# Rule of replacement

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Inference rule that may be applied to only a particular segment of an expression

In logic, a **rule of replacement**[1][2][3] is a [transformation rule](/source/Transformation_rule) that may be applied to only a particular segment of an [expression](/source/Well-formed_formula). A [logical system](/source/Logical_system) may be constructed so that it uses either [axioms](/source/Axiom), [rules of inference](/source/Rules_of_inference), or both as transformation rules for [logical expressions](/source/Well-formed_formula) 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](/source/Logical_proof), [logically equivalent](/source/Logically_equivalent) expressions may replace each other. Rules of replacement are used in [propositional logic](/source/Propositional_logic) to manipulate [propositions](/source/Proposition).

Common rules of replacement include [de Morgan's laws](/source/De_Morgan's_laws), [commutation](/source/Commutative_property), association, distribution, [double negation](/source/Double_negation),[a] [transposition](/source/Transposition_(logic)), [material implication](/source/Material_implication_(rule_of_inference)), [logical equivalence](/source/Logical_equivalence), [exportation](/source/Exportation_(logic)), and [tautology](/source/Tautology_(rule_of_inference)).

## Table: Rules of Replacement

The rules above can be summed up in the following table.[4] The "[Tautology](/source/Tautology_(logic))" 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}}} ( ( p ∨ q ) ∨ r ) → ( p ∨ ( q ∨ r ) ) {\displaystyle ((p\vee q)\vee r)\rightarrow (p\vee (q\vee r))} Associative p ∧ q ∴ q ∧ p ¯ {\displaystyle {\begin{aligned}p\wedge q\\\therefore {\overline {q\wedge p}}\\\end{aligned}}} ( p ∧ q ) → ( q ∧ p ) {\displaystyle (p\wedge q)\rightarrow (q\wedge p)} Commutative ( p ∧ q ) → r ∴ p → ( q → r ) ¯ {\displaystyle {\begin{aligned}(p\wedge q)\rightarrow r\\\therefore {\overline {p\rightarrow (q\rightarrow r)}}\\\end{aligned}}} ( ( p ∧ q ) → r ) → ( p → ( q → r ) ) {\displaystyle ((p\wedge q)\rightarrow r)\rightarrow (p\rightarrow (q\rightarrow r))} Exportation p → q ∴ ¬ q → ¬ p ¯ {\displaystyle {\begin{aligned}p\rightarrow q\\\therefore {\overline {\neg q\rightarrow \neg p}}\\\end{aligned}}} ( p → q ) → ( ¬ q → ¬ p ) {\displaystyle (p\rightarrow q)\rightarrow (\neg q\rightarrow \neg p)} Transposition or contraposition law p → q ∴ ¬ p ∨ q ¯ {\displaystyle {\begin{aligned}p\rightarrow q\\\therefore {\overline {\neg p\vee q}}\\\end{aligned}}} ( p → q ) → ( ¬ p ∨ q ) {\displaystyle (p\rightarrow q)\rightarrow (\neg p\vee q)} 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}}} ( ( p ∨ q ) ∧ r ) → ( ( p ∧ r ) ∨ ( q ∧ r ) ) {\displaystyle ((p\vee q)\wedge r)\rightarrow ((p\wedge r)\vee (q\wedge r))} Distributive p q ∴ p ∧ q ¯ {\displaystyle {\begin{aligned}p\\q\\\therefore {\overline {p\wedge q}}\\\end{aligned}}} ( ( p ) ∧ ( q ) ) → ( p ∧ q ) {\displaystyle ((p)\wedge (q))\rightarrow (p\wedge q)} Conjunction p ∴ ¬ ¬ p ¯ {\displaystyle {\begin{aligned}p\\\therefore {\overline {\neg \neg p}}\\\end{aligned}}} p → ( ¬ ¬ p ) {\displaystyle p\rightarrow (\neg \neg p)} Double negation introduction ¬ ¬ p ∴ p ¯ {\displaystyle {\begin{aligned}{\neg \neg p}\\\therefore {\overline {p}}\\\end{aligned}}} ( ¬ ¬ p ) → p {\displaystyle (\neg \neg p)\rightarrow p} Double negation elimination

## See also

- *[Salva veritate](/source/Salva_veritate)*

## Notes

1. **[^](#cite_ref-4)** not admitted in [intuitionistic logic](/source/Intuitionistic_logic)

## References

1. **[^](#cite_ref-1)** Copi, Irving M.; Cohen, Carl (2005). *Introduction to Logic*. Prentice Hall.

1. **[^](#cite_ref-2)** Hurley, Patrick (1991). [*A Concise Introduction to Logic 4th edition*](https://archive.org/details/studyguidetoacco00burc). Wadsworth Publishing. [ISBN](/source/ISBN_(identifier)) [9780534145156](https://en.wikipedia.org/wiki/Special:BookSources/9780534145156).

1. **[^](#cite_ref-3)** Moore and Parker [*[full citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citing_sources#What_information_to_include)*]

1. **[^](#cite_ref-5)** Kenneth H. Rosen: *Discrete Mathematics and its Applications*, Fifth Edition, p. 58.

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Adapted from the Wikipedia article [Rule of replacement](https://en.wikipedia.org/wiki/Rule_of_replacement) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Rule_of_replacement?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.