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Full Idea
Universal Generalization: If we can prove P(x), only assuming what sort of object x is, we may conclude ∀xP(x) for the same x.
Gist of Idea
Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x)
Source
Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.3)
Book Ref
Wolf,Robert S.: 'A Tour Through Mathematical Logic' [Carus Maths Monographs 2005], p.20
A Reaction
This principle needs watching closely. If you pick one person in London, with no presuppositions, and it happens to be a woman, can you conclude that all the people in London are women? Fine in logic and mathematics, suspect in life.
17745 | For Frege, 'All A's are B's' means that the concept A implies the concept B [Frege, by Walicki] |
13905 | If there is a finite domain and all objects have names, complex conjunctions can replace universal quantifiers [Lemmon] |
13522 | Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS] |
13521 | Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS] |