# Universal generalization

Transformation rules |
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Propositional calculus |

Rules of inference |

Rules of replacement |

Predicate logic |

In predicate logic, **generalization** (also **universal generalization** or **universal introduction**,^{[1]}^{[2]}^{[3]} **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:** is derivable from and .

**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.