The Journal of Philosophy

Gödel argued that his incompleteness theorems imply that either “the mind cannot be mechanized” or “there are absolutely undecidable sentences.” In the precursor to this paper I examined the early arguments for the first disjunct. In the present paper I examine the most sophisticated argument for the first disjunct, namely, Penrose’s new argument. It turns out that Penrose’s argument requires a type-free notion of truth and a type-free notion of absolute provability. I show that there is a natural such system, DTK. I prove a series of results which show that (1) Gödel’s disjunction is provable in the system, (2) Penrose’s argument is invalid in the system, (3) there can be no proof or refutation of either disjunct in the system, (4) the independence results are robust in that they persist when one strengthens the principles governing absolute provability, and (5) there are reasons to believe that the situation will not improve under any plausible alteration of the underlying theory of truth.

It is widely thought in philosophy and elsewhere that parsimony is a theoretical virtue in that if a theory T1 is more parsimonious than another theory T2, then T1 is preferable to T2, other things being equal. This thesis admits of many distinct precisifications. I focus on a relatively weak precisification on which preferability is a matter of probability, and argue that it is false. This is problematic for various alternative precisifications, and even for Inference to the Best Explanation as standardly understood.