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This is a homework question, but I just have no clue how to approach it. I'm trying to find the step in the following "derivation" which violates the allowed rules of inference. Sorry if the formatting is terrible.

    1 Show ∃x(Fx ---> ∀xFx)

      2 Show Fx ---> ∀xFx

         3 Fx

         4 Show ∀xFx           (by Rep of line 3)

         5 Fx

    6 ∃x(Fx ---> ∀xFx)         (Existential Generalization of Line 2)
  • So you're supposed to show without any assumptions that ∃x (Fx ---> ∀x Fx)? Are your parenthesis correct? Are both variables supposed to be x? – virmaior Oct 23 '17 at 1:47
  • ..... no..... I used the quotation marks (in "derivation") in an ironic sense. Sorry if that wasn't clear. This is NOT a proof, and I have to find the step where it fails. I mean OBVIOUSLY this isn't a proof because the conclusion is self-evidently false. But I can't find the illogical step in the proof. – Clclstdnt Oct 23 '17 at 1:54
  • (yes, both variables are supposed to be x) – Clclstdnt Oct 23 '17 at 1:59
  • 2
    The conclusion is true (at least in classical logic), see drinker paradox. – Arno Oct 23 '17 at 10:24
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You are trying to prove: ∃x (Fx → ∀x Fx) and your derivation starts with the assumption Fx.

If so, the step with the Universal Generalization, deriving ∀xFx from Fx, is wrong.

The rule has the proviso that the variable to be quantified, in this case x, must not be free in any assumption; in the proof, we have Fx as assumption, and x is free in it.

The proviso formalizes the intuitive rule that "what holds for any, holds for all". I.e. if we consider a "generic" object x and we prove that P holds of it, we can infer that Px holds for every x.

In order to be sound, the above rule must be applied to an x whatever: an x for which no presupposition has been stated.

This fact will be formalized with the restriction that no assumption regarding x must be present in the proof. If so, we can safely generalize it.

In the (wrong) proof above, we assume Fx, and thus we cannot assert that x is "generic", because we do not know if every object is F or not.


For a discussion, see Drinker paradox.

For proofs, see the posts: Why is this true and Proof of Drinker paradox.

  • Thanks! Could you please elaborate on why deriving ∀xFx from Fx is wrong? It clearly is wrong.... It's like saying "Here is a green thing..... therefore all things are green". I know you explain this further in "The rule has the ..... free in it." But I don't quite follow this... could you phrase it slightly different? Thanks – Clclstdnt Oct 23 '17 at 21:50

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