Saturday, September 15, 2012

Deflation of the foundations of probability

I don't really want to commit to the following, but it has some attraction.

Question 1: What is probability?

Answer: Any assignment of values that satisfies the Kolmogorov axioms or an appropriate analogue of them (say, a propositional one).

Question 2: Are probabilities to be interpreted along frequentist, propensity or epistemic/logical lines?

Answer: Frequency-based, propensity-based and epistemically-based assignments of weights are all probabilities when the assignments satisfy the axioms or an appropriate analogue of them. In particular, improved frequentist probabilities are genuine probabilities when they can be defined, but so are propensity-based objective probabilities if they satisfy the axioms, and likewise logical probabilities. Each of these may have a place in the world.

Question 3: But what about the big metaphysical and epistemological questions, say about the grounds of objective tendencies and epistemic probabilities?

Answer: Those questions are intact. But they are not questions about the interpretation of probability as such. They are questions about the grounds of objective propensity or about the grounds of epistemic assignments. Thus, the former question belongs to the philosophy of science and the metaphysics of causation and the latter to epistemology.

Question 4: But surely one of the interpretations of probability is fundamental.

Answer: Maybe, but do we need to think so? Take the axioms of group theory. There are many kinds of structures that satisfy these axioms. Why think one kind of structure satisfying the axioms of group theory is fundamental?

Question 5: Still, couldn't there be connections, such as that logical probabilities ultimately derive from propensities via some version of the Principal Principle, or the other way around?

Answer: Maybe. But even if so, that doesn't affect the deflationary theory. There are plenty more structures that satisfy the probability calculus that do not derive from propensities.

Question 6: But shouldn't we think there is a focal Aristotelian sense of probability from which the others derive?

Answer: Maybe, but unlikely given the wide variety of things that instantiate the axioms. Maybe instead of an Aristotelian pros hen analogy, all we have is structural resemblance.


Heath White said...

Well, here are some disjointed thoughts.

Probability is probability of truth. Deflationists about truth want to say something very similar to what you are saying about probability; they want to say there is this word, ‘true’, which figures in a certain set of inferences, and that’s all there is to it. You are saying there is this assignment of numbers which obeys certain axioms, and that’s it. Probability deals with more complicated inferences than truth simpliciter but the gist is the same.

Realist theorists of truth reply that, for various reasons, the deflationary story about truth is insufficient. I have a lot of sympathy with that, but I would guess that the probability deflationist might be okay with it too. He (you) could just say that probability is probability of truth, and there is a serious question about the reality of truth, but we can be deflationists about the probability bit. That might work.

At any rate, one way to think of the classical theories of probability is as versions or extensions of truthmaker theory. They are looking for what makes something probable (i.e. probably true) and truthmaker theorists are looking for what makes it true, full stop.

Jonah Schupbach said...

Question 7. But what about all of those additive measures that satisfy the axioms, but which clearly are not probabitlities -- like an appropriately normalized measure of mass?

Alexander R Pruss said...

That's a very good objection and perhaps decisive. On the other hand mmathematicians do say that all you need for a measure to be a probability measure is that the total measure be 1. But maybe not all probability measures measure probabilities.

Alexander R Pruss said...


Right: it is a search for grounds. But my suggestion is that "probable" is said in many ways, and different ways call for different grounds.


I'm still thinking about your objection. Maybe the thing to go for then is a family resemblance view. Tendencies, rationally required degrees of belief and frequencies all resemble each other in respect of formal features and in terms of some non-formal features as well. This family resemblance makes it appropriate to call them all probabilities.

But this isn't quite a deflation, because there now is a need to analyze the family resemblance. Maybe it has something to do with truth or justification.

There is something smug about deflationary stories like the one I had given, and thank you for puncturing it.

Jonah Schupbach said...

Well, I wish I could say that I thought of the objection ... this observation is floating around the interpretations of probability literature -- e.g., Alan Hajek says something to this effect in his SEP entry.

I'm actually sympathetic to the deflationary line. For a while, I've wanted to be a pluralist about interpretations of prob. It seems to me that all the standard interpretations are getting at concepts of probability. And what makes them all concepts of *probability* is the fact that these things all satisfy the axioms. So Question 7 actually bothers me quite a bit!

I'd be curious to hear your thoughts on one way that I've thought of trying to respond -- which is quite different than the family resemblance approach: Can't we get around this sort of objection to the deflationary line if we get into more detail about the domain of probability functions?? The domain is some algebra (or sigma-algebra) of, say, propositions. But doesn't this requirement rule out the annoying cases of measures with total measure 1 that are, e.g., measures of volume or area? Maybe not, but it seems to me that this might do the trick.

Alexander R Pruss said...

I am not sure domain restrictions help. Consider some finite sigma-algebra of propositions. Imagine a possible world where there is some community of N individuals such that the function F(p)=(number of individuals who hate p)/N happens to satisfy the Kolmogorov axioms. Seems clearly wrong to say that these are probabilities!