One question about subjective probability that I’ve received a few times is how necessarily false propositions can get a non-zero probability. It’s a consequence of any form of the axioms of probability that any tautologically false proposition must get probability zero, but depending on which form of the axioms one takes, this won’t have to be the case even for first-order logically false propositions, much less metaphysically impossible ones. The reason I say “depending on which form of the axioms one takes” is because there are several different forms that are generally considered to be equivalent.
For instance, Kolmogorov says that probability (1) is a non-negative function on sets of possible worlds (2) that assigns the value 1 to the complete set (3) and is (finitely or countably) additive on disjoint sets of worlds. Another formulation replaces (2) and (3), saying instead that logical truths get probability 1, and that probability adds on logically incompatible statements. Yet another (which I get from Brian Weatherson) replaces all four by saying that logical truths and falsehoods get probability 1 and 0 respectively, that if A entails B then P(A)≤P(B), and that P(A)+P(B)=P(A and B)+P(A or B).
The reason for this equivalence (and also, I think, the motivation behind the original questions) is that most people take a probability function (subjective, epistemic, logical, objective, or whatever) to be a function from sets of possible worlds to real numbers. (I’ve also heard of some proposals that would restrict to rationals, expand to complex numbers, or change to p-adics or some other number system. However, I think most of these suggestions occur for purely mathematical reasons and just encode certain similarities to probabilities, but aren’t really considered to be probabilities themselves.) The later formulations I give above talk about logical truths and the like, but it is very easy (and relatively standard) to talk about propositions as just being sets of worlds. And if the set of all possible worlds is taken to include all logically possible worlds (rather than metaphysically possible worlds), then I believe all the above formulations are equivalent. This slip between logical and metaphysical possibility is I think why people worry about whether impossible statements must get probability 0.
However, in many cases it seems natural to take a more neutral stance (as Brian Weatherson did) and just think of a probability function as an assignment of real numbers to propositions or even just statements in a language rather than sets of possible worlds. This was pursued by Karl Popper in appendices *iv and *v to The Logic of Scientific Discovery, where he tries to give an account of conditional probability on antecedents of probability zero. (I happen to think his particular approach is wrong, but he has the right general idea.) Popper’s axioms are quite nice, in that he can prove for any two tautologically equivalent statements that they get the same probability from the probability axioms alone, without making any assumptions about what the logical connectives mean or what propositions are.
I think that for dealing with tricky logics, or statements containing essential indexicals, this sort of distinction is important to make. People try to modify the “sets of possible worlds” approach by talking about “sets of centered possible worlds”, but I’m skeptical about how much can be accomplished in this way.
Antimeta 
Expanding to complex numbers is the essence of quantum mechanics and they are used because they actually work in the real world. One of the most bizarre things about quantum mechanics is that the “possible worlds” become a little more actual than possible and participate in interactions in an observable way. Hence phenomena like the double slit experiment or even “counterfactual computation”.
I think in an earlier draft I had “purely mathematical or physical reasons”. At any rate, I don’t know enough about quantum mechanics myself to say, but is there really a meaningful sense in which the complex numbers themselves are probabilities? I thought I had generally heard that you take the squared modulus of the number to get the probability, but maybe that’s mistaken.
I suppose though that there is no core sense of what probability really is – it’s just a family of related notions, like credence, partial entailment, partial belief, propensity, chance, and perhaps other notions that happen to satisfy (something very much like) the Kolmogorov axioms. These others with strange sets of numbers are just slightly more exotic members of the family.
The square modulus of the complex numbers (‘amplitudes’) gives you the probabilities. But it’s not as simple as that.
The ‘calculus’ that you use to derive these amplitudes is formally very similar to probability theory. In particular these amplitudes act formally like probabilities up until the very last step where you translate them into ‘normal’ probabilities by using |.|^2. For example, speaking crudely, the amplitude of “A or B”, where A and B are disjoint, is the sum of the separate amplitudes, and the amplitude of “A and B”, where A and B are ‘independent’, is the product of the amplitudes. So we’re not simply doing ordinary probability theory but working with some kind of complex square root of the ordinary probability. An amplitude really is a different kind of thing. For example P(A or B) can be 0 even though P(A)>0 and P(B)>0, something impossible for ordinary probabilities.
If you have any interest in alternative probability theories I strongly urge you to investigate QM – or maybe quantum computing, because that isolates the part of QM that has consequences for the philosophy of probability without you having to get bogged down in the subjects of traditional physics like masses, momenta and energies.
Actually I started writing a document a while back giving a basic introduction to quantum mechanics from the point of view of QM being formally almost identical to a probability theory. The intention was to give people with no physics background, but at least some intuition about probability, an easy “way in” to understand QM. I must finish it some time. I think what I was writing was completely uncontroversial, I was just highlighting the formal mathematical similarity without making any philosophical claims. The simlarity is obscured by most physics textbooks.
