Definition
A proposition is what a declarative sentence asserts — the content that can be true or false, independently of the language or wording used to express it. The English sentence "snow is white", the French "la neige est blanche", and the German "Schnee ist weiß" express the same proposition. Two different sentences can express the same proposition, and a single sentence can in some contexts express different propositions depending on who is speaking and when. Propositions are abstract: they are not sentences, not utterances, not mental states. They are the truth-bearers that logic takes as its objects.
In propositional logic, propositions are treated as the atomic units of analysis. Their internal structure is set aside; what matters is how they combine via connectives — and, or, not, if-then — to form complex statements whose truth values depend on the values of their components.
Why it matters
How it works
Propositional logic begins by abstracting away from the particular sentences that express propositions and using single letters to stand for them. The proposition that snow is white might be written as P, the proposition that grass is green as Q, and so on. From these atomic letters and the truth-functional connectives, every complex statement of propositional logic is built up. Each connective has a truth table that specifies how the truth value of the compound depends on the truth values of its components. Once propositions are represented this way, every question of validity, equivalence, and contradiction in propositional logic becomes a mechanical question about truth-table columns.
The distinction between proposition and sentence matters most when context, indexicals, or translation enter the picture. The sentence "I am hungry" expresses different propositions depending on who utters it; the sentence "the present king of France is bald" arguably expresses no proposition at all if there is no present king of France. These complications are the entry point to the philosophy of language and to more sophisticated logical systems such as modal logic and predicate logic. But for propositional calculus, the working assumption is simple: each atomic letter stands for one definite proposition with a definite truth value, and everything else follows from the connectives.