Was Hegel an Authoritarian Thinker? Reading Hegel’s Philosophy of History on the Basis of his Metaphysics

With Hegel’s metaphysics attracting renewed attention, it is time to address a long-standing criticism: Scholars from Marx to Popper and Habermas have worried that Hegel’s metaphysics has anti-individualist and authoritarian implications, which are particularly pronounced in his Philosophy of History, since Hegel identifies historical progress with reason imposing itself on individuals. Rather than proposing an alternative non-metaphysical conception of reason, as Pippin or Brandom have done, this article argues that critics are broadly right in their metaphysical reading of Hegel’s central concepts. However, they are mistaken about what Hegel’s approach entails, when one examines the specific types of states discussed by the philosopher in his Philosophy of History. Even on a traditional metaphysical reading, Hegel is not only non-authoritarian; he also makes a powerful argument concerning freedom, whereupon the freest society involves collective oversight and the shaping of social structures so as to ensure that they benefit everybody

How unprovable is Rabin's decidability theorem?

We study the strength of set-theoretic axioms needed to prove Rabin's theorem on the decidability of the MSO theory of the infinite binary tree. We first show that the complementation theorem for tree automata, which forms the technical core of typical proofs of Rabin's theorem, is equivalent over the moderately strong second-order arithmetic theory $\mathsf{ACA}_0$ to a determinacy principle implied by the positional determinacy of all parity games and implying the determinacy of all Gale-Stewart games given by boolean combinations of ${\bf \Sigma^0_2}$ sets. It follows that complementation for tree automata is provable from $\Pi^1_3$- but not $\Delta^1_3$-comprehension. We then use results due to MedSalem-Tanaka, M\"ollerfeld and Heinatsch-M\"ollerfeld to prove that over $\Pi^1_2$-comprehension, the complementation theorem for tree automata, decidability of the MSO theory of the infinite binary tree, positional determinacy of parity games and determinacy of $\mathrm{Bool}({\bf \Sigma^0_2})$ Gale-Stewart games are all equivalent. Moreover, these statements are equivalent to the $\Pi^1_3$-reflection principle for $\Pi^1_2$-comprehension. It follows in particular that Rabin's decidability theorem is not provable in $\Delta^1_3$-comprehension.Comment: 21 page