# Propositional Logic: Quality, Quantity, and Relationships

## Propositional Logic

### Quality and Quantity

**Propositions** are specific uses of language that convey information about the world and can be either true or false.

**Quality:** The quality of a proposition is either affirmative (includes a class in another) or negative (excludes a class from another).

**Quantity:** The quantity of a proposition refers to the extent of inclusion or exclusion. It is universal if full, and particular if partial.

**Quantifiers:** Words like “all,” “none,” and “some” determine the quantity of a proposition.

### Relationships Between Propositions

**Opposition:** Incompatibility between propositions with the same subject and predicate but differing in quantity, quality, or both.

**Contradictory:**Propositions that differ in both quality and quantity (e.g., “All S are P” and “Some S are not P”). One is always true, and the other is always false.**Contrary:**Universal propositions that differ in quality (e.g., “All S are P” and “No S are P”). Both cannot be true simultaneously, but both can be false.**Subcontrary:**Particular propositions that differ in quality (e.g., “Some S are P” and “Some S are not P”). Both can be true simultaneously, but both cannot be false.**Subalternation:**A relationship between a universal proposition and a particular proposition with the same quality. The truth of the universal implies the truth of the particular, but the falsity of the universal does not imply the falsity of the particular.

#### Examples of Propositional Relationships

**Universal Affirmative (A):**All judges are aware of the law.**Singular Affirmative:**Pedro Lopez is secretary of agreements.**Particular Affirmative (I):**Some documents are executive.**Universal Negative (E):**No fact is obvious testing ground.**Singular Negative:**Juan Campos has no personality in this trial.**Particular Negative (O):**Some laws are not constitutional.

### Immediate Inferences

Immediate inferences are drawn from a single premise.

**A:**All S is P**E:**No S is P**I:**Some S is P**O:**Some S is not P

The relationship between A and E is contrary. They cannot both be true, but they can both be false.

The relationship between A and O, and E and I, is contradictory. One is true, and the other is false.

The relationship between A and I, and E and O, is subalternation. The truth of the universal implies the truth of the particular.

The relationship between I and O is subcontrary. They can both be true, but they cannot both be false.