Propositional logic explains how to work with the most basic elements of logic, propositions. A proposition is any statement that is either true or false but not both. Declarative sentences such as "The sky is blue," "I have one arm," and "1 + 1 = 3" are all propositions. Commands ("Close the door!"), interrogatives ("What are you doing?"), and sentences that include variables ("Someone is watching me.") are not propositions.

The truth or falsity of a proposition might change depending on the speaker and the time it is uttered. The proposition "I have one arm" is true only when spoken by a person with one arm and false when spoken by those with two or no arms. A proposition that is always true, for example, "Either it is raining or it is not raining," is called a tautology; a proposition that is always false, as with "1 + 1 = 3," is called a contradiction. The truth value of a proposition is T if it is true and F if it is false.

Compound propositions can be created through the use of logical operators. For example, if p is any proposition, the negation of p can be written, "It is not the case that p." Other logical operators include "and," "or," "if, then," and "if and only if." The truth or falsity of a compound proposition is determined by examining the truth or falsity of each of its component propositions. The truth of the compound proposition "p and q," for example, is true only when both p is true and q is true.

A statement that contains a variable, such as "x is divisible by 6," can be turned into a proposition through the use of a quantifier and a universe of discourse. The quantifiers come in two forms: universal ("For all x, ...") and existential ("There exists an x such that..."). The universe of discourse tells us what values x can possibly be. Through quantifiers the statement above can become the (false) proposition, "For all x such that x is a whole number, x is divisible by 6." Using the existential quantifier on the above statement produces a true proposition since there does exist at least one whole number that is divisible by six, six itself being one example.