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I have often read in logic books that some arguments, like the famous classic argument, "All men are mortal; Socrates is a man. Therefore, Socrates is mortal"

  1. $\forall x\;$(Man(x) $\implies$ Mortal(x))
  2. Man(Socrates)
  3. Mortal(Socrates)

can be shown valid in predicate logic but not in propositional logic. How does one rigorously prove this assertion? That is, how does one rule out the possibility of showing the above argument's validity using some clever application of propositional logic?

ryang
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user107952
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    It's not that it can't be proved, it can't even be stated since propositional logic lacks quantifiers. This is a pre-mathematical observation, not something you can prove (although it should also be the case that there is no encoding of predicate logic into propositional logic that preserves provability). – Naïm Camille Favier Jun 30 '25 at 16:22

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