Statement and Its Truth Value

Communication mostly happens through sentences, which come in several types: declarative (assertive), imperative (a command or request), exclamatory (expressing emotion), interrogative (a question), and open (truth depends on an unstated condition).

Definition 12.7. A statement or proposition is any declarative sentence that is either true or false, but never both. An imperative, exclamatory or interrogative sentence is never a proposition.

The truth value of a statement is the fact of its being true or false: $T$ (or $1$) for a true statement, $F$ (or $0$) for a false one.

Open sentences are not (yet) propositions. An open sentence is one whose truth varies with a condition not stated in the sentence itself -- e.g. "$x\times7=35$" is true only when $x=5$ and false otherwise; "He is a bad person" varies with opinion. Such a sentence only becomes a genuine proposition once the missing condition is supplied (e.g. quantified over a fixed domain, or $x$ given a specific value). …