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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)
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.
