Sequent calculus is, in essence, a style of formal logical
argumentation where every line of a proof is a conditional
tautology (called a
sequent by
Gerhard Gentzen) instead of an unconditional tautology. Each conditional tautology is inferred from other conditional tautologies on earlier lines in a formal argument according to rules and procedures of
inference, giving a better approximation to the style of natural deduction used by mathematicians than
David Hilbert's earlier style of formal logic where every line was an unconditional tautology. There may be more subtle distinctions to be made; for example, there may be non-logical axioms upon which all propositions are implicitly dependent. Then sequents signify conditional
theorems in a
first-order language rather than conditional tautologies.