# Universal generalization

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.

Transformation rules |
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Propositional calculus |

Rules of inference |

Rules of replacement |

Predicate logic |

## 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:

- (Hypothesis)
- (Existential instantiation)
- (Existential instantiation)
- (Faulty universal generalization)

This purports to show that which is an unsound deduction. Note that is permissible if is not mentioned in (the second restriction need not apply, as the semantic structure of is not being changed by the substitution of any variables).

## 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.

## References

- Copi and Cohen
- Hurley
- Moore and Parker