Please amplify on "#1 is wrong".
If lottery tickets were bought after paying off debts and after loss of government benefits, no one who was in debt, or who was receiving government benefits, could buy lottery tickets. Unless I misunderstand.
A->B, C->B, C steals the evidence that C provides for A. You need to provide new D with A->D, C!->B. Where the implication from C to B is imperfect then B goes on providing some trickle of evidence to A but if the implications are equally strong then the trickle does not distinguish between A and C as opposed to other hypotheses and the prior odds win out.
I tried to explain in my previous comment why I think this is the wrong way of looking at it. You're speaking as if B is a proposition with a truth-value that has a single cause. However, I think my explanation was not quite right either.
The weakest, most obviously true reply is that this is not a Boolean net; B does not have a single cause; and A => B and C => B can both be having an effect. It's even possible, in the real-valued non-Boolean world, to have (remember this is not Boolean; this is more like a metabolic network) A > 0, C > 0, A => B, C => B, B < 0.
A reply that is a little stronger ( = has more consequences), and a little less clearly correct, is that your argument for C => B is not as good as my argument for A => B, so who's stealing whose evidence?
The strongest, least-clear reply is that we have priors in favor of both A => B and C => B. Because they're both just-so stories, and we have no quantitative expectations of how much of an increase in B either would provide; and, unlike when B is a truth-value, there's no upper limit on how large B can get; A or C can't steal much evidence from each other without some quantitative prediction. All the info you have is that A and C would both make B > 0, and B > 0. If C accounts for x points of B, and B = x + y, then this knowledge can increase the probability of A. C, C => B diminishes the probability of A in the absence of knowledge about the value of B and the value of B explained by C, but by so little compared to the priors, that presenting it as an argument against an argument from principles is misleading.
The lottery came up in a recent comment, with the claim that the expected return is negative - and the implicit conclusion that it's irrational to play the lottery. So I will explain why this is not the case.
It's convenient to reason using units of equivalent value. Dollars, for instance. A utility function u(U) maps some bag of goods U (which might be dollars) into a value or ranking. In general, u(kn) / u(n) < k. This is because a utility function is (typically) defined in terms of marginal utility. The marginal utility to you of your first dollar is much greater than the marginal utility to you of your 1,000,000th dollar. It increases the possible actions available to you much more than your 1,000,000th dollar does.
Utility functions are sigmoidal. A serviceable utility function over one dimension might be u(U) = k * ([1 / (1 + e-U)] - .5). It's steep around U=0, and shallow for U >> 0 and U << 0.
Sounds like I'm making a dry, academic mathematical point, doesn't it? But it's not academic. It's crucial. Because neglecting this point leads us to make elementary errors such as asserting that it isn't rational to play the lottery or become addicted to crack cocaine.
For someone with $ << 0, the marginal utility of $5 to them is minimal. They're probably never going to get out of debt; someone has a lien on their income and it's going to be taken from them anyway; and if they're $5 richer it might mean they'll lose $4 in government benefits. It can be perfectly reasonable, in terms of expected utility, for them to play the lottery.
Not in terms of expected dollars. Dollars are the input to the utility function.
Rationally, you might expect that u(U) = 0 for all U < 0. Because you can always kill yourself. Once your life is so bad that you'd like to kill yourself, it could make perfect sense to play the lottery, if you thought that winning it would help. Or to take crack cocaine, if it gives you a few short intervals over the next year that are worth living.
Why is this important?
Because we look at poor folks playing the lottery, and taking crack cocaine, and we laugh at them and say, Those fools don't deserve our help if they're going to make such stupid decisions.
When in reality, some of them may be making <EDITED> much more rational decisions than we think. </EDITED>
If that doesn't give you a chill, you don't understand.
(I changed the penultimate line in response to numerous comments indicating that the commenters reserve the word "rational" for the unobtainable goal of perfect utility maximization. I note that such a definition defines itself into being irrational, since it is almost certainly not the best possible definition.)