Operations (Transformations)On Categorical Sentences

• Conversion (Simple & Limited)

• Obversion

• Contraposition

Equivalent Sentences*

Sentences are logically equivalent if (and only if) it is impossible for them to differ in truth value. In every situation they are either both true or both false.

Examples:

Some S are P. Some P are S.

No S are P. No P are S.

Conversion To form the converse of a categorical sentence simply exchange its terms.

*** S *** P

*** P *** S

Example: Some cats are not males. Some males are not cats.

Converse of Universal Affirmative (A)

All S are P S P

Converse of Universal Affirmative (A)’

• No Logical Connection. Logically independent.

S P

All S are P

All P are SS P

Converse of Universal Negative (E)

S PNo S are P

No P are S

Equivalent

Converse of Universal Negative (E)

No S are P

No P are S

---------- Equivalent ----------

S P

Converse of Particular Affirmative (I)

Some S are P

Some P are S

-------- Equivalent ----------

S P

*

Converse of Particular Negative (O)’

Some S are not P Some P are not S

---- Not Equivalent ----(Truth values not always the same.)

Some cats are not males. --- Some males are not cats.

Some mammals are not cats. ---Some cats are not mammals.

S P S P

* *

LogiCatCats ?Males ?

CATegorical Logic !

Limited Converse

To form the limited converse of a form A sentence (All S are P) limit the quantity to particular and then convert. This requires an existential assumption that there is at least one S.

Example: All freshmen are logicians.

Some logicians are freshmen.(This is implication,not equivalence.)

Conversion by Limitation

All S are P Converse: All P are S

S P S PNo Logical Relationship

Assume S exist..

*

T

Implication

*S P

Some S are P

Some P are S

Converse (equivalent)

Obversion

To form the obverse of a categorical sentence simply change the quality of that sentence and negate its predicate term.

Example:

No Republicans are Gaulists.

All Republicans are non-Gaulists.

Negating a term

To negate a term is to refer to all things other than what the term designates.

T

nonT (things other than T).

Negating Terms in English

To negate a single-word term, simply prefix it with “non.” To negate a complex term, prefix it with “things other than...”

Examples:

tomatoes --- non-tomatoes

funny stories --- things other than funny stories.

Negating English Terms (Further Examples)

1) non-citizen

non-non-citizen

2) onions picked by Miss America

non-onions picked by Miss America

things other than onions picked by Miss America

3) truths told by Bill Clinton

non-truths told by Bill Clinton

things other than truths told by Bill Clinton

4) theme parks

things other than theme parks.

Examples of Obversion

(1) Some wealthy women are politicians.

Some wealthy women are not non-politicians.

(2) Every U.S. Senator is sworn to uphold the Constitution.

All U.S. Senators are people sworn to uphold the Constitution. (Standard Form)

No U.S. Senators are things other than people sworn to uphold the Constitution.

A and Its Obverse

A: All S are P

(No S are outside P)

Obverse: No S are non-P

(No S are outside P)

----- Equivalent ----

S P

E and Its Obverse

E: No S are P.

Obverse: All S are non-P.

(No S are in P)

--- Equivalent ---

S P

I and Its Obverse

I: Some S are P

Obverse: Some S are not non-P

(Some S are not things outside P)

--- Equivalent ---

S P

*

O and Its Obverse

O: Some S are not P

Obverse: Some S are non-P

(Some S are things outside P)

--- Equivalent ---

S P*

Equivalent ?

A E I O

Converse No Yes Yes No

Obverse Yes Yes Yes Yes

S P

negate

non-P non-S

Contraposition

Contraposition

To form the contrapositve of a categorical sentence simply exchange and negate its terms.

Examples:

No tables are chairs --- No non-chairs are non-tables.

Some Republicans are not capitalists. ---

Some non-capitalists are not Republicans.

Contraposing a Categorical Sentence

**** S ******* P

**** non-P ******* non-S

negate

A and Its Contrapositve

A: All S are P.S P

Contrapositive: All non-P are non-S

(nothing outside P is inside S)S P

--- Equivalent ---.

E and Its Contrapositve

E: No S are P.

S P

Contrapositive: No non-P are non-S

(nothing outside P is outside S)S P

--- No Logical Relationship ---

I and Its Contrapositve

I: Some S are P.

S P

Contrapositive: Some non-P are non-S

(Something outside P is outside S)

S P

--- No Logical Relationship ---

*

*

O and Its Contrapositve

O: Some S are not P.S P

Contrapositive: Some non-P are not non-S

(Something outside P is inside S)S P

--- Equivalent ---

*

*

Equivalent ?

A E I O

Converse No Yes Yes No

Obverse Yes Yes Yes Yes

Contrapositive Yes No No Yes