According to Ramsey's Foundations of Mathematics, chapter III, suppose $'a', 'b', ..., 'z'$ were all the individuals, then $\phi{a}.\phi(b)...\phi(z)$ expresses the same proposition as $(x)\phi(x)$ because they express agreement and disagreement with the same possibilities. (see bottom half of page 34 of first edition.)

But according to W&R's PM 1st ed, General Judgments, (Chapter II, page 47), "all men are mortal" is a radically different kind of judgement, because from "Socrates is mortal and Plato is mortal and ..." one can infer that "Socrates is mortal," but from "all men are mortal" there is no way one can deduce "Socrates is mortal" before knowing that "Socrates is a man."

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    $\begingroup$ But Ramsey's point is not that $\phi(a) \wedge \phi(b) \dots \phi(z)$ is equivalent to $\forall x (\phi(x) \implies \psi(x))$ for some $\psi$. Rather, it is equivalent to $\phi(x)$. Thus, the conjunction would be analogous to "Socrates is a man and Plato is a man and ... and Zeno is a man" being equivalent to "Everything is a man". $\endgroup$ – Nagase Nov 9 '15 at 22:48
  • $\begingroup$ @Nagase, good point! $\endgroup$ – George Chen Nov 9 '15 at 22:55
  • $\begingroup$ @Nagase - Your comment was short but was right on point. Could you please post it as an answer? Thanks. $\endgroup$ – George Chen Jan 22 '18 at 18:21
  • $\begingroup$ I was thinking of studying Principia Mathematica and write my PhD thesis on it. Would you mind if I ask you some questions regarding it (for example a possible reading guideline to PM)? $\endgroup$ – user 170039 Jun 9 '18 at 14:41
  • $\begingroup$ @user170039 - Not at all. Those who answered my questions know infinitely more than I do. $\endgroup$ – George Chen Jun 9 '18 at 16:00

As I mentioned in the comments to the question, Ramsey's point is not that $\phi(a) \wedge \phi(b) \wedge \dots \wedge \phi(z)$ is equivalent to $\forall x (\phi(x) \rightarrow \psi(x))$. Rather, $\phi(a) \wedge \phi(b) \wedge \dots \phi(z)$ is equivalent to $\forall x \phi(x)$. So "Socrates is a man and Plato is a man and (...) and Zeno is a man" is not equivalent to "Every man is mortal", but rather to "Everything is a man" (supposing, of course, that we list all the objects of our domain in the conjunction!).


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