Penrose’s New Argument and Paradox

Abstract

In this paper we take a closer look at Penrose’s New Argument for the claim that the human mind cannot be mechanized and investigate whether the argument can be formalized in a sound and coherent way using a theory of truth and absolute provability. Our findings are negative; we can show that there will be no consistent theory that allows for a formalization of Penrose’s argument in a straightforward way. In a second step we consider Penrose’s overall strategy for arguing for his view and provide a reasonable theory of truth and absolute provability in which this strategy leads to a sound argument for the claim that the human mind cannot be mechanized. However, we argue that the argument is intuitively implausible since it relies on a pathological feature of the proposed theory.

Appendix

In this appendix we show that the first disjunct of GD cannot be proved on the basis of the principles (T _K), (T-I n), (T B), and (N e c).³⁴ We introduce some new terminology to state the result in a precise way. We write W _e to denote the output of the Turing machine with index e. Using this terminology the first disjunct of GD can be formulated as

$$\displaystyle \begin{aligned}&(\ast)&&\neg\exists e\forall x(W_e(x)\leftrightarrow Kx).&&\end{aligned} $$

We will show that (∗) is not a consequence of the principles (T _K), (T-I n), (T B), and (N e c).

Theorem 4

Let P A T K be the extension of PA in the language containing a truth and an absolute provability predicate ( $\mathbb {L}_{\mathsf {PATK}}$ ). Then

$$\displaystyle \begin{aligned}(\mathsf{T_K}), (T\mathit{\mbox{-}}\mathsf{In}), (\mathsf{TB}),(\mathsf{Nec})&\not\vdash_{\mathsf{PATK}}\neg\exists e\forall x(W_e(x)\leftrightarrow Kx).\end{aligned} $$

Proof

The proof of Theorem 4 is an compactness argument: if we can prove (∗) in P A T K on the basis of (T _K), (T-I n), (T B), and (N e c), then there must be a finite proof of (∗) in P A T K and, in particular, a proof with only finitely many applications of the rule (N e c). The argument we give shows that there cannot be such a proof of finite length for (∗). To this end we define a family of theories by recursion:

where C n denotes the operation that closes these sets of sentences under logical consequence. Σ _n allows for proofs with n-many applications of the rule (N e c). Notice also that by construction Σ _n+1 ⊆ Σ _n+1. By our compactness argument it follows that if

$$\displaystyle \begin{aligned}(\mathsf{T_K}), (T\mbox{-}\mathsf{In}), (\mathsf{TB}),(\mathsf{Nec})&\vdash_{\mathsf{PATK}}\neg\exists e\forall x(W_e(x)\leftrightarrow Kx),\end{aligned} $$

then there must be a proof of (∗) in some Σ _n for n ∈ ω. We now construct a suitable model M _n for each Σ _n with n ∈ ω such that

$$\displaystyle \begin{aligned}\mathsf{M}_n\models\exists e\forall x(W_e(x)\leftrightarrow Kx).\end{aligned} $$

Let $\mathsf {M}_0=(\mathbb {N}, \|T\|{ }_0,\|K\|{ }_0)$ with

$$\displaystyle \begin{aligned} \|T\|{}_0&:=\{\phi:(\mathbb{N}\models\phi\,\&\,\phi\in\mathbb{L}_{\mathsf{PA}})\text{ or }(\phi\in\mathbb{L}_{\mathsf{PATK}}\,\&\,\phi\not\in\mathbb{L}_{\mathsf{PA}})\}\\ \|K\|{}_0&:=\{\phi:\mathsf{PATK}\vdash\phi\}.\end{aligned} $$

Clearly, M ₀⊧Σ ₀ and M ₀⊧∀x(P r _PATK(x) ↔ Kx). But the theorems of P A T K are recursively enumerable and we thus know that M ₀⊧∃e(W _e(x) ↔ Kx). Now, for M _n we define

$$\displaystyle \begin{aligned}\|T\|{}_{n+1}&:=\|T\|{}_n\\ \|K\|{}_{n=1}&:=\{\phi:\varSigma_n\vdash\phi\}.\end{aligned} $$

It trivially follows that M _n+1⊧Σ _n+1 and $\mathsf {M}_{n+1}\models \forall x(\mathsf {Pr}_{\varSigma _n}(x)\leftrightarrow Kx)$, which implies M _n⊧∃e(W _e(x) ↔ Kx). This concludes the proof of Theorem 4. There cannot be a proof establishing the first disjunct of GD on the basis of the principles (T _K), (T-I n), (T B), and (N e c).