[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?

[kgwgk]

>> 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?

That's how we postulate [1] that the numeric representations of degrees of belief are added. Doesn't that look like a property that you want a numeric representation of degrees of belief to have?

If you have some degree of belief about A, some degree of belief about B, and you believe that A and B are mutually exclusive, wouldn't you want the number representing the degree of belief of "any of them" p(A or B) to be the sum p(A)+p(B)?

>> 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?

Consider p(H or T) = p(H) + p(T) = 2 p(H) = 2 p(T). Isn't it natural to quantify the degree of belief that I got any outcome with a number that is the sum of the numeric representations of the degrees of belief that I got each outcome?

Or say that, instead of flipping a coin, I toss two of them. They're lying flat on my desk right now. The number of heads up is 0, 1, or 2.

How would you describe your degree of belief about the statements "X=0: there are no heads", "X=1: there is one" and "X=2: there are two"?

Wouldn't you say that your degree of belief about "X=1" is of the same magnitude as your degree of belief about "X=0 or X=2"?

Wouldn't you say that your degree of belief about "X=0" is of the same magnitude as your degree of belief about "X=2"?

Wouldn't that make the numerical representation of your degree of belief about "X=1" twice as large as the numerical representations of your degrees of belief about each of "X=0" and "X=2"? (Where you assign numbers to degrees of belief using the representation we're discussing.)

p(X=1) = p(X=0) + p(X=2) = 2 p(X=0) = 2 p(X=2)

[1] in fact I think this is what we get from postulates which are a bit more general, but for the sake of the discussion we may stay in this level

[lottin]

Addition is required by the axioms of probability, not by the interpretation of it. When interpreting probability as a degree of belief this property is not only not useful, but is particularly troublesome, because adding and subtracting degrees of beliefs doesn't appear to make a lot of sense.

All in all, to me it's clear that these degrees of belief are a theoretical construct, not an empirical reality. I don't think people assess the truth value of a statement on a continuum from truth to false. This is not how the human psyche works. Personally, no, it's not natural for me to have a degree of belief (in the way that you have defined them) about a statement, and I have no idea how to interpret arithmetic operations involving these "things".

[kgwgk]

(Note: I enjoy the discussion but I won't be offended if I get no reply! You're free to consider probability just a construct and not accept its applicability to real-world reasoning.)

Ok, what can you say then regarding your degrees of belief about the statements "X=0: there are no heads", "X=1: there is one", "X=2: there are two" and “X={0,2}: there are none or two”?

Of course, if you don't think you have degrees of belief to start with there is no way you can make sense of assigning numbers to them. But I thought we had progressed to the point where you could accept that they existed and that they could be ordered.

> I don't think people assess the truth value of a statement on a continuum from truth to false.

So they cannot be ordered? Or there are no extremes?

> When interpreting probability as a degree of belief this property is not only not useful, but is particularly troublesome, because adding and subtracting degrees of beliefs doesn't appear to make a lot of sense.

Are the rules I proposed problematic in some specific way? They make a lot of sense as far as I can tell.