[kgwgk]

And what I try to explain is that one way to define what it means to be X% sure about A is to say that

- you put a number on it p(A)

- which is between 0 and 1

- and allows you to compare how sure you are about different things p(A) and p(B)

This number can be used to compute how sure you are about composite things:

p(A or B) = p(A) + p(B) - p(A and B)

p(A and B) = p(A given B) p(B) = p(B given A) p(A)

That number p happens to correspond to the notion of probability, but it has not been defined using a pre-existing notion of probability: https://en.wikipedia.org/wiki/Cox%27s_theorem

[lottin]

What do you mean "you put a number on it"? What number? If the number is arbitrary, which is what your explanation suggests, it cannot mean anything.

[kgwgk]

It's not completely arbitrary, it represents the degree of plausability you assign to the event.

These numbers have to obey some rules if you require that a set of beliefs is consistent.

The number you assign to the plausability of A and the number you assign to the plausability of not-A have to sum 1.

If you think A and B are equally plausible, you have to put the same number on them.

If you think that A and not-A are equally plausible, you have to assign the number 0.5 to both.

If you put the number p(head)=p(tails)=0.5 as your degree of plausability that the coin I just flipped (I actually did it!) is showing head or tails it's not an "arbitrary" number. It means that you think both (exhaustive) outcomes are equally plausible. Why do you say it cannot mean anything?

[lottin]

It seems to me that if a probability is a quantity representing a degree of belief, and it's only meaningful in relation to another quantity representing another degree of belief, in the sense that we can only say than being X% sure is being more sure than being Y% sure, if X > Y, or equally sure, if X = Y, then such quantity only has an ordinal value, which is to say the quantity itself is meaningless. For it to be meaningful it has to have an interpretation that does not always refer us to another degree of belief. It also must not refer to a "degree of plausibility" since this is just another expression for "probability", and probability is what we are trying to define.

[kgwgk]

I'm not sure I understand where do you see a problem.

In the example of the degree of belief (between 0 and 1) that you have that the coin on my desk is showing one face or the other, don't you agree that the right numbers that represent your indifference are 0.5 and 0.5? The quantity itself is not meaningless.

[lottin]

You say indifference, but indifference with regards to what? In economics, an individual is said to be indifferent between two alternatives if those alternatives result in the same level of utility for him or her, utility being an abstract concept representing well-being. But you don't say why this person is indifferent to the coin showing one face or the other. Because if it's because he or she thinks both options are equally likely, then we again have a problem, since we don't know what "likely" means.

[kgwgk]

At what point does the following argument derail for you?

0) I tossed a coin, it lies flat on my desk

1) You have some degree of belief about the statements H:“the coin shows heads” and T:“the coin shows tails”

2) You want to quantify that degree of belief

3) You postulate that you can put a number on your degree of belief about some statement A with the following properties:

3a) p(A) it is between 0 (false) and 1 (true)

3b) p(A or B) = p(A) + p(B) - p(A and B)

3c) p(A and B) = p(A given B) p(B) = p(B given A) p(A)

4) p(H) + p(T) = 1

5) Unless your degree of belief about H is higher than your degree of belief about T

or your degree of belief about T is higher than your degree of belief about H ...

6) ... it follows that p(H) = p(T) = 0.5

[lottin]

(in reply to @kgwgk's comment https://news.ycombinator.com/item?id=25313531)

The argument is flawless, the problem is with the interpretation.

> p(A or B) = p(A) + p(B) - p(A and B)

How does one add degrees of belief and what sense do we make out of the result?

> p(H) = p(T) = 0.5

Sure, two equal quantities representing degrees of belief must mean the degrees of belief are of the same magnitude. But what about P(H) = 2P(T)? What does it mean for one degree of belief to be twice as large as the other?