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Natural deduction in its modern form was independently proposed by the German mathematician Gerhard Gentzen in 1933, in a dissertation delivered to the faculty of mathematical sciences of the University of Göttingen. [3] The term natural deduction (or rather, its German equivalent natürliches Schließen) was coined in that paper:
Gerhard Karl Erich Gentzen (24 November 1909 – 4 August 1945) was a German mathematician and logician. He made major contributions to the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus. He died of starvation in a Czech prison camp in Prague in 1945.
Dag Prawitz (born 1936, Stockholm) is a Swedish philosopher and logician.He is best known for his work on proof theory and the foundations of natural deduction. [1] [2]Prawitz is a member of the Norwegian Academy of Science and Letters, [3] of the Royal Swedish Academy of Letters and Antiquity and the Royal Swedish Academy of Science.
In mathematical logic, a deduction theorem is a metatheorem that justifies doing conditional proofs from a hypothesis in systems that do not explicitly axiomatize that hypothesis, i.e. to prove an implication A → B, it is sufficient to assume A as a hypothesis and then proceed to derive B.
In proof theory and mathematical logic, sequent calculus is a family of formal systems sharing a certain style of inference and certain formal properties. The first sequent calculi systems, LK and LJ, were introduced in 1934/1935 by Gerhard Gentzen [1] as a tool for studying natural deduction in first-order logic (in classical and intuitionistic versions, respectively).
An inference of natural deduction is a normal form, according to Dag Prawitz, if no formula occurrence is both the principal premise of an elimination rule and the conclusion of an introduction rule. [1]
Below, the left-hand side formalizes intuitionistic implicational natural deduction as a calculus of sequents (the use of sequents is standard in discussions of the Curry–Howard isomorphism as it allows the deduction rules to be stated more cleanly) with implicit weakening and the right-hand side shows the typing rules of lambda calculus.
A mathematical proof is a deductive ... mathematical induction is a method of deduction, ... be a mathematical statement involving the natural number n ...