogs are smelly. WE CAN EVALUATE CATEGORICAL SYLLOGISMS BY USING ….. • VENN DIAGRAMS • INDUCTIVE METHODS • SYLLOGISTIC COGENCE • VALIDITY ANALYSIS • CONSTRUCTIVE CRITICISM

21. VENN DIAGRAMS All dogs have fur. All animals with fur are smelly. Thus, all dogs are smelly.

22. VENN DIAGRAMS All dogs have fur. All animals with fur are smelly. Thus, all dogs are smelly. IF ALL DOGS HAVE FUR, we eliminate all of the dogs that do NOT have fur!

23. VENN DIAGRAMS All dogs have fur. All animals with fur are smelly. Thus, all dogs are smelly. IF ALL furry animals are smelly, we have to get rid of the non-smelly ones!

24. VENN DIAGRAMS All dogs have fur. All animals with fur are smelly. Thus, all dogs are smelly. NOW WE CAN SEE that the only dogs left are the SMELLY ONES, so ALL DOGS are in-fact smelly!

25. BUT WAIT JUST A MINUTE! “All kittens are cute.” WHAT IF an animal is NOT cute? What do we know? It ain't no kitten!

26. BUT WAIT JUST ANOTHER MINUTE! “No lizards are cute.” WHAT IF an animal IS cute? What do we know? NOT a lizard.

27. INFERENCES WHAT IS AN INFERENCE? AN INFERENCE IS A CONCLUSION. So, “inferring” is the act or process of drawing a LOGICAL conclusion!

28. INFERENCES If we know the following: “Nancy is a human being.” Can we draw any inferences from this?

29. REMEMBER THIS? • A E • I O • “A” = ALL B are T = “Universal Affirmative” • “E” = NO B are T = “Universal Negative” • “I” = Some B are T = “Particular Affirmative” • “O” = Some B are not T = “Particular Negative” We can do “OPERATIONS” to the various statements, and when we do these operations, we can automatically draw conclusions from the result.

30. REMEMBER THIS? HELL NO! • A E • I O • “A” = ALL B are T = “Universal Affirmative” • “E” = NO B are T = “Universal Negative” • “I” = Some B are T = “Particular Affirmative” • “O” = Some B are not T = “Particular Negative” CONVERSION: FLIP THINGS AROUND: ALL BOOGERS ARE TASTY OBJECTS “Converts” to ALL TASTY OBJECTS ARE BOOGERS. DOES THIS MAKE ANY SENSE?

31. REMEMBER THIS? • A E • I O • “A” = ALL B are T = “Universal Affirmative” • “E” = NO B are T = “Universal Negative” • “I” = Some B are T = “Particular Affirmative” • “O” = Some B are not T = “Particular Negative” CONVERSION DOES NOT WORK ON “A” STATEMENTS, nor does it work for “O” statements.

32. “A” = ALL B are T = “Universal Affirmative” • “E” = NO B are T = “Universal Negative” • “I” = Some B are T = “Particular Affirmative” • “O” = Some B are not T = “Particular Negative” • A E • I O • CONVERSION: FLIP THINGS AROUND: • “No boogers are tasty objects,” “Converts” to : • No people who like tasty things eat boogers • No tasty objects are eaten by boogers • No tasty objects are boogers • Boogers are tasty • Some boogers are not tasty

33. REMEMBER THIS? • A E • I O • “A” = ALL B are T = “Universal Affirmative” • “E” = NO B are T = “Universal Negative” • “I” = Some B are T = “Particular Affirmative” • “O” = Some B are not T = “Particular Negative” CONVERSION DOES WORK ON “E” STATEMENTS! IF “NO BOOGERS ARE TASTY OBJECTS” IT CAN AUTOMATICALLY BE CONCLUDED (INFERRED) THAT THERE ARE NO TASTY OBJECTS WHICH ARE BOOGERS!

34. REMEMBER THIS? • A E • I O • “E” = NO B are T, so “No T are B” MEANS THE SAME THING! WE CAN SEE THIS VISUALLY:

35. REMEMBER THIS? • A E • I O • “I” = SOME B are T • CONVERSE: SOME T ARE B SAME THING GOES FOR “I” STATEMENTS X

36. THE “CONVERSE” OF “E” & “I” • They are logically equivalent: they mean the same thing, so we can “infer” one from the other.

