is a statement that is either true or false.
is a proposition, as is . "Professor George is taller than 5'" is a proposition. If is known to be 5, then " " is a proposition.
If a fact depends on a variable, we will not consider it to be a proposition. For example, "properties of , or predicates on . Similarly, we can have properties of two variables (e.g. ).
" by itself is not a proposition. Neither is " ". These statements are called
Predicates can be turned into propositions by quantifying them; stating that they must be true either for all or for some .
For example, "for any sets and , " is a proposition, as is "there exists some with ".
This process can be repeated: "for all , " is a predicate, because it depends on (but not !). But "there exists such that for every , " is a proposition.
We are often lazy and leave off the quantification of a variable in a statement. When we have an undefined variable in a claim, we implicitly mean "for all ". For example, if I ask you to prove or disprove that , I implicitly mean to prove or disprove that, for all sets and , .