Propositions and Truth Values

Mathematical Logic · Axiom Academy

LESSON Propositions and Truth Values The foundation of mathematical logic Mathematical Logic • Unit 1 - Propositional Logic A proposition is the fundamental building block of logic. It's a declarative statement that is either true or false—but never both and never neither. Formal Definition: Proposition A proposition is a declarative sentence that has exactly one of two possible truth values : true or false. In symbols: Every proposition satisfies exactly one of: Every proposition has exactly one truth value. We use multiple notations to represent these values: Simple Propositions and Propositional Variables A simple or atomic proposition cannot be broken down into simpler parts. We use lowercase letters to represent them as variables. We denote propositions with letters: Each variable stands for a complete proposition with a truth value. Examples of Atomic Propositions Let's examine concrete examples of propositions and their truth values: TRUE: "5 is a prime number" — This is factually correct FALSE: "4 is a prime number" — This is factually incorrect TRUE: "Paris is the capital of France" — Verifiable fact FALSE: "0 = 1" — Mathematical falsehood Questions: "Is 5 prime?" — No truth value Commands: "Calculate 2 + 2" — No truth value Open sentences: "x > 5" — Truth depends on x Paradoxes: "This sentence is false" — No consistent truth value This fundamental principle states that every proposition must have exactly one truth value—true or false. No middle ground, no ambiguity.

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