# The Laws of Belief by Wolfgang Spohn

### Richard Baron forms some beliefs about Wolfgang Spohn’s book.

We have beliefs, and we change our beliefs as new evidence comes in. The Laws of Belief sets out one way to analyse that process of belief-change. Take, for example, a detective who is trying to work out who committed a murder. The suspects are the butler, the gardener, and the doctor. At first, the detective thinks it was probably the gardener. Then he finds a fleck of blood on a kitchen knife, so he suspects the butler. Then he learns that the maid is careless about cleaning knives that are used to cut up meat, so maybe that fleck of blood proves nothing. The gardener is back in the frame. Then it turns out that the doctor would inherit money from the victim, so he becomes the prime suspect. And so on.

We could analyse this process in terms of probabilities. The detective starts by assigning probabilities of different suspects being the murderer: 60% to the gardener, 20% to the butler and 20% to the doctor. When the detective notices the blood on the knife, these then change to gardener 30%, butler 60%, doctor 10%; then back to 60-20-20 when the detective learns about the maid’s carelessness; then to gardener 10%, butler 10%, doctor 80% when he learns about the inheritance. We could even draw on some well-established mathematics (Bayes’ Theorem) that tells us how to change probabilities when new evidence comes in.

Wolfgang Spohn proposes a different method. He steps back from probabilities to a ranking of propositions by the extent to which we believe them. Which proposition, out of ‘the gardener did it’, ‘the butler did it’, and ‘the doctor did it’ does the detective most believe at any given time? Which proposition comes second? To work out his ranking of suspects, the detective only needs to work out which proposition best fits the evidence, which comes second, and so on. That is, after all, the sort of result we want in practice – we want to end up believing whatever fits best with the evidence.

We do, however, want to work our way to the proposition that fits best in some systematic way, taking account of different bits of evidence as we get them. Some evidence is very weighty, and should push some propositions right to the bottom of the rankings as totally unbelievable. Other evidence makes a smaller difference, and could be counteracted by fresh evidence. So because of the different weightings of evidence, we cannot banish numbers entirely.

Thus Spohn introduces numbers that reflect degrees of disbelief. If we assign zero disbelief to some proposition, we believe it: if we assign infinite disbelief to a proposition, we utterly and decisively reject it. Spohn gives us mathematical rules that allow us to change the numbers attached to different propositions as we get new evidence. He also sets up a mathematical framework for propositions – a stage on which the rules of change can play their parts.

Why make the effort, when probability theory is so well established? Spohn is perfectly clear that his approach parallels probability theory. He does not tell us that we have been believing the wrong propositions for centuries because we were doing the wrong kind of mathematics. Rather, his approach helps us to avoid some nasty traps with probability theory. Probability theory is not good at representing some of the sorts of claim we make everyday, such as “I believe the economy will recover next year.” That is a firm statement of a definite belief; but it’s not a statement that there is (for instance) a 99% probability that the economy will recover next year. The difference between “I believe that…” and “there is a 99% probability that…” is a source of paradox for probability theory. If we join Spohn in talking only about what we believe or disbelieve, we can avoid this sort of difficulty.

The focus of all this theorizing is not what is true, but what we believe. However, we do want to work towards the truth. It would be nice if we could know that the ways in which we rank propositions, from believed to unbelievable, have some objectivity to them. Spohn tackles this problem towards the end of the book. Causation is a central example. Spohn applies his approach to analyse puzzles about multiple causes and complex causal chains; but his analysis translates causes into beliefs about causes. It does not create a direct link between our process of belief-formation and what really goes on in the world. This part of Spohn’s work is fascinating, challenging, and, as he freely admits, inconclusive.

Overall, the book is a tough but rewarding read. It is quite mathematical. The reader needs to be at ease with a modest amount of set theory and graph theory. In places, the style is also that of a loosely disciplined conversation. It is sometimes unclear what will eventually form part of the author’s conclusions, and what is an aside. But that is a minor quibble about a major achievement. The reader does not come away with all the answers, but with a head full of important new questions.