Philosophy and Probability

Published in

Science and Philosophy

11 min readJul 5, 2020

When I tell people that one of the things I work on is the philosophy of probability, they are usually surprised. ‘Probability? Isn’t that part of maths — what’s a philosopher studying it for?’ In part, this reaction betrays the misconception, shared by many people, that philosophy consists mostly of ethics and political philosophy (maybe with a bit of philosophy of religion thrown in). But there might be a more serious intellectual query behind the surprise. What, exactly, is of philosophical interest in what seems to be a purely mathematical theory?

When mathematicians look at probability theory, they see a formal system, characterised by a collection of quite simple axioms that have just as much in common with the abstract theory of measures (like length, volume, and especially, proportion) as with our ordinarily concept of probability. The vast and imposing edifice of mathematical probability theory consists in drawing out the consequences of these axioms: sometimes highly non-obvious and interesting consequences.

But what of the axioms themselves — what is their status? We could say they are merely formal postulates, in which case mathematical probability theory need not have any justification beyond its intrinsic mathematical interest. But the many successful applications of probability theory might lead us to think there is something more to these axioms than just mathematical elegance. If a mathematical theory is successfully applied, generally that indicates some feature of the physical world, or our beliefs about that physical world, must correspond in some way to the structures characterised by that theory. So we can ask: what features of the physical world are adequately modelled by these axioms (if any)? And what are the consequences for our fundamental conception of reality if probabilities have a basic role to play? It is these questions with which philosophers of probability concern themselves.

Philosophers of probability aren’t asking, ‘which particular specific probability claims are correct?’ Answering that question would be a job for physical science, because to assign probabilities in many cases will involve detailed experimental investigation of some system of interest. Philosophers are concerned with questions that are prior to empirical inquiry, such as: ‘what, in general, are the kinds of features of reality that would make a probabilistic physical theory appropriate?’ This really is a question of ‘meta-physics’ — it is about understanding the nature and concept of probability in general, not offering some specific probability function to model a system of interest.

These metaphysical questions are by no means easy. For example: Probabilities, it seems, attach to outcomes — it is some particular die roll which has a 1/6 probability of landing with 4 uppermost. Yet when we look around, every potential outcome in the world either comes to happen or it does not. When the die roll takes place, and (let’s suppose) lands 4, we see no ‘shadow’ of the other outcomes, or indeed anything about that particular roll from which we might uniquely derive the number 1/6. (I’ll come back to the role of there being six faces in a second.) Whereas the duration or location of an outcome are directly observable features of that outcome, its probability is not at all apparent. If we enumerate the total history of all rolls of some die and their outcomes, we will see the die land 6, 2, 3, etc., but we never observe any physical quality of the die or the individual rolls which is measured with the value 1/6. Despite the evident utility of probability theory in many spheres of human endeavour, it is difficult to find a place for probabilities in the measured qualities of any of the outcomes that occur in physical reality.

This tension, between the usefulness of probability theory and the difficulty of understanding what exactly it is describing about the world, is a pressing concern for anyone who wishes to give a systematic metaphysical theory of how reality is, and how it could be — philosophers included.

The Classical Theory

Perhaps the simplest theory of probability takes the hint from the formal analogies with measure theory in the axioms of probability. Recall classroom exercises in probability, where to discover the probability of some outcome (for example, tossing two heads in a row with a fair coin), one begins by enumerating all the possible ways the experiment could go, and calculating what proportion of these ways are instances of the outcome in question. (So in this example, there are four possible ways things could go, only one of which involves two consecutive heads, so the probability would be 1/4.) This idea that probability is a measure of how possible an outcome is, defined as the proportion of all possible outcomes in which it occurs, is the classical theory of probability, dating at least to Laplace’s work in the early 19th century. Unfortunately, it doesn’t quite work. Consider a loaded die, which has a higher probability of landing 6 up — say, 1/5. Yet there remain six possible outcomes of a single roll, only one of which is 6, so the proportion of possible outcomes which correspond to 6 uppermost is still 1/6. The probability assigned by the classical theory is incorrect. Outcomes which are equally possible needn’t be equally probable.

