Presentation on theme: "Operations (Transformations) On Categorical Sentences"— Presentation transcript:
1 Operations (Transformations) On Categorical Sentences Conversion (Simple & Limited)ObversionContraposition
2 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.
3 ConversionTo form the converse of a categorical sentence simply exchange its terms.*** S *** P *** P *** SExample: 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.
4 Converse of Universal Affirmative (A) All S are P S P
5 Converse of Universal Affirmative (A)’ All S are PS PS PAll P are SNo Logical Connection. Logically independent.
6 Converse of Universal Negative (E) No S are PS PNo P are SEquivalent
7 Converse of Universal Negative (E) No S are PNo P are SEquivalentS P
8 Converse of Particular Affirmative (I) Some S are PSome P are SEquivalentS P*
9 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.
11 Limited ConverseTo 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.)
12 Conversion by Limitation Converse:All P are SAll S are PNo Logical RelationshipS P S P*TAssume S exist..ImplicationS P*Some S are PConverse (equivalent)Some P are S
13 ObversionTo 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.
14 Negating a termTo negate a term is to refer to all things other than what the term designates.TnonT (things other than T).
15 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-tomatoesfunny stories --- things other than funny stories.
16 Negating English Terms (Further Examples) 1) non-citizennon-non-citizen2) onions picked by Miss Americanon-onions picked by Miss Americathings other than onions picked by Miss America3) truths told by Bill Clintonnon-truths told by Bill Clintonthings other than truths told by Bill Clinton4) theme parksthings other than theme parks.
17 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.
18 A and Its Obverse A: All S are P (No S are outside P) S PA: All S are P(No S are outside P)Obverse: No S are non-PEquivalent ----
19 E and Its Obverse E: No S are P. Obverse: All S are non-P. S PE: No S are P.Obverse: All S are non-P.(No S are in P)--- Equivalent ---
20 I and Its Obverse I: Some S are P Obverse: Some S are not non-P S P*I: Some S are PObverse: Some S are not non-P(Some S are not things outside P)--- Equivalent ---
21 O and Its Obverse O: Some S are not P Obverse: Some S are non-P S P*O: Some S are not PObverse: Some S are non-P(Some S are things outside P)--- Equivalent ---
22 Equivalent ?A E I OConverse No Yes Yes NoObverse Yes Yes Yes Yes
24 ContrapositionTo 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.
25 Contraposing a Categorical Sentence **** S ******* Pnegate**** non-P ******* non-S
26 A and Its Contrapositve S PA: All S are P.Contrapositive: All non-P are non-SS P(nothing outside P is inside S)--- Equivalent ---.
27 E and Its Contrapositve S PE: No S are P.Contrapositive: No non-P are non-SS P(nothing outside P is outside S)--- No Logical Relationship ---
28 I and Its Contrapositve S PI: Some S are P.*Contrapositive: Some non-P are non-SS P(Something outside P is outside S)*--- No Logical Relationship ---
29 O and Its Contrapositve S PO: Some S are not P.*Contrapositive: Some non-P are not non-SS P(Something outside P is inside S)*--- Equivalent ---
30 Equivalent ? A E I O Converse No Yes Yes No Obverse Yes Yes Yes Yes Contrapositive Yes No No Yes