37. OBVERSION • We take a CATEGORICAL statement and we : 1) Turn the “distributed” thing into its opposite: “All” becomes “NO” and vice-versa. • “All dogs” becomes “No dogs” • “No chickens” becomes “All chickens”

38. OBVERSION • We take a CATEGORICAL statement and we : • Turn the “distributed” ting into its opposite: “All” becomes “NO” and vice-versa. • Turn whatever the distributed thing “is” into its opposite. • “All dogs are furry animals” becomes “No dogs are non-furry animals.”

39. MUY IMPORTANTE! • “All dogs are furry animals” becomes “No dogs are non-furry animals.” • WE USE “NON-FURRY” NOT “Furless” or “fleshy” • No bankers are honorable people becomes • “ALL bankers are non-honorable people.” • THAT MEANS SOMETHING QUITE DIFFERENT THAN “ALL BANKERS ARE DISHONORABLE PEOPLE.”

40. RECALL FROM LAST CLASS: A and O statements CONTRADICT each other: If one is TRUE the other MUST be FALSE. All B are TSome B are not T Same for E and I Statements: they CONTRADICT, so if one is true, the other must be false. No B are TSome B are T

41. RECALL FROM LAST CLASS: A and O statements CONTRADICT each other: If one is TRUE the other MUST be FALSE. All B are TSome B are not T X SO CONTRADICTING RELATIONSHIPS CAN BE CONCLUDED or INFERRED BASED UPON THE STATEMENT TYPE!

42. CONTRAPOSITION FIRST: CONVERT ALL CATS ARE STUPID ANIMALS becomes “ALL STUPID ANIMALS ARE CATS” THEN WE ADD A “non” to each of the two categories: “All non-stupid animals are non-cats”

43. WHICH IS THE CONTRAPOSITIVE? “NO KARATE FIGHTERS ARE BRAVE PEOPLE” • All non-brave people are karate fighters • No non-brave people are non karate fighters • All brave people are non-karate fighters • All non-brave people are non-karate fighters • DAMN, I am sooooo lost. BRAVE KARATE THIS MAKES NO FLIPPIN’ SENSE! CONTRAPOSING “E” and “I” STATEMENTS DOES NOT WORK!

44. La Grande Point • SO WHEN WE HAVE CERTAIN KINDS OF STATEMENTS, WE CAN AUTOMATICALLY CONCLUDE, JUST BASED UPON THEIR “TYPE” (A,E,I,O) certain yummy things about them: “OPERATING” ON THEM TURNS THEM INTO LOGICALLY EQUIVALENT STATEMENTS, SO WE CAN INFER THE EQUIVALENT FROM THE ORIGINAL. • PAGE 222 HAS A CHART OF ALL THAT WE NEED TO KNOW

45. PG 222 TELLS US THAT WE GET LOGICALLY EQUIVALENT STATEMENTS WHEN: • “A” STATEMENTS ARE OBVERTED OR CONTRAPOSED • “E” STATEMENTS ARE CONVERTED OR OBVERTED • “I” STATEMENTS ARE CONVERTED OR OBVERTED • “O” STATEMENTS ARE CONTRAPOSED OR OBVERTED.

46. IN THE EVENT YOU ARE GETTING YOUR BUTT KICKED BY THIS STUFF…. • RE-READ CHAPTER 7 • LISTEN TO THIS LECTURE AND THE LAST ONE ON-LINE while watching the PowerPoints • COME TO THE NEXT CLASS PREPARED TO DO A WHOLD BUNCH OF PRACTICE PROBLEMS. • BRING YOUR QUESTIONS WITH YOU!

47. WHAT KIND OF CLAIM (A,E,I,O)? • Some people who have been tested cannot give blood. Some [people who have been tested] are [people who cannot give blood] (I-claim)

48. WHAT IS THE CONVERSE? • Some people who have been tested cannot give blood. Some [people who have been tested] are [people who cannot give blood] (I-claim) Converse: Some [people who cannot give blood] are [people who have been tested]

49. WHAT IS THE OBVERSE? • Some people who have been tested cannot give blood. Some [people who have been tested] are [people who cannot give blood] (I-claim) Converse: Some [people who cannot give blood] are [people who have been tested] Obverse: Some [people who have been tested] are not [people who can give blood]

50. 1. Some people who have not been tested can give blood. Some [people who have not been tested] are [people who can give blood] (I-claim)Converse: Some [people who can give blood] are [people who have not been tested]Obverse: Some [people who have not been tested] are not [people who cannot give blood] Translate the claim below into a standard-form categorical claim. Then, do 2 immediate inferences on it (either a conversion or a contraposition AND an obversion).