Substitution (logic)

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. For any formal system that is closed under substitution, any substitution of a tautology will also produce a tautology.

Definition

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 ${\displaystyle \leftrightarrow }$ A) ${\displaystyle \leftrightarrow }$ (A ${\displaystyle \leftrightarrow }$ A)

is a substitution instance of:

(A ${\displaystyle \leftrightarrow }$ 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.

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

Tautologies

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.