Mathematical proposition

A mathematical proposition is a statement considered under specified hypotheses. In classical logic it has a truth value once the meanings of its symbols and any free parameters are fixed. A sentence has no free variables; a formula such as $x=x$ has a free variable until it is assigned a value or quantified.

Hypotheses and quantifiers

The conditional proposition $P\Rightarrow Q$ says that $Q$ follows whenever $P$ holds. Proving that conditional does not prove its hypothesis $P$ . Similarly, $\forall x\exists y\,R(x,y)$ permits a different witness for each $x$ , whereas $\exists y\forall x\,R(x,y)$ requires a common witness.

Role in exposition

A result called “Proposition” is usually a proved statement; its heading alone is not evidence of a proof. The logical content comes from its complete hypotheses, quantifiers and conclusion.

References

Jonathan Pila, Set Theory, Sections 2–9 and 15.