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:
“We shall do better to abandon the whole attempt to learn the truth ... unless we can trust to the human mind’s having such a power of guessing right that before very many hypotheses shall have been tried, intelligent guessing may be expected to lead us to one which will support all tests, leaving the vast majority of possible hypotheses unexamined.”
—Charles S. Pierce (1839–1914)