rule, replacement, logic, rule, replacement, transformation, rule, that, applied, only, particular, segment, expression, logical, system, constructed, that, uses, either, axioms, rules, inference, both, transformation, rules, logical, expressions, system, wher. In logic a rule of replacement 1 2 3 is a transformation rule that may be applied to only a particular segment of an expression A logical system may be constructed so that it uses either axioms rules of inference or both as transformation rules for logical expressions in the system Whereas a rule of inference is always applied to a whole logical expression a rule of replacement may be applied to only a particular segment Within the context of a logical proof logically equivalent expressions may replace each other Rules of replacement are used in propositional logic to manipulate propositions Common rules of replacement include de Morgan s laws commutation association distribution double negation a transposition material implication logical equivalence exportation and tautology Contents 1 Table Rules of Replacement 2 See also 3 Notes 4 ReferencesTable Rules of Replacement EditThe rules above can be summed up in the following table 4 The Tautology column shows how to interpret the notation of a given rule Rules of inference Tautology Name p q r p q r displaystyle begin aligned p vee q vee r therefore overline p vee q vee r end aligned nbsp p q r p q r displaystyle p vee q vee r rightarrow p vee q vee r nbsp Associativep q q p displaystyle begin aligned p wedge q therefore overline q wedge p end aligned nbsp p q q p displaystyle p wedge q rightarrow q wedge p nbsp Commutative p q r p q r displaystyle begin aligned p wedge q rightarrow r therefore overline p rightarrow q rightarrow r end aligned nbsp p q r p q r displaystyle p wedge q rightarrow r rightarrow p rightarrow q rightarrow r nbsp Exportationp q q p displaystyle begin aligned p rightarrow q therefore overline neg q rightarrow neg p end aligned nbsp p q q p displaystyle p rightarrow q rightarrow neg q rightarrow neg p nbsp Transposition or contraposition lawp q p q displaystyle begin aligned p rightarrow q therefore overline neg p vee q end aligned nbsp p q p q displaystyle p rightarrow q rightarrow neg p vee q nbsp Material implication p q r p r q r displaystyle begin aligned p vee q wedge r therefore overline p wedge r vee q wedge r end aligned nbsp p q r p r q r displaystyle p vee q wedge r rightarrow p wedge r vee q wedge r nbsp Distributivep q p q displaystyle begin aligned p q therefore overline p wedge q end aligned nbsp p q p q displaystyle p wedge q rightarrow p wedge q nbsp Conjunctionp p displaystyle begin aligned p therefore overline neg neg p end aligned nbsp p p displaystyle p rightarrow neg neg p nbsp Double negation introduction p p displaystyle begin aligned neg neg p therefore overline p end aligned nbsp p p displaystyle neg neg p rightarrow p nbsp Double negation eliminationSee also EditSalva veritateNotes Edit not admitted in intuitionistic logicReferences Edit Copi Irving M Cohen Carl 2005 Introduction to Logic Prentice Hall Hurley Patrick 1991 A Concise Introduction to Logic 4th edition Wadsworth Publishing ISBN 9780534145156 Moore and Parker full citation needed Kenneth H Rosen Discrete Mathematics and its Applications Fifth Edition p 58 nbsp This logic related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Rule of replacement amp oldid 1169595875, wikipedia, wiki, book, books, library,