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I was reading this question and I was curious, since I haven't quite grasp this topic being fairly new and all, how much would it differ the "tree" or list of deductions if instead of $\vdash \exists x (Px \rightarrow \forall x Px)$ we had $\vdash \exists x (Px \rightarrow \forall y Py)$. With the same axioms from the aforementioned question:

  • Tautologies
  • $\forall x \alpha \rightarrow \alpha_t^x$, where $t$ is substitutable for $x$ in $\alpha$.
  • $\forall x (\alpha \rightarrow \beta)\rightarrow(\forall x \alpha \rightarrow \forall x \beta)$

And if the language includes equality,

  • $x=x$
  • $x=y \rightarrow (\alpha \rightarrow \alpha')$ , where $ \alpha$ is atomic and $\alpha'$ is obtained from $\alpha$ by replacing $x$ in zero or more places by $y$.

It seems like maybe we could use a substitution but I'm not sure its possible since $x$ occurs freely in $P(x)$. Any light shed on this problem would be highly appreciated, many thanks.

Ben-ZT
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  • Yes, $y$ will be substitutable for $x$ if and only if $y$ does not occur free in $P(x)$ (otherwise a free variable may be bound during the substitution which is forbidden). – Graham Kemp Apr 05 '18 at 01:22
  • Since you said you are new....Hilbert's logic (which you are citing above) wasn't made to be user friendly. It is more of a proof of concept logic, designed to show that the idea of logic is possible. Miserable to work with. I'd suggest starting with natural deduction instead (even though historically Hilbert's approach came first). – DanielV Apr 05 '18 at 05:13
  • Nothing changes, because $∀xPx$ and $∀yPy$ and $∀zPz$ ... are all equivalent: they all mean e.g. "all men are mortals" and there are no difference between the $x$-men and the $y$-men. – Mauro ALLEGRANZA Apr 05 '18 at 06:11
  • @Mauro I think this change of variable can be done only when the variable in question (the one that is being changed) isn't present in any other instance in the formula, right? – Ben-ZT Apr 06 '18 at 17:56

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