“Categories” of Stuff

All boogers are tasty objects.No tasty objects are unhealthy

snacks.Therefore, no boogers are

unhealthy snacks.

“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”.

http://dictionary.reference.com/browse/category

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

• 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.

• Govier says, “That a person is a sister deductively entails that she is a female.”

• DO YOU BUY THAT?

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”

THE MAGIC OF CATEGORICAL FORMS

• Categorical forms revolve around 4 kinds of statements:

1) “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”

2) “E” = NO B are T = “Universal Negative”1) 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”

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

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

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.

Let’s look a little closer:

• EXAMPLE: ALL CATS are stupid animals.

• We know something about every member of the category “Cat”….which is……



• We know something about every member of the category “Cat”….which is……

• SO “CAT” is a “DISTRIBUTED” category.



• Do we know ANYTHING about 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”?

• NO!

• COULD WE MAKE A PREMISE ABOUT STUPID ANIMALS WHICH DISTRIBUTES IT?

• SURE!• “ALL STUPID ANIMALS GET RUN

OVER.”

EVERYONE GETTING IT?EVERYONE GETTING IT?

• WHAT KIND OF STATEMENT IS THIS?• “No supervisors are brilliant.”

A) AB) EC) ID) OE) NONE OF THE ABOVE

LET’S TRY ANOTHERLET’S TRY ANOTHER

• WHAT KIND OF STATEMENT IS THIS?• “Some LaCross players are not smart.”



• WHAT KIND OF STATEMENT IS THIS?• “Korean cars are low quality.”



• WHAT KIND OF STATEMENT IS THIS?• “Lots of lizards are cute.”


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 …..

A)VENN DIAGRAMSB) INDUCTIVE METHODSC)SYLLOGISTIC COGENCED)VALIDITY ANALYSISE) CONSTRUCTIVE CRITICISM

VENN DIAGRAMS

All dogs have fur.

All animals with fur are smelly.

Thus, all dogs are smelly.

VENN DIAGRAMS

All dogs have fur.



IF ALL DOGS HAVE FUR, we eliminate all of the dogs that do NOT have fur!

VENN DIAGRAMS

All dogs have fur.



IF ALL furry animals are smelly, we have to get rid of the non-smelly ones!

VENN DIAGRAMS

All dogs have fur.



NOW WE CAN SEE that the only dogs left are the SMELLY ONES, so ALL DOGS are in-fact smelly!

BUT WAIT JUST A BUT WAIT JUST A MINUTE!MINUTE!

“All kittens are cute.”

WHAT IF an animal is NOT cute? What do we know?

BUT WAIT JUST ANOTHER BUT WAIT JUST ANOTHER MINUTE!MINUTE!

“No lizards are cute.”

WHAT IF an animal IS cute? What do we know?

INFERENCES

WHAT IS AN INFERENCE?

AN INFERENCE IS A CONCLUSION.

So, “inferring” is the act or process of drawing a LOGICAL conclusion!

If we know the following:

“Nancy is a human being.”

Can we draw any inferences from this?

INFERENCES

• 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”

REMEMBER THIS?REMEMBER THIS?

We can do “OPERATIONS” to the various statements, and when we do these operations, we can automatically draw conclusions from the result.

• A E

• I O




CONVERSION: FLIP THINGS AROUND:

ALL BOOGERS ARE TASTY OBJECTS “Converts” to

ALL TASTY OBJECTS ARE BOOGERS.

DOES THIS MAKE ANY SENSE?

• A E

• I O




CONVERSION DOES NOT WORK ON “A” STATEMENTS, nor does it work for “O” statements.

• A E

• I O



CONVERSION: FLIP THINGS AROUND:

“No boogers are tasty objects,” “Converts” to :

A)No people who like tasty things eat boogers

B)No tasty objects are eaten by boogers

C)No tasty objects are boogers

D)Boogers are tasty

E)Some boogers are not tasty

• A E

• I O




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!

• A E

• I O

“E” = NO B are T, so “No T are B” MEANS THE SAME THING!


