I am just learning about Universal Generalization and don't understand why it is even brought up. There is nothing to prove in "∀xP(x) is true, given the premise that P(c) is true for all elements c in the domain", the ∀ already literally says that the proposition is valid for all elements. I could accept it as a "definition", fine, but how is this a "rule"???
To me that sounds just like, "For all means for all! Bam, proof!" - and I'm so done with this ---