Another problem with the classical theory involves how to divide the outcome space into basic outcomes. It might seem natural enough in the case of a coin toss to think that the basic equally possible outcomes are just heads and tails (though why isn’t landing on the edge a possible outcome?). But in other cases things are not so simple. Consider van Fraassen’s ‘cube factory’:

A factory produces cubes with side-length between 0 and 1 foot; what is the probability that a randomly chosen cube has side-length between 0 and 1/2 a foot? The classical … answer is apparently 1/2, as we imagine a process of production that is uniformly distributed over side-length. But the question could have been given an equivalent restatement: A factory produces cubes with face-area between 0 and 1 square-feet; what is the probability that a randomly chosen cube has face-area between 0 and 1/4 square-feet? Now the answer is apparently 1/4, as we imagine a process of production that is uniformly distributed over face-area. … And so on for all of the infinitely many equivalent reformulations of the problem (in terms of the fourth, fifth, … power of the length, and indeed in terms of every non-zero real-valued exponent of the length). What, then, is the probability of the event in question? (Hájek, ‘Interpretations of Probability’, §3.1)

The classical theory gives no unique answer in many cases as to what the probability of an outcome is; and even when it gives a natural answer, that answer is prone to being undermined by further facts about the situation.

But perhaps we can retain the idea that probability is a kind of measure of possibility without indulging in the crude arithmetic of possibility involved in the classical theory, by looking at another sense of possibility.

Subjectivism

Note that many judgements of possibility reflect our beliefs: we judge that something is possible when it might happen compatibly with all we believe. These ‘doxastic’ possibilities may make a suitable base for a probability measure, because another feature of our beliefs is that we are more or less confident in them, and these different levels of confidence can apply even to equally possible outcomes. So, for example, it is consistent with what you believe (I assume) that a hurricane could knock down the flèche on Exeter College chapel next February — but even though it is possible, you are surely right to be quite confident that such a hurricane will not eventuate.

It is not entirely obvious that this proposal will work. It’s not clear we can even assign numerical values to our level of confidence in various future possibilities—or that if we do, that those values should meet the conditions to be probabilities. But this pessimism is misplaced, as a remarkable theorem proved in the 1920s by the Cambridge philosopher Frank Ramsey shows (it was independently discovered by the statistician Bruno de Finetti in the early 1930s, so is sometimes called the Ramsey-de Finetti theorem). Ramsey had two key insights. Firstly, that we can measure your or my level of confidence (more or less) by our fair betting prices. The simple bets Ramsey considers consist in a ticket which pays £1 if an outcome p occurs, and nothing if not (the stakes are small to ensure that our natural aversion to risk doesn’t interfere). The fair price for such a bet is the most we would pay to own such a ticket (or equivalently, the least we would accept to sell such a ticket). The fair price is not a price we would actually pay — as it is the price at which you are indifferent between buying and selling the bet, and who would engage in a bet unless they had some confidence in making a profit? But, Ramsey reasoned, the fair price does reflect how confident you are in the outcome p. You wouldn’t pay anywhere near £1 for a bet which pays £1 if a hurricane knocks down the flèche — you’d probably be hesitant to pay even 1p. But you might be confident enough that the sun will rise tomorrow that you would pay nearly £1 for a bet which pays £1 when it does. Again, these aren’t bets you would actually make, not least because it would be hard to find a counterparty! But in other cases, where the outcome is something we are unsure of but where different people might reasonably have different levels of confidence, my fair betting price might be different from yours — and in that case, I might sell a bet for (what I believe to be) an advantageous price, and you might similarly buy it from me for what you believe to be an advantageous price. This is the rational basis of gambling.

