7.2 KB · ⇊ 4 parts · ⤓ Micron 7.2 KB · ⤓ HTML 55.2 KB

top

In logic, a rule of replacementcite-ref-1[1]cite-ref-2[2]cite-ref-3[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,cite-ref-4[a] transposition, material implication, logical equivalence, exportation, and tautology.

• Table: Rules of Replacement

• See also

• Notes

• References

The rules above can be summed up in the following table.cite-ref-5[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}}} | ( ( 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 |

• Salva veritate

cite-note-4a. ↑ not admitted in intuitionistic logic

cite-note-11. ↑ citerefcopicohen2005Copi, Irving M.; Cohen, Carl (2005). Introduction to Logic. Prentice Hall.

cite-note-22. ↑ citerefhurley1991Hurley, Patrick (1991). A Concise Introduction to Logic 4th edition. Wadsworth Publishing. ISBN 9780534145156.

cite-note-33. ↑ Moore and Parker

cite-note-54. ↑ Kenneth H. Rosen: Discrete Mathematics and its Applications, Fifth Edition, p. 58.