Generalization

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In mathematical logic, generalization (also universal generalization, GEN) is an inference rule of predicate calculus. 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, φ a formula, and has been derived. The generalization rule states that can be derived if x is not mentioned in Γ and x does not occur in φ.

These restrictions are necessary for soundness. Without the first restriction, one could conclude from the hypothesis P(y). Without the second restriction, one could make the following deduction:

1. (Hypothesis)
2. (Existential instantiation)
3. (Existential instantiaion)
4. (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		Axiom PRED-1
4		From (1) and (3) by Modus Ponens
5		Axiom PRED-1
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, the deduction theorem was applicable in steps 10 and 11 because the formulas being moved have no free variables.

See also

• First-order logic
• Universal instantiation
• Hasty generalization
• Generalization error
• Categorical imperative redirected from "Generalization in Ethics"
• Generalizations of Fibonacci numbers
• Generalizations of Pauli matrices
• Derivative (generalizations)