Substitution rule

From Encyclopedia of Mathematics
Jump to: navigation, search


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).


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


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