Not quite. I outlined the things that have to be going on for me to be making a decision.

Let's assume that every test has the same probability of returning the correct result, regardless of what it is (e.g., if + is correct, then Pr[A returns +] = 12/20, and if - is correct, then Pr[A returns +] = 8/20).

The key statistic for each test is the ratio Pr[X is positive|disease] : Pr[X is positive|healthy]. This ratio is 3:2 for test A, 4:1 for test B, and 5:3 for test C. If we assume independence, we can multiply these together, getting a ratio of 10:1.

If your prior is Pr[disease]=1/20, then Pr[disease] : Pr[healthy] = 1:19, so your posterior odds are 10:19. This means that Pr[disease|+++] = 10/29, just over 1/3.

You may have obtained 1/2 by a double confusion between odds and probabilities. If your prior had been Pr[disease]=1/21, then we'd have prior odds of 1:20 and posterior odds of 1:2 (which is a probability of 1/3, not of 1/2).

If you're looking for high-risk activities that pay well, why are you limiting yourself to legal options?

On the subject of Arimaa, I've noted a general feeling of "This game is hard for computers to play -- and that makes it a much better game!"

Progress of AI research aside, why should I care if I choose a game in which the top computer beats the top human, or one in which the top human beats the top computer? (Presumably both the top human and the top computer can beat me, in either case.)

Is it that in go, you can aspire (unrealistically, perhaps) to be the top player in the world, while in chess, the highest you can ever go is a top human that will still be defeated by computers?

Or is it that chess, which computers are good at, feels like a solved problem, while go still feels mysterious and exciting? Not that we've solved either game in the sense of having solved tic-tac-toe or checkers. And I don't think we should care too much about having solved checkers either, for the purposes of actually playing the game.

This ought to be verified by someone to whom the ideas are genuinely unfamiliar.

I know that's what you're trying to say because I would like to be able to say that, too. But here's the problems we run into.

1. Try writing down "For all x, some number of subtract 1's cause it to equal 0". We can write the "∀x. ∃y. F(x,y) = 0" but in place of F(x,y) we want "y iterations of subtract 1's from x". This is not something we could write down in first-order logic.

2. We could write down sub(x,y,0) (in your notation) in place of F(x,y)=0 on the grounds that it ought to mean the same thing as "y iterations of subtract 1's from x cause it to equal 0". Unfortunately, it doesn't actually mean that because even in the model where pi is a number, the resulting axiom "∀x. ∃y. sub(x,y,0)" is true. If x=pi, we just set y=pi as well.

The best you can do is to add an infinitely long axiom "x=0 or x = S(0) or x = S(S(0)) or x = S(S(S(0))) or ..."

Repeating S n times is not addition: addition is the thing defined by those axioms, no more, and no less. You can prove the statements:

∀x. plus(x, 1, S(x))

∀x. plus(x, 2, S(S(x)))

∀x. plus(x, 3, S(S(S(x))))

and so on, but you can't write "∀x. plus(x, n, S(S(...n...S(x))))" because that doesn't make any sense. Neither can you prove "For every x, x+n is reached from x by applying S to x some number of times" because we don't have a way to say that formally.

From outside the Peano Axioms, where we have our own notion of "number", we can say that "Adding N to x is the same as taking the successor of x N times", where N is a real-honest-to-god-natural-number. But even from the outside of the Peano Axioms, we cannot convince the Peano Axioms that there is no number called "pi". If pi happens to exist in our model, then all the values ..., pi-2, pi-1, pi, pi+1, pi+2, ... exist, and together they can be used to satisfy any theorem about the natural numbers you concoct. (For instance, sub(pi, pi, 0) is a true statement about subtraction, so the statement "∀x. sub(x, x, 0)" can be proven but does not rule out pi.)

What makes you think that decision making in our brains is free of "regular certainty in physics"? Deterministic systems such as weather patterns can be unpredictable enough.

To be fair, if there's some butterfly-effect nonsense going on where the exact position of a single neuron ends up determining your decision, that's not too different from randomness in the mechanics of physics. But I hope that when I make important decisions, the outcome is stable enough that it wouldn't be influenced by either of those.

I´d say this is not needed, when people say "Snow is white" we know that it really means "Snow seems white to me", so saying it as "Snow seems white to me" adds length without adding information.

Ah, but imagine we're all-powerful reformists that can change absolutely anything! In that case, we can add a really simple verb that means "seems-to-me" (let's say "smee" for short) and then ask people to say "Snow smee white".

Of course, this doesn't make sense unless we provide alternatives. For instance, "er" for "I have heard that", as in "Snow er white, though I haven't seen it myself" or "The dress er gold, but smee blue."