12- Standard Categorical Statements

12- Standard Categorical Statements •

Categorical statements are statements which affirm or deny something about a given subject

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4 forms –

All S are P.

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No S are P.

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Some S are P.

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Some S are not P.

12- Standard Categorical Statements •

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Nobody shuts the door. –

or- Nobody is a door-shutter.

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No person is a door-shutter

2 Parts –

Subject (S)

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Predicate (P)

A statement shows the relationship between an S and a P

12- Standard Categorical Statements •

Quantity and a Quality –

Quantity • •

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Universal (all and no) Particular (some and some...not)

Quality • •

Affirmative (all and some) Negative (no and some....not)

Statement

Quantity

Quality

All S are P

Universal

Affirmative

No S are P

Universal

Negative

Some S are P

Particular

Affirmative

Some S are not P

Particular

Negative

12- Standard Categorical Statements •

Rules –

The statements must begin with the words all, no , or some.

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The verb must be the verb of being: is, as, was, were, will be....

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Both the subject and the predicate must be a noun or a noun phrase.

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Example:

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All dogs are brown.

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All dogs are brown animals (All D are B)

13- The Square of Opposition •

A statements (Universal affirmative)

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Translation

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1. Everyone who comes to school studies.

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2. All our school attenders study.

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3. All our school attenders are studiers.

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4. All A are S.

13- The Square of Opposition •

E statements (Universal negative)

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Translation

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1. None of my friends understand Algebra.

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2. No friends of mine understand Algebra

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3. No friends of mine are Algebra-understanders

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4. No F are A.

13- The Square of Opposition •

I statements (Particular Affirmative)

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Translation

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1. Many students know a lot about Statistics

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2. Some students know a lot about Satistics.

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3. Some students are Staistics-knowers

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4. Some S are K

13- The Square of Opposition •

O statements (Particular Negative)

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Translation

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1. Many books do not have words.

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2. Some books do not have words

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3. Some books are not word-books.

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4. Some N are not W.

13- The Square of Opposition •

Square of Opposition

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All S are P (A)

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Some S are P (I)

No S are P (E)

Some S are not P (O)

14- Contradiction •

Square of Opposition presents 5 different relationships

1. Contradiction

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2. Contrariety

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3. Subcontrariety

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4. Subimplication

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5. Superimplication

Contradiction- Relationship between A and O and I and E

14- Contradiction •

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A and O –

All S are P – Some S are not P.

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All apples are fruit- Some apples are not fruit.

E and I –

Some S are P- No S are P

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Some apples are fruit. No apples are fruit.

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Both cannot be true and both cannot be false.

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One must be true and the other false.

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These are inconsistent.

14- Contradiction •

A

E

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I

O

15- Contrariety Contrary- relationship between A and E. –

Both cannot be true but both can be false.

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All astronauts are men. No astronauts are men.

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All A are M. No A are M. •

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Either one could be true or they both could be false but they can't both be true.

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All snakes are green reptiles.

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No snakes are green reptiles

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All S are G. No S are G.

Discussion over if all Caesers were cruel or some were cruel.

16- Subcontrariety Subcontrariety- relationship between I and O. –

Both cannot be false but both can be true.

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Some teachers are boring speakers.

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Some teachers are not boring speakers.

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Some students are intelligent.

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Some students are not intelligent.

16- Subcontrariety Think about it: •

1. If it is false that some students are not intelligent, that is the same as saying that all are intelligent (by contradiction)

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2. If all students are intelligent, then it must be true that some students are intelligent (by implication)

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3. And if it is true that some students are intelligent, then it cannot be false that some are (by the laws of thought)

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4. Thus they cannot both be false.

17- Subimplication Subimplication- The relationship of A to I ; E to O •

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The truth of a universal implies the truth of a particular –

If A is true then I is true

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If E is true the O is true

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All apples are red. Some apples are red.

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No apples are purples. Some apples are not purple.

Nothing to do with falsity but with implication of truth.

18- Superimplication Superimplication- The relationship of I to A ;O to E •

The falsity of the particular implies the falsity of the universal –

If I is false then A is false

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If O is false then E is false • •

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Some apples are red.All apples are red. Some apples are not purple.No apples are purples.

1. If it is false that some apples are not purple then it must be true that all apples are purple (by contradiciton)

18- Superimplication •

2. If all of them are purple then it must be false that none are purples (by contrariety)