Strict Logical Entailments of Categorical Propositions Immediate Inferences Conversion Obversion Contraposition

Conversion Simply Switch Positions of the Terms No Cats are Dogs E SP No Dogs are Cats E PS Valid operation for E and I statements

Obversion 1. Change the Quality 2. Form the complement of the predicate term All Popes are Men No Popes are non-men women?

In stating the complement of the predicate class, choose in the way that makes the best sense for the example. No sky-divers are people who wear glasses. All sky-divers are non-people who wear glasses ? ? All sky-divers are people who do not wear glasses.

Obversion is valid for all four forms (use ~ to mean “non”) A SP E S~P E SP A S~P I SP O S~P O SP I S~P

These operations can be combined No cats are dogs No dogs are cats -conversion All dogs are non-cats - obversion

But some moves are invalid, e.g., All dogs are non-cats (true) All non-cats are dogs (false) Conversion is not valid for A or O statements; nothing follows from converting an A or an O.

Contraposition Obversion of the Converse of an Obverse Since conversion is not valid for A or O, contraposition is not valid for E or I Why is that? Because if you begin with an E, obverting makes it an A, which cannot be converted validly

Obverse: No Popes are Women Contraposition Obverse of the converse of an obverse All Popes are Men Obverse: No Popes are Women Converse: No Women are Popes Obverse: All Women are non-Popes

Contraposition State the complement of each term and 2. Switch the position of the terms All dogs are mammals All non-mammals are non-dogs

Switch position of terms Conversion Valid for E and I only Change quality; state complement of the predicate Obversion valid for all forms Contraposition Form the complement of both terms, and switch position Valid: A, O

Logical relationships between statement forms The Square of Opposition A E I O

CONTRADICTORIES: opposite in truth-value A: universal affirmative O: particular negative E: universal negative I: particular affirmative CONTRADICTORIES: opposite in truth-value

CONTRARIES: A, E A: universal affirmative E: universal negative Cannot both be true, but can both be false. No dogs are birds T All dogs are birds F No women are presidential candidates All women are presidential candidates CONTRARIES: A, E Both false.

I, O: SUBCONTRARIES I: particular affirmative O: particular negative Cannot both be false, but can both be true (“some” means “there is at least one”) Some Senators are Republicans true Some Senators are not Republicans true Some cats are Great Danes false Some cats are not Great Danes true I, O: SUBCONTRARIES

A: universal affirmative I: particular affirmative E: universal negative O: particular negative If a universal statement is true, the particular of the same quality is necessarily true as well. SUBALTERNATION

A: universal affirmative I: particular affirmative E: universal negative O: particular negative If a particular statement is false, the universal of the same quality must also be false –necessarily. SUPERALTERNATION

The Square of Oppositions A contrary E t subalternation contradiction superalternation f I subcontrary O

Logical relations of Categorical Propositions Conversion, Obversion, Contraposition (generate immediate inferences) Square of Oppositions Allowed inferences from one propositional form to another