Once Ramsey has extracted your fair betting prices, his next insight comes into play. Let your credence in an outcome be your fair betting price (in pence) divided by 100p. (The units cancel, so it is just a number.) Ramsey showed that if your credences don’t behave mathematically as probabilities, you are guaranteed to be exploitable, and thus irrational. Specifically, he shows that if your credences aren’t probabilities, there is a collection of bets (a ‘Dutch book’), each of which you regard as fair (or even advantageous) by your own standards, but which is such that if you accept them all, you are absolutely foreseeably guaranteed to make a sure loss, no matter what happens. What makes you irrational is that you have a disposition (whether or not you act on it) to exchange some positive amount of money for a package of bets that leaves you worse off. Such a disposition

would be inconsistent in the sense that it violated the laws of preference between options… if anyone’s mental condition violated these laws, his choice would depend on the precise form in which the options were offered him, which would be absurd. (Ramsey, ‘Truth and Probability’)

If your credences are probabilities, on the other hand, no Dutch book can be made against you, and your betting dispositions are not exploitable. So you are not irrational—at least, not in this respect!

Probability theory, on this subjectivist interpretation, is a measure of rational confidence — your confidence — in how likely an outcome is. One might quibble with some of the details (maybe you don’t have a precise fair price for a bet, but rather will do no more than quote a buy-sell spread), but the consensus is that people really do have credences, or degrees of belief, which can be measured well enough by fair betting prices, and that these credences are probabilities. Problem solved! The feature of the world which makes probability claims true is to be found in our heads, in our beliefs.

Chances

Not so fast. There really do seem to be some objective constraints on reasonable credences. Someone whose fair betting price on a fair coin toss landing heads was 99p might obey the probability axioms but be unreasonable for all that. For, we will say, such a person is ignoring the chance of heads; if they were not, their fair price would be 50p, and their credence would be 0.5. The philosopher David Lewis encapsulated this in his Principal Principle: roughly, if all that is known about an outcome is its chance, then a rational credence in that outcome is equal to the chance. More precisely:

Assume we have a number x, proposition A, time t, rational agent whose evidence is entirely about times up to and including t, and a proposition E that (a) is about times up to and including t and (b) entails that the chance of A at t is x. In any such case, the agent’s credence in A given E is x. (Weatherson ‘David Lewis’ §5.1)

The subjectivist de Finetti denied that there were any such things as chances, but most have recognised them as another kind of probability in addition to credences. They are particularly linked to what our best physical theory tells us about the possible outcomes of an experiment. But to what features of these experiments are the chances linked? We could, as the propensity theory says, simply take these chances as a new basic kind of physical thing. But doing so, quite apart from other problems with propensities, hardly answers the questions with which we began. Simply re-naming chance ‘propensity’ is not explanatory.

One common view is that the chance of an outcome is linked to the frequency with which other outcomes of the same type have occurred in similar experiments. Yet some unrepeatable events have non-trivial chances: Joe Biden might have more of a chance of winning the next US presidential election than Donald Trump, though that experiment will only happen once. The so-called law of large numbers shows that, in the limit, frequencies will be close to the chances. But this is a near-guarantee that frequencies are generally a good way of measuring the chances, not any sort of argument that we should identify the chances with the frequencies. (Indeed, another consequence of the law of large numbers is that it is very likely that the chance will not be exactly equal to the observed frequency — if you tossed a fair coin one million times, you would expect about 500,000 heads, but should be surprised if you got exactly 500,000 heads!).

Propensities can’t explain chances, and frequencies aren’t equal to them. The current state of the debate has offered further candidates. But in my view, none of these is particularly promising. What is a promising project, and what my own work has contributed to, is producing a correct account of what kind of thing chance could be. Rather than picking some specific thing to be the chances, we describe the role that chance must play. We know that chance guides credence in line with the Principal Principle; it is close to frequency; and it is linked to (objective) possibility. So maybe chance is whatever thing that it turns out actually plays this chance role, or plays it near enough. In a slogan: chance is as chance does.

In this vein, some of my own work has focussed on the connection between chances and abilities. Perhaps surprisingly, in looking at the complicated and vexed relationship between chance and determinism, I’ve found a potential connection between the debate over probability and the more traditional philosophical problem of free will. This prospect shows once again the importance of philosophical debates over probability, and the rich promise of future research in this area.

Philosophy and Probability

The Classical Theory

Subjectivism

Chances

Further reading

Written by Antony Eagle 🐜🦅