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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 ---

Rautermann
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    That isn’t really what universal generalization says. An informal version of what it actually says is that if we can prove $P(x)$ without assuming anything about $x$ beyond the fact that it’s in the domain of discourse, then we can conclude $\forall x,P(x)$. You can see the actual formal statement of universal generalization here. – Brian M. Scott Apr 12 '21 at 20:30
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    Universal generalization is the rule corresponding to the "arbitrary element" technique in mathematical arguments: to show that $\forall xP(x)$ is true, we introduce an "arbitrary" object $c$, show that $P$ holds of $c$, and then conclude that $P$ in fact holds of every element since $c$ was arbitrary. (In the context of formal proof, we want precise rules which let us build "valid sequents" (or similar), and natural-language proof strategies need to be reflected by such rules (or combinations thereof). Based on your last couple sentences, I think this may be a point of confusion.) – Noah Schweber Apr 12 '21 at 20:34
  • Your argument applies to all the rules of first-order logic: they are all "obviously" correct, in some sense. However, that does not make logic uninteresting, because what logic is trying to do is to model our reasoning processes in a rigorous way that avoids appealing to any intuitions about the meaning of symbols. In saying you are "done"with mathematical logic, you are giving up on a lot o interesting ideas, both mathematical and philosophical. – Rob Arthan Apr 13 '21 at 00:02
  • First of all, thanks for all your comments (I can't upvote yet)! Just to make it clear, I am seriously confused by this and am indeed trying to understand it - not throwing in the towel yet. I think my main concern is that I cannot think of any sensible definition of ∀ (and it has to be defined first, right?) that does not already explicitly include being true for arbitrary elements, making this a somewhat circular reasoning without any value added.
    I can - to a degree - enjoy the art of first-order logic, this is the first time I really fail to see the point.     – Rautermann Apr 14 '21 at 10:15

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