**Substitution** is a fundamental concept in logic. Substitution is a syntactic transformation on strings of symbols of a formal language.

In propositional logic, a **substitution instance** of a propositional formula is a second formula obtained by replacing symbols of the original formula by other formulas. A key fact is that for any consistent formal system, any substitution of a tautology will also produce a tautology.

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:

- (R S) (T S)

is a substitution instance of:

- P Q

and

- (A A) (A A)

is a substitution instance of:

- (A A)

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.

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.