LOGIC 4 (Categorical Interference)

Assert or deny something about a class or category of things.

The foundation of classical logic developed by Aristotle.

Another: No birds are reptiles.

We're connecting: BIRDS ↔ REPTILES

Another: Some students are athletes.

We're connecting: STUDENTS ↔ ATHLETES

CATEGORICAL PREPOSITION

Each categorical proposition relates two classes or terms, namely, a subject and a predicate, and claims their relationship.

connects two classes/groups.

So whenever you see a categorical proposition, look for:

SUBJECT + relationship + PREDICATE

 a statement that can be judged TRUE or FALSE.

PROPOSITION

A statement that asserts what is or is not the case—also called assertoric.

Standard Form of Categorical Proposition

QUANTIFIER + SUBJECT + COPULA + PREDICATE

Identify the four parts of Categorical Proposition

EXAMPLE: All cats are mammals.

Part	Word	Meaning
Quantifier	All	How many cats?
Subject	cats	What are we talking about?
Copula	are	Connects the two
Predicate	mammals	What are cats being related to?

QUANTIFIER

tells you HOW MUCH OF THE SUBJECT we're talking about.

ALL = everyone
NO = nobody
SOME = at least one/some

THREE QUANTIFIERS

ALL All cats are mammals.

Means: Every cat

NO No cats are reptiles.

Means: Zero cats

SOME Some cats are black.

Means: At least some cats

SUBJECT

What are we talking about

Example: All cats are mammals.

Subject = cats

Some students are athletes.

Subject = students

No politicians are immortal.

Subject = politicians

PREDICATE

What we say about the subject

Example: All cats are mammals.

Predicate = mammals.

Some students are athletes.

Predicate = athletes.

COPULA

Connects the subject and the predicate.

Usually: are Or are not

Examples:

Cats are mammals.

Cats are not reptiles.

Every categorical proposition has TWO characteristics:

QUANTITY

Asks: HOW MANY?

QUALITY

Asks: CONNECTED OR SEPARATED?

Quantity: universal / particular / singular

Quality: affirmative / negative

HOW MANY?

 universal / particular / singular

QUANTITY

Three Types of Quantity

Universal = ALL/NO

Particular = SOME

Singular = ONE NAMED THING

Talks about the whole subject class.

Words: ALL, NO

UNIVERSAL

Examples:

All cats are mammals.

No cats are reptiles.

You're talking about cats as a whole class.

Talks about some members of the subject class.

Keyword: SOME

PARTICULAR

Examples:

Some cats are black.

Some cats are not friendly.

You're NOT talking about every cat. Only some.

Talks about one specific individual/group.

SINGULAR

Example: Jose Rizal is a national hero.

You're talking about one particular individual.

AFFIRMATIVE or NEGATIVE

QUALITY

Does the proposition affirm or deny the relationship?

ARE = AFFIRM

ARE NOT = NEGATE

It says the subject belongs with the predicate.

All cats are mammals.

Some cats are black.

AFFIRMATIVE

It says the subject doesn't belong with the predicate.

No cats are reptiles.

Some cats are not white.

NEGATIVE

Symbol	Quantity	Quality	Basic pattern
A	Universal	Affirmative	All S are P
E	Universal	Negative	No S are P
I	Particular	Affirmative	Some S are P
O	Particular	Negative	Some S are not P

FOUR standard forms.

Universal + Affirmative

A = ALL

All S are P

Example: All cats are mammals.

Universal + Negative

E = NONE

No S are P

Example: No cats are reptiles.

Particular + Affirmative

I = SOME ARE

Some S are P

Example: Some cats are black.

Particular + Negative

O = SOME ARE NOT

Some S are not P

Example: Some cats are not black.

easiest A-E-I-O memory trick

A = ALL are

E = NONE are

I = SOME are

O= SOME are NOT

Why are they called A, E, I, O?

It comes from Latin words related to:

affirmo = I affirm And nego = I deny

The letters were taken from those words.

A + I = Affirmative

E + O = Negative

DISTRIBUTED vs UNDISTRIBUTED

Does the proposition talk about EVERY MEMBER of a class?

If yes: DISTRIBUTED

If no: UNDISTRIBUTED

ALL cats are mammals.

Imagine: 🐱🐱🐱🐱🐱 = all cats

All of those cats are being included inside: 🍼 mammals

So we're talking about every cat.

Therefore: Subject = DISTRIBUTED

But are we talking about every mammal? No!

We're only saying that cats belong to the larger group of mammals.

There are many mammals that aren't cats:

• dogs

• humans

• whales

• cows

Therefore: Predicate = UNDISTRIBUTED

So: A preposition

All S are P

S = distributed

P = undistributed

NO cats are reptiles.

We're saying:

Every single cat is excluded from reptiles.

So we're talking about the whole class of cats.

Therefore: Subject = DISTRIBUTED

But we're also saying cats are excluded from the whole class of reptiles.

Therefore: Predicate = DISTRIBUTED

So: E preposition

No S are P

S = distributed

P = distributed

Some cats are black.

We're talking about: SOME cats. Not all cats.

Therefore: Subject = UNDISTRIBUTED

And we're not talking about all black things either.

There are black:

• cars

• shoes

• dogs

• clothes

We're only saying that some cats overlap with the group of black things.

Therefore: Predicate = UNDISTRIBUTED

So: I preposition

Some S are P

S = undistributed

P = undistributed

Some cats are NOT black.

We're talking about: SOME cats

So: Subject = UNDISTRIBUTED

But look at the predicate: not black

We're saying those particular cats are excluded from the entire category of black things.

Therefore: Predicate = DISTRIBUTED

So: O

Some S are NOT P

S = undistributed

P = distributed

Memorize DISTRIBUTION like this

	Subject	Predicate
A All S are P	D	U
E No S are P	D	D
I Some S are P	U	U
O Some S are not P	U	D

A = D U
E = D D
I = U U
O = U D

 There's actually a beautiful pattern

Affirmative propositions

A → DU

I → UU

Negative propositions

E → DD

O → UD

And there's a rule: 

AFFIRMATIVE → predicate is usually UNDISTRIBUTED

NEGATIVE → predicate is DISTRIBUTED

 Euler diagrams

Circles representing groups.

The whole lesson in ONE picture

Think of two circles: SUBJECT ↔ PREDICATE

QUESTION 1:

HOW MANY SUBJECTS?

ALL/NO → Universal

SOME → Particular

ONE NAMED THING → Singular

QUESTION 2:

ARE THEY CONNECTED OR SEPARATED?

ARE → Affirmative

ARE NOT → Negative

Quantity	Quality	Form
Universal	Affirmative	A
Universal	Negative	E
Particular	Affirmative	I
Particular	Negative	O