In predicate logic, generalization (also universal generalization, universal introduction,[1][2][3] GEN, UG) is a valid inference rule.

The full generalization rule allows for hypotheses to the left of the turnstile, but with restrictions.

Assume

is a set of formulas,

a formula, and

The generalization rule states that

does not occur in

These restrictions are necessary for soundness.

Without the first restriction, one could conclude

{\displaystyle \forall xP(x)}

from the hypothesis

Without the second restriction, one could make the following deduction: This purports to show that

which is an unsound deduction.

(the second restriction need not apply, as the semantic structure of

is not being changed by the substitution of any variables).

Proof: In this proof, universal generalization was used in step 8.

The deduction theorem was applicable in steps 10 and 11 because the formulas being moved have no free variables.