# Substitution rule

#### Logic

One of the derivation rules (cf. Derivation rule) in logico-mathematical calculi. By a "substitution rule" one may mean various forms of the rule. For example, in the propositional calculus a substitution rule is a formula together with all occurrences of the propositional variable. In predicate calculus it is: a) a formula together with the predicate variable (here one has to obey a series of constraints on the occurrence of the individual variables in order to avoid variable collisions, i.e. a situation were a variable that is free in the formula becomes bounded as a result of the substitution); b) a substitution rule for a term together with free occurrences of an individual variable of the corresponding type (here also it is necessary to avoid variable collisions).

#### Analysis

The term Substitution rule is often used to denote the formula for the change of variables in an integral, see also Integration by substitution.

#### References

 [1] P.S. Novikov, "Elements of mathematical logic" , Oliver & Boyd and Acad. Press (1964) (Translated from Russian) [2] J.R. Shoenfield, "Mathematical logic" , Addison-Wesley (1967) [3] D. Hilbert, P. Bernays, "Grundlagen der Mathematik" , 1–2 , Springer (1968–1970)
How to Cite This Entry:
Substitution rule. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Substitution_rule&oldid=28785
This article was adapted from an original article by S.N. Artemov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article