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
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