The first systematic formulations of the propositional and predicate calculi were presented axiomatically, on the analogy of certain branches of mathematics. In 1934, Gerhard Gentzen (1909–1945), a logician of Hilbert's school, published a formalization of logical principles more in accordance with the way in which these principles are customarily applied. (A similar approach was developed independently by S. Jaśkowski; see below, section on Polish logicians.) In illustrating his technique Gentzen considered how we might establish as valid the schema (X ∨ (Y & Z)) ⊃ (X ∨ Y) & (X ∨ Z). Assuming that the antecedent holds, either X is true, or Y & Z is true. In the former case we can pass to each of X ∨ Y and X ∨ Z and hence to their joint assertion. Assuming now Y & Z, we may infer Y, whence X ∨ Y, and likewise Z, whence X ∨ Z. In this case the conjunction is once more derivable. Since it...