Editing Deductive reasoning (section)

=== Natural deduction ===
The term "[[natural deduction]]" refers to a class of proof systems based on self-evident rules of inference.<ref name="IEPNatural" /><ref name="StanfordNatural">{{Cite encyclopedia |year=2021 |title=Natural Deduction Systems in Logic |encyclopedia=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |url=https://plato.stanford.edu/entries/natural-deduction/ |access-date=15 March 2022 |last2=Hazen |first2=Allen |last1=Pelletier |first1=Francis Jeffry}}</ref> The first systems of natural deduction were developed by [[Gerhard Gentzen]] and [[Stanislaw Jaskowski]] in the 1930s. The core motivation was to give a simple presentation of deductive reasoning that closely mirrors how reasoning actually takes place.<ref>{{Cite journal |last=Gentzen |first=Gerhard |date=1934 |title=Untersuchungen über das logische Schließen. I |url=https://gdz.sub.uni-goettingen.de/id/PPN266833020_0039?tify={%22pages%22:[180],%22panX%22:0.559,%22panY%22:0.785,%22view%22:%22info%22,%22zoom%22:0.411} |journal=Mathematische Zeitschrift |volume=39 |issue=2 |pages=176–210 |doi=10.1007/BF01201353 |s2cid=121546341 |quote=Ich wollte nun zunächst einmal einen Formalismus aufstellen, der dem wirklichen Schließen möglichst nahe kommt. So ergab sich ein "Kalkül des natürlichen Schließens. (First I wished to construct a formalism that comes as close as possible to actual reasoning. Thus arose a "calculus of natural deduction".) |lang=de|url-access=subscription }}</ref> In this sense, natural deduction stands in contrast to other less intuitive proof systems, such as [[Hilbert-style deductive systems]], which employ axiom schemes to express [[logical truth]]s.<ref name="IEPNatural">{{Cite encyclopedia |title=Natural Deduction |encyclopedia=Internet Encyclopedia of Philosophy |url=https://iep.utm.edu/natural-deduction/ |access-date=15 March 2022 |last1=Indrzejczak |first1=Andrzej}}</ref> Natural deduction, on the other hand, avoids axioms schemes by including many different rules of inference that can be used to formulate proofs. These rules of inference express how [[logical constant]]s behave. They are often divided into [[Natural deduction#Introduction and elimination|introduction rules and elimination rules]]. Introduction rules specify under which conditions a logical constant may be introduced into a new sentence of the [[Formal proof|proof]].<ref name="IEPNatural" /><ref name="StanfordNatural" /> For example, the introduction rule for the logical constant {{nowrap|"<math>\land</math>"}} (and) is {{nowrap|"<math>\frac{A, B}{(A \land B)}</math>"}}. It expresses that, given the premises {{nowrap|"<math>A</math>"}} and {{nowrap|"<math>B</math>"}} individually, one may draw the conclusion {{nowrap|"<math>A \land B</math>"}} and thereby include it in one's proof. This way, the symbol {{nowrap|"<math>\land</math>"}} is introduced into the proof. The removal of this symbol is governed by other rules of inference, such as the elimination rule {{nowrap|"<math>\frac{(A \land B)}{A}</math>"}}, which states that one may deduce the sentence {{nowrap|"<math>A</math>"}} from the premise {{nowrap|"<math>(A \land B)</math>"}}. Similar introduction and elimination rules are given for other logical constants, such as the propositional operator {{nowrap|"<math>\lnot</math>"}}, the [[Logical connective|propositional connectives]] {{nowrap|"<math>\lor</math>"}} and {{nowrap|"<math>\rightarrow</math>"}}, and the [[Quantifier (logic)|quantifiers]] {{nowrap|"<math>\exists</math>"}} and {{nowrap|"<math>\forall</math>"}}.<ref name="IEPNatural" /><ref name="StanfordNatural" />

The focus on rules of inferences instead of axiom schemes is an important feature of natural deduction.<ref name="IEPNatural" /><ref name="StanfordNatural" /> But there is no general agreement on how natural deduction is to be defined. Some theorists hold that all proof systems with this feature are forms of natural deduction. This would include various forms of [[sequent calculi]]{{efn|name=natDeduc |1= In natural deduction, a simplified [[sequent]] consists of an environment <math>\Gamma</math> that yields (<math>\vdash</math>) a single conclusion <math>C</math>; a single sequent would take the form
:"''Assumptions'' A1, A2, A3 etc. yield ''Conclusion'' C1"; in the symbols of [[natural deduction]], <math>\Gamma A_1, A_2, A_3 ... \vdash C_1</math> 
*However if the premises were true but the conclusion were false, a hidden assumption could be intervening; alternatively, a hidden process might be coercing the form of presentation, and so forth; then the task would be to unearth the hidden factors in an ill-formed syllogism, in order to make the form valid. 
*''see [[Deduction theorem]]''
}}  or [[Method of analytic tableaux#Tableau calculi and their properties|tableau calculi]]. But other theorists use the term in a more narrow sense, for example, to refer to the proof systems developed by Gentzen and Jaskowski. Because of its simplicity, natural deduction is often used for teaching logic to students.<ref name="IEPNatural" />