That would be great if you could do that! I’d certainly be interested to read it at some point, as I’m sure plenty of others would be too.
As a philosopher of mathematics aren’t you interested in varieties of subjective probability theory which depart from logical omniscience? Ian Hacking in his 1967 paper ‘Slightly More Realistic Personal Probability Theory’ Philosophy of Science 34: 311-323, outlined various strengths of such a theory, right down to the case where you may know P and P->Q, but not put them together to derive Q. If you want to think about Bayesianism in Mathematics, and extend Polya’s program, you have to move in this direction to some extent.
That’s definitely right – I’d like a system of probability that doesn’t imply logical omniscience. However, tautological omniscience is not so terrible as a starting point. And (as Branden told me Garber once suggested, to avoid the old evidence problem), one can separate the logic into an “internal” and an “external” component, by representing some complex sentences as atomic, so that their logical relations aren’t known.
But I’ll certainly check out the Hacking at some point! (And I’ll have to check out your paper too.)
I wonder how you’d make a principled separation between “internal” and “external”. It would need to allow different degrees of belief for “pi = 4(1 – 1/3 + 1/5 – …) & its hundred trillionth decimal digit is 7” and “pi = 4(1 – 1/3 + 1/5 – …)” (or whatever that digit is). Then as our supercomputers reach that number, should we require the rational person to have equal degrees of belief. If not, would it be rational to have nonzero degree of belief in “pi = 4(1 – 1/3 + 1/5 – …) & its integer part is 5”? Fiddling about on rather uninteresting examples like this one distracts from the richer cases one ought to be considering, such as the one mentioned towards the beginning of my Thursday 3 November post.
More thoughts on QM and probability — how I tend to think about it:
It seems that 20th C philosophers of science have traditionally demanded that predictions be worded as real-valued probabilities of outcomes. I tend to think that ultimately QM will be understood differently — I believe, with only intuition and casual hearsay to support me — that a deeper physical understanding will involve discarding the reals in favor of much more intricate algebraic structures. (Some of this move is happening, and I don’t know enough about Quantum Geometry to comment on what the underlying structures actually are.) It’s certainly the case, though, that human experience is well (although not perfectly) described and approximated by the reals, so demanding that probabilities be real seems reasonable.
So what? Understanding sigfpe’s points at face value, it seems that fundamental particles don’t experience the world in real probabilities, but in complex ones. Somehow, of course, these complexities, so to speak, disentangle as the number of particles approaches infinity, so that we get classical events.
A big part of this is that thinking of complex probabilities models one too many dimensions: for any given event, there’s an overall non-physical phase factors. So if I have a bunch of particles, except in a carefully constructed situation on average all the individual particles, etc., will be come with randomly distributed phases, which will ultimately cancel out.
Why do we take norms at the end? I say this is exactly _because_ of this overall phase factors: at the end of the day, the argument of your “probability” is meaningless. And the only good way, with current mathematical technology, of throwing away that one piece of information is by taking absolute values.
The situation is similar, in my mind at least, to gauge (and other symmetry) transformations. To wit: in EM, for instance, physicists use the (lorentz)vector field A_\mu to describe the state of the universe, even though A_\mu + \d_\mu\Phi will describe the state exactly as well, for any (lorentz) scalar field \Phi. During a calculation, which \Phi (which “gauge”) you pick is important, and there are “interference patterns” that you would not predict if you through away the gauge information for each constituent part (roughly equivalent to calculating as if you have a bunch of “classical” particles instead of “quantum” particles).
To make another physics analogy, GR notwithstanding the total potential energy is non-physical (in fact, passing from Lagrangians to Actions a la Feynman translates any constant shift of the potential into a phase shift): only the relationships between different potential energies matter. For potential energies, there is _no_ natural way to through away that bit, except by taking derivatives, and so its no wonder that Newton talked about the patently physical Force rather than Potential Energy. And yet the mathematics is so nice if potential energies had full real meaning that all of physics these days is worded in terms of lagrangians and hamiltonians rather than forces, that it’s almost as if potentials really exist.
Just like QM makes it seem that complex probabilities almost exist in their own right.
P.S. I was going to mention, too, that complex amplitudes can’t be completely physical because there’s no way of picking i or –i. Which is sortof true, but not a good proof. For one: places like Spec(\R[x]) (i.e. the orbifold of \C under complex conjugation) do a good job of collapsing such concerns, and still preserving enough underlying structure that one might construct a natural and coherent system of arithmetic in which to do calculations (it hasn’t been done yet, to my knowledge). But also: I would read any (unpatched) CPT violation as essentially the universe distinguishing between i and –i, and I don’t think this is an out-of-the-question possibility.