Proposition

Definition

A proposition is a declarative statement that is either true or false, but not both.

Why It Matters

Propositions are the atoms of mathematical reasoning. Every theorem, lemma, and corollary is ultimately a proposition. Before we can prove anything, we must express it as a proposition.

Formal Statement

A proposition \(P\) is a statement with a definite truth value:

\[P \in \{\text{true}, \text{false}\}\]

Properties

Bivalence

Every proposition has exactly one truth value: true or false. There is no third option.

Law of Excluded Middle

For any proposition \(P\):

\[P \lor \neg P\]

This is always true. Either \(P\) is true, or its negation is true.

Law of Non-Contradiction

For any proposition \(P\):

\[\neg(P \land \neg P)\]

\(P\) and \(\neg P\) cannot both be true simultaneously.

Examples

Example 1: A Proposition

"The number 7 is prime."

This is a proposition. Its truth value is true.

Example 2: A Proposition

"The number 9 is even."

This is a proposition. Its truth value is false.

Example 3: Not a Proposition

"Is 7 prime?"

This is a question, not a proposition. It has no truth value.

Example 4: Not a Proposition

"This statement is false."

This is a paradox (the liar paradox), not a valid proposition. If it is true, then it is false. If it is false, then it is true.

Example 5: A Proposition with Unknown Truth Value

"There are infinitely many twin primes."

This is a proposition, but its truth value is currently unknown. It is either true or false; we simply do not know which yet.

Logical Connective — How to combine propositions
Direct Proof — How to prove propositions