Universal generalization

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:

  1. (Hypothesis)
  2. (Existential instantiation)
  3. (Existential instantiation)
  4. (Faulty universal generalization)

This purports to show that which is an unsound deduction.

Example of a proof

Prove: .


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