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.
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 y is not mentioned in Γ and x does not occur in φ....
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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.
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 y 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:
This purports to show that which is an unsound deduction.
Prove: .
Proof:
In this proof, the deduction theorem was applicable in steps 10 and 11 because the formulas being moved have no free variables.
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