Proof system
A proof system or proof calculus is a formal system used to prove statements.
à la Hilbert
Proof systems à la Hilbert leave little distinction between propositions and judgements.
à la Gentzen
Proof systems à la Gentzen are distinguished by their use of contexts, so that every line in a deduction (i.e. every judgement) is conditional on zero or more assumptions, having the form
Examples of such systems include