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We're in first-order logic, using the Hilbert system (with inference rule of modus ponens), and the problem is to show $\vdash \exists x (Px \rightarrow \forall x Px)$.

We just learned about this stuff, so I'm really not sure where to begin. I believe we should create a sequence of deductions from $\exists x Px$ in order to arrive to the "proof of" $\forall x Px$. And we can use the following "axioms" along with modus ponens in order to arrive there:

  • Tautologies

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

  • $\forall x (\alpha \rightarrow \beta)\rightarrow(\forall x \alpha \rightarrow \forall x \beta)$

  • $\alpha \rightarrow \forall x \alpha$, where $x$ does not occur free in $\alpha$

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 $y$.
Git Gud
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Bobby Lee
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  • Do you mean $\exists x(Px\rightarrow \forall yPy)$? – Arthur Nov 09 '14 at 19:34
  • No, it is written with just $x$ in my book problem. – Bobby Lee Nov 09 '14 at 19:43
  • I just think it's strange to quantify twice over $x$ is all. – Arthur Nov 09 '14 at 19:54
  • Anyway, an intuitive "proof" goes like this: either $\forall xPx$, in which case the sentence $Px\rightarrow \forall xPx$ is a tautology. Otherwise, there is at least one $x$ such that $\lnot Px$. If you chose such an $x$ from the $\exists x$-part, the sentence is of the form "false $\to$ false", which is also a tautology. – Arthur Nov 09 '14 at 20:00

1 Answers1

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In this post you can find a proof of :

$\vdash (∀xβ→α)↔∃x(β→α)$, if $x$ is not free in $\alpha$.

The proof uses the axioms you have listed.

In the above theorem we replace $\beta$ with $Px$ and $\alpha$ with $\forall xPx$. Clearly, $x$ is not free in $\forall xPx$: thus, the proviso of the theorem applies :

$(∀xPx→∀xPx)↔∃x(Px→∀xPx)$.

The LHS is a tautology; thus, by modus ponens :

$\vdash ∃x(Px→∀xPx)$.

We can prove it directly with the above axioms and rules (see Herbert Enderton, A Mathematical Introduction to Logic (2nd - 2001)) :

(1) $\lnot Px \vdash Px \rightarrow \forall x Px$ --- by $\vDash_{TAUT} \lnot \phi \rightarrow (\phi \rightarrow \psi)$

(2) $\lnot Px \vdash \exists x (Px \rightarrow \forall x Px)$ --- from (1) by the "derived rule" : $\varphi_t^x \rightarrow \exists x \varphi$ [provable from Ax.2] and modus ponens

(3) $\vdash \lnot Px \rightarrow \exists x (Px \rightarrow \forall x Px)$ --- from (2) by Deduction Theorem

(4) $\lnot \exists x (Px \rightarrow \forall x Px) \vdash \lnot Px \rightarrow \lnot \exists x (Px \rightarrow \forall x Px)$ --- by $\vDash_{TAUT} \phi \rightarrow (\psi \rightarrow \phi)$

(5) $\lnot \exists x (Px \rightarrow \forall x Px) \vdash Px$ --- from (3) and (4) by $\vDash_{TAUT} (\lnot \phi \rightarrow \lnot \psi) \rightarrow ((\lnot \phi \rightarrow \psi) \rightarrow \phi)$

(6) $\lnot \exists x (Px \rightarrow \forall x Px) \vdash \forall x Px$ --- from (5) by GENERALIZATION THEOREM [page 117] : $x$ is not free in the assumption

(7) $\lnot \exists x (Px \rightarrow \forall x Px) \vdash Px \rightarrow \forall x Px$ --- from (6) by $\vDash_{TAUT} \phi \rightarrow (\psi \rightarrow \phi)$

(8) $\lnot \exists x (Px \rightarrow \forall x Px) \vdash \exists x(Px \rightarrow \forall x Px)$ --- From (7) by the "derived rule" of (2) and modus ponens

(9) $\vdash \lnot \exists x (Px \rightarrow \forall x Px) \rightarrow \exists x(Px \rightarrow \forall x Px)$ --- from (9) by Deduction Theorem

(10) $\vdash \lnot \exists x (Px \rightarrow \forall x Px) \rightarrow \lnot \exists x(Px \rightarrow \forall x Px)$ --- by $\vDash_{TAUT} \phi \rightarrow \phi$

(11) $\vdash \exists x (Px \rightarrow \forall x Px)$ --- from (9) and (10) by $\vDash_{TAUT} (\lnot \phi \rightarrow \lnot \psi) \rightarrow ((\lnot \phi \rightarrow \psi) \rightarrow \phi)$

Community
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Mauro ALLEGRANZA
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  • Wait you showed $\neg Px \vdash \exists x (Px \rightarrow \forall x Px)$ in line 3. Then you showed $\neg Px$ in line 4. Isn't this enough to show $\vdash \exists x (Px \rightarrow \forall x Px)$? – Bobby Lee Nov 10 '14 at 19:50
  • @Bobby Lee - NO - In line (3) we have $¬Px \vdash ∃x(Px→∀Px)$ but in line (4) we have $¬∃x(Px→∀Px)⊢¬Px→¬∃x(Px→∀Px)$ ... – Mauro ALLEGRANZA Nov 11 '14 at 07:18
  • How does line 2 follow from modus ponens? We would need $Pt \rightarrow \forall x Px$. – CuriousKid7 Mar 10 '17 at 05:00
  • Could you please explain how the derived rule is being used? I don't see how it's applicable here. – CuriousKid7 Mar 10 '17 at 17:11
  • @CuriousKid7 - No, we do not need (the invalid) $Pt→∀xPx$; I've used the valid $φ→∃xφ$, where $φ$ is : $Px→∀xPx$. – Mauro ALLEGRANZA Mar 10 '17 at 19:17
  • But we need to get $\phi_t^x$ in order to use MP to get $\exists x \phi$. That is, we need to instantiate $\phi$ with some constant that does not appear in $\phi$, correct? If so, I don't see how we do this here. – CuriousKid7 Mar 10 '17 at 19:36
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    NO; $t$ is a term and not a constant and $x$ is a term; thus $\varphi^x_x$ is $\varphi$. – Mauro ALLEGRANZA Mar 10 '17 at 19:44
  • Ah, I see. I was thinking of existential instantiation in Enderton. But what you're saying is something different Thank you! – CuriousKid7 Mar 10 '17 at 19:51