Frege argued that number concept are, in the first place, second-order predicates. When we talk about numbers as objects, we use a logical device of "nominalization" that introduces object-level representations of higher-level properties. In Grundgesetze, he assumed that every first-order predicate can be nominalized: for every first-order predicate F, there is an associated object – the "extension" of F – such that F and G are associated with the same object iff ∀x(Fx ↔︎ Gx). The number N is then identified with the extension of 'having an extension with N elements'. Unfortunately, the assumption that every predicate has an extension turned out to be inconsistent, so the whole approach collapsed.
In "Quasi-Realism is Fictionalism" (Lewis (2005)), Lewis seems to suggest that Blackburn's quasi-realism about moral discourse is a kind of fictionalism. The suggestion is bizarre. Has Lewis made silly mistake? (Spoiler: No.)
Let's compare what quasi-realism and fictionalism say about moral discourse.
Blackburn's quasi-realism (as presented, e.g., in Blackburn (1984, ch.6) and Blackburn (1993)) is a brand of expressivism. According to Blackburn, moral statements like (1) don't serve to describe special facts, but to express moral attitudes.
Bruno de Finetti (de Finetti (1970)) suggested that chance is objectified credence. The suggestion is explained and defended in Jeffrey (1983, ch.12), Skyrms (1980 ch.I), Skyrms (1984, ch.3), and Diaconis and Skyrms (2017, ch.7), but I still find it hard to understand. It seems to assume that rational credence functions are symmetrical in a way in which I think they shouldn't be.
The Best-Systems Account of chance promises to explain why beliefs about chance should affect our beliefs about ordinary events, as formalized by the Principal Principle. In this post, I want to discuss a challenge to any such explanation.
First, some background.
For any candidate chance function f, let [f] be the set of worlds of which f is (part of) the best system. According to the Best-Systems Account (BSA), the hypothesis "Ch=f" that f is the true chance function expresses the proposition [f]. In what follows, I'll assume that a world is simply a history of "outcomes", and that the candidate systems can be compressed into a single (possibly parameterized) chance function.
In 2009, at the ANU, Mike Titelbaum organized a small workshop on the Sleeping Beauty problem. I gave a talk in which I argued that the answer to the problem depends on whether we accept genuinely diachronic norms on rational belief: if yes, halfing is the most plausible answer; if no, we get thirding. A successor of this talk is now forthcoming in Noûs. Here's a PDF. In this post, I want to discuss a surprisingly hard question Kenny Easwaran raised in the Q&A after my talk:
How confident should Beauty be on Wednesday that the coin has landed heads?