Editing Method of analytic tableaux (section)

{{Short description|Tool for proving a logical formula}}
[[File:Partially built tableau.svg|thumb|200px|A graphical representation of a partially built propositional tableau]]
In [[proof theory]], the '''semantic tableau'''<ref name=":1">{{Cite book |last=Howson |first=Colin |title=Logic with trees: an introduction to symbolic logic |date=1997 |publisher=Routledge |isbn=978-0-415-13342-5 |location=London; New York |pages=ix, x,24–29, 47}}</ref> ({{IPAc-en|t|æ|ˈ|b|l|oʊ|,_|ˈ|t|æ|b|l|oʊ}}; plural: '''tableaux'''), also called an '''analytic tableau''',<ref name=":0">{{Cite book |last=Restall |first=Greg |url= |title=Logic: an introduction |date=2006 |publisher=Routledge |isbn=978-0-415-40067-1 |series=Fundamentals of philosophy |location=London; New York |pages=5, 42, 55 |oclc=ocm63115330}}</ref> '''truth tree''',<ref name=":1" /> or simply '''tree''',<ref name=":0" /> is a [[decision procedure]] for [[sentential logic|sentential]] and related logics, and a [[proof procedure]] for formulae of [[first-order logic]].<ref name=":1" /> An analytic tableau is a tree structure computed for a logical formula, having at each node a subformula of the original formula to be proved or refuted. Computation constructs this tree and uses it to prove or refute the whole formula.{{sfn|Howson|2005|page=27}} The tableau method can also determine the [[satisfiability]] of finite sets of [[formula]]s of various logics. It is the most popular [[proof procedure]] for [[modal logic]]s.{{sfn|Girle|2014}}

A method of truth trees contains a fixed set of rules for producing trees from a given logical formula, or set of logical formulas. Those trees will have more formulas at each branch, and in some cases, a branch can come to contain both a formula and its negation, which is to say, a contradiction. In that case, the branch is said to '''close'''.<ref name=":1" /> If every branch in a tree closes, the tree itself is said to close. In virtue of the rules for construction of tableaux, a closed tree is a proof that the original formula, or set of formulas, used to construct it was itself self-contradictory,<ref name=":1" /> and therefore false. Conversely, a tableau can also prove that a logical formula is [[Tautology (logic)|tautologous]]: if a formula is tautologous, its negation is a contradiction, so a tableau built from its negation will close.<ref name=":1" />