Thought for the Evening: Probability as a Measure
When we measure something of type X, it's fairly common for us to do it by comparison to something else of type X as a reference point. So, for instance, our most basic way of measuring length is to use an extension to measure an extension. Likewise, time-measurement is the measurement of changes by a change, which latter we often call generically a 'clock'. Weight, at least as a measure, is ordinarily measured by comparison of mass to mass, as on a balance scale. And so forth. This isn't universal, by any means, and you can do all sorts of more complicated kinds of measurements (measuring length by clock, for instance). But these are often derivative or else depend on complicated theory that needs already to be in place; and, at the very least, the most commonly accessible kind of measurement is to measure things of type X by relating them to a thing of type X.
If we are thinking in these terms, and think of probability as a measure of something, it makes sense to think of a probability as using a possibility to measure possibilities. Take for instance the standard dice roll example. We have identified a possible event -- the upfacing result of a roll -- and use that to measure a possible kind of state, the sides of the die that can face up. And thus each gets a measure of 1/6, all other things being equal, just as a length might get the measure of being 1/6 of the standard ruler.
This raises an interesting question. In most cases of using a thing of type X to measure things of type X, it looks like the comparison opens the possibility of some kind of relativity, for lack of a better term; the measurement can change given on how the measuring thing is related to the measured. Thus length and time measurements depend on how the measuring thing and the measured thing are moving with respect to each other; weight measurements depend on the gravitational field, so your measurements will depend on the gravitational forces acting on the measuring mass and the measured mass -- if there is a disparity between the two, it will directly affect your measurement. So is there something like this for probability, as well?
Here is a reason to think that measuring possibility and measured possibility can come apart due to an external factor in a way that at least makes room for this. If we roll two dice, we are usually trying to add the numbers assigned to each dice, which is more complicated than simply considering the single die. But we can easily handle this to calculate that the probability of rolling a 2 is a bit under three percent, the probability of rolling a 3 is a tad over three and a half percent, the probability of rolling a 4 is somewhat over eight percent, and so forth. But consider St. Olaf's Roll. St. Olaf rolls two six-sided dice and rolls a thirteen, because one of the dice shows a 6 and the other die, when it hits the ground, splits in two and leaves its 1 and 6 faces up. What is the probability of rolling a 13 with two normal six-sided dice? People will sometimes get into complicated accounts of physics and what not in an attempt at least to gesture at some finite probability, but in fact this is all handwaving. The probability of rolling a 13 with two normal six-sided dice, as measured by the sides in the normal way, is zero, just as you would have expected beforehand. But regardless of the exact relation between possibility and causation, measurement of possibilities by possibilities depends on a presumed causal set-up, and modifying that causal set-up in a way that affects the measuring and the measured differently alters things -- just as measurements of length, time, and weight depend on some causal set-up which, if modified, modifies the measurements.
So St. Olaf's Roll seems to suggest that there are causal presuppositions in place for probability as well as for length, time, and the like, that at least allow for the possibility of relativity. The implications of this are, I think, well beyond my capacity to sound. In particular, you can have a fairly precise account of what the causal conditions are for length, time, and weight. I don't know what the corresponding account for probability would look like.
Various Links of Interest
* "Men" Is Not A Group Capable of Taking Action at "DarwinCatholic"
* Margee Kerr, Why is it fun to be frightened?
* Patrick McKenzie, Japan's Hometown Tax
* The homily Oscar Romero gave at the Mass at which he was killed.
* Charlotte Allen, Peter Damian's Counsel.
* The Record Acting Game with Vincent Price. ht: MrsD.
* Sortes Vigilianae
* Lauren N. Ross, Causal Explanation and the Periodic Table (PDF)
* Jeanne Peijnenburg and David Atkinson, “Till at last there remain nothing”: Hume’s Treatise 1.4.1 in contemporary perspective (PDF)
* I really like this story by Owen Stephens about when a British Lieutenant who had never played a Role Playing Game sat down for a Star Wars RPG (ht: Christopher Lansdown):
Mary Shelley, Frankenstein
Tim Button and Sean Walsh, Philosophy and Model Theory
G. E. M. Anscombe, Intention
Mary Midgley, Wickedness: A Philosophical Essay
Jules Verne, The Adventures of a Special Correspondent