# Difference between revisions of "Derived rule"

From Encyclopedia of Mathematics

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''of a derivation in a given calculus'' | ''of a derivation in a given calculus'' | ||

A [[Derivation rule|derivation rule]] whose conclusion is derivable from its premises in the calculus under consideration. For example, in [[Propositional calculus|propositional calculus]] the derivation rule | A [[Derivation rule|derivation rule]] whose conclusion is derivable from its premises in the calculus under consideration. For example, in [[Propositional calculus|propositional calculus]] the derivation rule | ||

− | + | $$\frac{A\supset B,B\supset C}{A\supset C}$$ | |

is a derived rule, since in this calculus there is derivability from the premises: | is a derived rule, since in this calculus there is derivability from the premises: | ||

− | + | $$A\supset B,B\supset C\vdash A\supset C.$$ | |

Every derived rule is a [[Sound rule|sound rule]], but not every sound rule is a derived rule. For example, the [[Substitution rule|substitution rule]] in propositional calculus is a sound but not a derived rule. | Every derived rule is a [[Sound rule|sound rule]], but not every sound rule is a derived rule. For example, the [[Substitution rule|substitution rule]] in propositional calculus is a sound but not a derived rule. | ||

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====References==== | ====References==== | ||

<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S.C. Kleene, "Introduction to metamathematics" , North-Holland (1951)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> S.C. Kleene, "Introduction to metamathematics" , North-Holland (1951)</TD></TR></table> | ||

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+ | [[Category:Logic and foundations]] |

## Latest revision as of 19:13, 17 October 2014

*of a derivation in a given calculus*

A derivation rule whose conclusion is derivable from its premises in the calculus under consideration. For example, in propositional calculus the derivation rule

$$\frac{A\supset B,B\supset C}{A\supset C}$$

is a derived rule, since in this calculus there is derivability from the premises:

$$A\supset B,B\supset C\vdash A\supset C.$$

Every derived rule is a sound rule, but not every sound rule is a derived rule. For example, the substitution rule in propositional calculus is a sound but not a derived rule.

#### References

[1] | S.C. Kleene, "Introduction to metamathematics" , North-Holland (1951) |

**How to Cite This Entry:**

Derived rule.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Derived_rule&oldid=15149

This article was adapted from an original article by S.N. Artemov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article