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Difference between revisions of "Derived rule"

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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031300/d0313001.png" /></td> </tr></table>
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$$\frac{A\supset B,B\supset C}{A\supset C}$$
  
 
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031300/d0313002.png" /></td> </tr></table>
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$$A\supset B,B\supset C\vdash A\supset C.$$
  
 
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.

Revision as of 10:11, 27 September 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