proposition

Think of a proposition as a sentence about which it makes sense to ask “true or false?” — and to which there is, in principle, exactly one answer. “7 is a prime number” is a proposition (true). “The square root of 2 is rational” is also a proposition (false). By contrast “Close the door”, “What time is it?”, and “x + 1” are not propositions: a command, a question, and a bare expression cannot be true or false.

Formally, a proposition is a declarative statement that is unambiguously either true or false but not both. This is the law of the excluded middle together with non-contradiction, and it is the bedrock on which classical logic — and therefore almost all of analysis — is built. The truth value of a proposition need not be known to us; “every even number greater than 2 is a sum of two primes” is a perfectly good proposition even though no one yet knows whether it is true.

A caveat worth stating plainly: open sentences containing free variables, such as “x > 0”, are not yet propositions. They become propositions only once the variable is given a definite value (“3 > 0” is true) or bound by a quantifier (“for all real x, x > 0” is a false proposition). Keeping straight the difference between an open sentence and a closed proposition prevents a great deal of confusion later.