Universal generalization
Transformation rules 

Propositional calculus 
Rules of inference 
Rules of replacement 

Predicate logic 

In predicate logic, generalization (also universal generalization or universal introduction,^{}^{}^{}GEN) 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 Γ is a set of formulas, a formula, and has been derived. The generalization rule states that can be derived if y is not mentioned in Γ and x 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:
 (Hypothesis)
 (Existential instantiation)
 (Existential instantiation)
 (Faulty universal generalization)
This purports to show that which is an unsound deduction.
Example of a proof
Prove: .
Proof:
Number  Formula  Justification 

1  Hypothesis  
2  Hypothesis  
3  Universal instantiation  
4  From (1) and (3) by Modus ponens  
5  Universal instantiation  
6  From (2) and (5) by Modus ponens  
7  From (6) and (4) by Modus ponens  
8  From (7) by Generalization  
9  Summary of (1) through (8)  
10  From (9) by Deduction theorem  
11  From (10) 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.
See also
 Firstorder logic
 Hasty generalization
 Universal instantiation