WE CAN SEE THIS VISUALLY:

• A E

• I O

“I” = SOME B are TCONVERSE: SOME T ARE B


SAME THING GOES FOR “I” STATEMENTS

X

THE “CONVERSE” OF “E” & “I”

• They are logically equivalent: they mean the same thing, so we can “infer” one from the other.

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”

OBVERSION

• We take a CATEGORICAL statement and we :

1) Turn the “distributed” ting into its opposite: “All” becomes “NO” and vice-versa.

2) Turn whatever the distributed thing “is” into its opposite.

• “All dogs are furry animals” becomes “No dogs are non-furry animals.”

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.”

RECALL FROM LAST CLASS:



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

RECALL FROM LAST CLASS:



X

SO CONTRADICTING RELATIONSHIPS CAN BE CONCLUDED or INFERRED BASED UPON THE STATEMENT TYPE!

CONTRAPOSITION

FIRST: CONVERTALL 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”

WHICH IS THE CONTRAPOSITIVE?

“NO KARATE FIGHTERS ARE BRAVE PEOPLE”A) All non-brave people are karate fightersB) No non-brave people are non karate fightersC) All brave people are non-karate fightersD) All non-brave people are non-karate fightersE) DAMN, I am sooooo lost.

THIS MAKES NO FLIPPIN’ SENSE!

KARATE BRAVE

CONTRAPOSING “E” and “I” STATEMENTS DOES NOT WORK!

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

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.

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!

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)

WHAT IS THE CONVERSE?



Converse: Some [people who cannot give blood] are [people who have been tested]

WHAT IS THE OBVERSE?



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]

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).

• 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]



• People who have not been tested cannot give blood.

All [people who have not been tested] are [people who cannot give blood] (A-claim)

Contrapositive: All [people who can give blood] are [people who have been tested]

Obverse: No [people who have not been tested] are [people who can give blood]



• Nobody who has been tested can give blood.

No [people who have been tested] are [people who can give blood] (E-claim)

Converse: No [people who can give blood] are [people who have been tested]

Obverse: All [people who have been tested] are [people who cannot give blood]


contraposition AND an obversion). • Nobody can give blood except those who have been

tested. • Nobody can give blood except those who have been

tested. = The only people who can give blood are those who have been tested. OR Only those who have been tested can give blood.

All [people who can give blood] are [people who have been tested]

Contrapositive: All [people who have not been tested] are [people who cannot give blood]

Obverse: No [people who can give blood] are [people who have not been tested]

Match the numbered claims with the lettered claims that are equivalent.

• 1. Some people who have not been tested can give blood.

• a. Some people who have been tested cannot give blood.

• b. Not everybody who can give blood has been tested.

• c. Only people who have been tested can give blood.

• d. Some people who cannot give blood are people who have been tested.

• e. If a person has been tested, then she cannot give blood.


• People who have not been tested cannot give blood.







• Nobody who has been tested can give blood.







• Nobody can give blood except those who have been tested.






BUT FIRST……

WHAT IS THIS:

• “There is no evidence that there is life on Mars, so we can conclude that ther is no life there!”

• A) Ad Hominem

• B) Appeal to Pity

• C) Appeal to Ignorance

• D) Straw Man

• E) Guilt by association

WHAT IS THIS:

• “Why should I listen to Al Gore about global warming when that guy drives a Hummer and lives in a huge, energy-wasting house?”

• A) Appeal to Popularity• B) Appeal to Pity• C) Appeal to Ignorance• D) Straw Man• E) Tu Quoque

WHAT IS THIS:

• “Over 10 MILLION people have purchased the ShamWow! It has to be great!”

• A) Ad Hominem

• B) Fallacy of Division

• C) Appeal to Ignorance

• D) Appeal to popularity

• E) Guilt by association

WHAT IS THIS:

• “Tom works with those guys at NASA. They are all a bunch of science goobers over there…so I can’t handle inviting a science goober like Tom to my party.”

• A) Ad Hominem• B) Fallacy of Division• C) Fallacy of Composition• D) Appeal to popularity• E) Guilt by association

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“Categories” of Stuff