Substitution (logic)

Substitution is a fundamental concept in logic. A substitution is a syntactic transformation on formal expressions. To apply a substitution to an expression means to consistently replace its variable, or placeholder, symbols by other expressions. The resulting expression is called a substitution instance of the original expression.

Where ψ and φ represent formulas of propositional logic, ψ is a substitution instance of φ if and only if ψ may be obtained from φ by substituting formulas for symbols in φ, always replacing an occurrence of the same symbol by an occurrence of the same formula. For example:

is a substitution instance of:

and

is a substitution instance of:

In some deduction systems for propositional logic, a new expression (a proposition) may be entered on a line of a derivation if it is a substitution instance of a previous line of the derivation (Hunter 1971, p. 118). This is how new lines are introduced in some axiomatic systems. In systems that use rules of transformation, a rule may include the use of a substitution instance for the purpose of introducing certain variables into a derivation.

In first-order logic, every closed propositional formula that can be derived from an open propositional formula a by substitution is said to be a substitution instance of a. If a is a closed propositional formula we count a itself as its only substitution instance.

A propositional formula is a tautology if it is true under every valuation (or interpretation) of its predicate symbols. If Φ is a tautology, and Θ is a substitution instance of Φ, then Θ is again a tautology. This fact implies the soundness of the deduction rule described in the previous section.

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