June 4, 2011

## The Projected Mind

This is a post about the “Projected Mind Fallacy”, as named by Edwin Jaynes. Roughly, the projected mind fallacy is the mistaking of uncertainty about the world for a property of the world itself. In other words, thinking that God plays dice. Unfortunately though, it’s not as simple as this and I feel that it’s interpretation deserves some discussion. It is not obvious exactly what should count ‘the world itself’/the non dice playing God/reality, the rest of this post is about how I think this question should be answered (and some other stuff).

Mind projection fallacies can often be spotted by their absurdity: If I had a bag of snooker balls it would be ridiculous to think that the balls exist in some kind of mixture of colours until I pick them out an look at them. Surely the balls are objectively some colour whether or not I decide to look at them. It is (if one accepts the mind projection fallacy) fallacious to say that that the balls that have an indeterminant colour, when in fact, it is just me who doesn’t know which one I will pick out. It is often stated as the confusion of ontology with epistemology, but these words don’t really help anyone understand it.

April 14, 2011

## Probability Theory and Logic, and Reasoning About Reasoning

Most of us who’ve studied probability theory at University level will have learned that it is formalised using the Kolmogorov axioms.  However, there is an interesting alternative way to approach the formalisation of probability theory, due to R. T. Cox.  You can get a quick overview from this Wikipedia page, although it doesn’t really motivate it very well, so if you’re interested you’re much better off downloading the first couple of chapters of Probability Theory: The Logic of Science by Edwin Jaynes, which is an excellent book (although sadly an incomplete one, because Jaynes died before he could write the second volume) and should be read by all scientists, preferably while they’re still impressionable undergraduates.

For Cox, probability theory is nothing less than the extension of logic to deal with uncertainty.  Probabilities, in Cox’s approach, apply not to “events” but to statements of propositional logic. to say p(A)=1 is the same as saying “A is true”, and saying p(A)=0.5 means “I really have no idea whether A is true or not”. A conditional probability p(A|B) can be thought of as the extent to which B implies A.

There are a couple of interesting differences between Cox’s probabilities and Kolmogorov’s.  Cox’s is more general, but also less formal (people are still working on getting it properly axiomatised).  One important difference is that in Cox’s approach a conditional probability p(A|B) can have a definite value even when p(B)=0 (this can’t happen in Kolmogorov’s formalisation because, for Kolmogorov, p(A|B) is defined as p(AB)/p(B)).  This means that, unlike the logical statement $B \to A$, the probabilistic statement p(A|B)=1 doesn’t mean that A is true if B is false.  So conditional probabilities are like logical implications only better, since they don’t suffer from that little weirdness.

Anyway, that’s cool but what I really wanted to write about was this: in Cox’s version of probability theory, it’s meaningful to talk about the probability of a probability. That is, you can write stuff like p(p(A|B)=1/2)=5/6 and have it make sense.  I’ll get to an example of this in a bit.