Proposition — Definition, Formula & Examples
A proposition is a declarative statement that is either true or false, but not both. It is the basic building block of mathematical logic, used to construct arguments and proofs.
A proposition is a well-formed declarative sentence that has a definite truth value — exactly one of true (T) or false (F). Propositions can be combined using logical connectives (such as conjunction, disjunction, and conditionals) to form compound propositions, and they serve as the atomic units of propositional logic.
How It Works
To determine whether a sentence is a proposition, check two things: it must be a declarative statement (not a question or command), and it must be possible to assign it a truth value of true or false. The sentence "7 is a prime number" is a proposition because it declares something that is definitively true. The sentence "Is 7 prime?" is not a proposition because questions lack truth values. Once you identify propositions, you can combine them with logical connectives like "and," "or," and "if...then" to build more complex logical expressions.
Example
Problem: Classify each sentence as a proposition or not a proposition: (A) "12 is divisible by 4." (B) "Close the door." (C) "x + 3 = 10."
Sentence A: "12 is divisible by 4" is a declarative statement. Since 12 ÷ 4 = 3 with no remainder, the statement is true. It has a definite truth value, so it is a proposition.
Sentence B: "Close the door" is a command, not a declarative statement. Commands cannot be true or false, so this is not a proposition.
Sentence C: "x + 3 = 10" contains a free variable x. Its truth value depends on what x is, so as written it is not a proposition — it is an open sentence. It becomes a proposition only when x is assigned a specific value or when a quantifier (like "for all x" or "there exists an x") is applied.
Answer: (A) is a proposition (true). (B) is not a proposition. (C) is not a proposition as written — it is an open sentence.
Why It Matters
Propositions are the foundation of every mathematical proof you will encounter in geometry, discrete mathematics, and beyond. Understanding what qualifies as a proposition helps you construct valid logical arguments and avoid reasoning errors in proof-based courses and computer science applications like programming conditionals.
Common Mistakes
Mistake: Treating open sentences (like "x > 5") as propositions.
Correction: An open sentence with a free variable has no fixed truth value. It only becomes a proposition when the variable is replaced by a specific value or bound by a quantifier such as "for all x" or "there exists an x."