Universal generalization

In predicate logic, generalization is a valid inference rule. It states that if has been derived, then can be derived.

Generalization with hypotheses

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 has been derived. The generalization rule states that can be derived if is not mentioned in and does not occur in.
These restrictions are necessary for soundness. Without the first restriction, one could conclude from the hypothesis. Without the second restriction, one could make the following deduction:

This purports to show that which is an unsound deduction. Note that is permissible if is not mentioned in .

Example of a proof

Prove: is derivable from and.
Proof:

Step	Formula	Justification
1		Hypothesis
2		Hypothesis
3		From by Universal instantiation
4		From and by Modus ponens
5		From by Universal instantiation
6		From and by Modus ponens
7		From and by Modus ponens
8		From by Generalization
9		Summary of through
10		From by Deduction theorem
11		From by Deduction theorem

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.