Substitution instance

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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 any substitution of a tautology is again a tautology.

[edit] 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 $\to$ S) & (T $\to$ S)

is a substitution instance of:

P & Q

and

(A $\leftrightarrow$ A) $\leftrightarrow$ (A $\leftrightarrow$ A)

is a substitution instance of:

(A $\leftrightarrow$ 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.

[edit] 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.

[edit] References

Hunter, G. (1971). Metalogic: An Introduction tothe Metatheory of Standard First Order Logic. University of California Press. ISBN 0-520-01822-2
Kleene, S. C. (1967). Mathematical Logic. Reprinted 2002, Dover. ISBN 0-486-42533-9

$\lnot(p\land\lnot p)$