Operations (Transformations) On Categorical Sentences

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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. Are the sentences in this example logically equivalent? Or do they just happen to be both true in virtue of the subject mater ? Stay tuned.

Converse of Universal Affirmative (A) All S are P
S P

Converse of Universal Affirmative (A)’
All S are P S P S P All P are S No Logical Connection. Logically independent.

Converse of Universal Negative (E)
No S are P S 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
S P S P * * 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.

LogiCat Cats ? 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
Converse: All P are S All S are P No Logical Relationship S P S P * T Assume S exist.. Implication S P * Some S are P Converse (equivalent) Some P are S

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)
S P A: All S are P (No S are outside P) Obverse: No S are non-P Equivalent ----

E and Its Obverse E: No S are P. Obverse: All S are non-P.
S P E: No S are P. Obverse: All S are non-P. (No S are in P) --- Equivalent ---

I and Its Obverse I: Some S are P Obverse: Some S are not non-P
S P * I: Some S are P Obverse: Some S are not non-P (Some S are not things outside P) --- Equivalent ---

O and Its Obverse O: Some S are not P Obverse: Some S are non-P
S P * O: Some S are not P Obverse: Some S are non-P (Some S are things outside P) --- Equivalent ---

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

A and Its Contrapositve
S P A: All S are P. Contrapositive: All non-P are non-S S P (nothing outside P is inside S) --- Equivalent --- .

E and Its Contrapositve
S P E: No S are P. Contrapositive: No non-P are non-S S P (nothing outside P is outside S) --- No Logical Relationship ---

I and Its Contrapositve
S P I: Some S are P. * Contrapositive: Some non-P are non-S S P (Something outside P is outside S) * --- No Logical Relationship ---

O and Its Contrapositve
S P O: Some S are not P. * Contrapositive: Some non-P are not non-S S P (Something outside P is inside S) * --- Equivalent ---

Equivalent ? A E I O Converse No Yes Yes No Obverse Yes Yes Yes Yes
Contrapositive Yes No No Yes