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 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 . 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.
Read more about this topic: Universal Generalization
Famous quotes containing the word hypotheses:
“But don’t despise error. When touched by genius, when led by chance, the most superior truth can come into being from even the most foolish error. The important inventions which have been brought about in every realm of science from false hypotheses number in the hundreds, indeed in the thousands.”
—Stefan Zweig (18811942)