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Kind of proof calculus
In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts with Hilbert style systems, which instead use axioms as much as possible to express the logical laws of deductive reasoning.

Natural deduction grew out of a context of dissatisfaction with the axiomatizations of deductive reasoning common to the systems of Hilbert, Frege, and Russell (see, e. g., Hilbert system). Such axiomatizations were most famously used by Russell and Whitehead in their mathematical treatise Principia Mathematica. Spurred on by a series of seminars in Poland in 1926 by Łukasiewicz that advocated a more natural treatment of logic, Jaśkowski made the earliest attempts at defining a more natural deduction, first in 1929 using a diagrammatic notation, and later updating his proposal in a sequence of papers in 1934 and 1935. His proposals led to different notations
such as Fitch style calculus (or Fitch's diagrams) or Suppes' method for which Lemmon gave a variant called system L.

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. The term natural deduction (or rather, its German equivalent natürliches Schließen) was coined in that paper:

Gentzen was motivated by a desire to establish the consistency of number theory. He was unable to prove the main result required for the consistency result, the cut elimination theorem—the Hauptsatz—directly for natural deduction. For this reason he introduced his alternative system, the sequent calculus, for which he proved the Hauptsatz both for classical and intuitionistic logic. In a series of seminars in 1961 and 1962 Prawitz gave a comprehensive summary of natural deduction calculi, and transported much of Gentzen's work with sequent calculi into the natural deduction framework. His 1965 monograph Natural deduction: a proof theoretical study was to become a reference work on natural deduction, and included applications for modal and second order logic. In natural deduction, a proposition is deduced from a collection of premises by applying inference rules repeatedly. The system presented in this article is a minor variation of Gentzen's or Prawitz's formulation, but with a closer adherence to Martin Löf's description of logical judgments and connectives.

Natural deduction has had a large variety of notation styles, indicated antecedent dependencies by line numbers in square brackets, anticipating Suppes' 1957 line number notation. 1950: In a textbook, demonstrated a method of using one or more asterisks to the left of each line of proof to indicate dependencies. This is equivalent to Kleene's vertical bars. (It is not totally clear if Quine's asterisk notation appeared in the original 1950 edition or was added in a later edition.) 1957: An introduction to practical logic theorem proving in a textbook by This indicated dependencies (i. e. antecedent propositions) by line numbers at the left of each line. 1963: uses sets of line numbers to indicate antecedent dependencies of the lines of sequential logical arguments based on natural deduction inference rules. 1965: The entire textbook by is an introduction to logic proofs using a method based on that of Suppes, what is now known as Suppes–Lemmon notation. 1967: In a textbook, briefly demonstrated two kinds of practical logic proofs, one system using explicit quotations of antecedent propositions on the left of each line, the other system using vertical bar lines on the left to indicate dependencies.