Recently I’ve been puzzling over the ‘coin puzzle’ recently spotlighted by Roger White. White uses it to raise trouble for a view of personal probabilities (variously called ‘mushy credences’, ‘fuzzy credences’, ‘imprecise probabilities’, ‘vague probabilities’, ‘thick confidences’) which characteristically represents personal probability states using sets of credence functions, such that each of the functions in the set individually conforms to the probability calculus and updates by conditionalization. The mushy credence view is often motivated by the desire to account for the difference between judgements of equiprobability and suspension of judgement.
I think White’s coin puzzle does serious damage to the mushy credence view. Properly understood, it is a way of making vivid the phenomenon of dilation, which results when mushy credences interact with sharp credences. Dilation isn’t news to the proponents of mushy credences, but as the coin puzzle shows, in combination with objective chance it becomes problematic. A known correlation of some p in which we have mushy credence with the result of some coin toss for which we know the sharp chance of heads leads to a mushy credence in heads, even when we knew the pre-toss chance. This mushification of our credence in chancy outcomes is repugnant and provides us with a reason to reject one of the premises that led us to it. But the only controversial premise was the mushy credence view.
This is bad news for proponents of mushy credences. Some I have spoken to would bite the bullet and accept the counter-intuitive consequences of known chance dilation, seeking to soften the impact by emphasizing that the coin case is unrealistic. Others (here I am thinking of Scott Sturgeon’s forthcoming paper in OSE) take this as reason to abandon the sets-of-credence-functions-individually-updated model. Others, like White, take it as reason to abandon mushy credences altogether and explain suspension of judgement in a different sort of way. Which route is the most promising?
I suspect that biting the bullet would prove too painful, and that once we abandon the formal model there wouldn’t be much left of the mushy credences view; but I won’t defend these claims here. Rather, my plan is to support the case against mushy credences by examining some cases which might have seemed amenable to mushy treatment, and providing an alternative non-mushy explanation of what is going on in these cases. So here are some example cases to ponder over. In the next part of this post, I’ll apply the mushy treatment to these cases and ask how well it fares with them.
[Percy] Someone you’ve never met emails to tells you that they use the name ‘Percy’ for some particular proposition; they don’t tell you anything about what Percy says. What is your credence that Percy is contingent? What is your credence that Percy is true?
[Quiz] In a tie-breaker round of the pub quiz you are asked how many times Australia have won the Ashes. You never watch sports or read the sports pages, and don’t know what the Ashes are or how often they are competed for. What is your credence that the number is larger than 10? What is your credence that the number is even?
[Constant] Fundamental physics reveals that the value of a certain ‘fundamental constant’ Q of nature is around 75. It turns out that a value of 74 or lower for Q would have resulted in a failure for stars to form; a value of 76 or higher would have resulted in a supergiant black hole sucking in the entire universe. Whether or not we take these results to be evidence for eg a benevolent God or multiple universes will depend on how antecedently unlikely we take a value of 75 for Q to be. What prior credence should we have had in Q being between 74 and 76?
[Cube] Bas van Fraassen locks you in a mystery cube factory. You discover that no cubes produced have edges longer than 2 metres. You have no other evidence about the distribution of cube size. What is your credence that the next cube produced will have a volume larger than a cubic metre?
[Ignorant tennis] You sit down to watch a tennis match. You have never heard of either player, they appear evenly-matched in fitness and physique. What is your credence that player A will win?
[Knowledgeable tennis] You sit down to watch a tennis match. You have coached both players, and have an detailed knowledge of their abilities and playing style. You consider them exactly evenly-matched. What is your credence that player A will win?
(thanks to John Cusbert for this last pair of examples.)
[To be continued]