Pascalian and non-Pascalian nonzero chances, definition: Any conceivable thing has a nonzero chance of being real -- we can call that the Pascalian nonzero chance. When we have some more solid reason to believe in something, the chance of it being real is also nonzero, is somehow higher, and can be called a non-Pascalian nonzero chance.
In estimating risk and reward, there are two kinds of nonzero chances I know of: what I will call "Pascalian" and "non-Pascalian".
Blaise Pascal proposed a famous Wager, which goes something like this:
If God exists:If God does not exist:
Being aligned with God promises eternal life. Value = infinity.
Not being aligned with God promises hell. Value = minus infinity.Expected value calculations are made by multiplying the expected payoff of a course of action by its likelihood. Investment is risky but can be rational if the expected profits are high enough. If you have good information about the likelihood and reward, if you keep pursuing investments with high expected values, in the long run you will win big at least a few times, and make up for all the times you don't. You can make expected value a part of your lifestyle, pursuing all kinds of rewards, so that at least one will take. This approach can be used when evaluating whether to invest in alignment with God.
Being aligned with God promises a finite life, followed by annihilation. (That life might be better or worse because of your alignment with the God-idea and God-community which must stand in for God if he doesn't exist.) Value = something finite.
Not being aligned with God promises a finite life, followed by annihilation. Value = something finite.
Is it ever rational to act as though God doesn't exist? Seemingly, not if there is a nonzero chance of God existing. Can anyone be 100% certain that God does not exist? Is there any possible way that God could have been the one to create and sustain the world? One would think there is. So if there is a tiny but nonzero chance that God exists, you multiply that by the infinity of eternal life and still get an infinite payout. So your expected value is infinite, no matter how small that tiny but nonzero chance is. Likewise with the negative payout that comes with not being aligned with God. But if you don't align yourself with God, the most you can expect is something finite, a finite life. Align yourself with God, then, no matter what the truth is.
Inspired by Pascal's Wager, some people have discussed Pascal's Mugging. It goes something like this:
A man walks up to you on the street. He says, "I know you're a rational person. So I have an offer to make you. If you give me $100, there's a nonzero chance that I will give you any finite amount of goodness you want -- just name your price. There's a nonzero chance that I have magical powers." A truly rational person, aware of expected value theory, will seemingly have to offer the mugger the money, in order to follow the dictates of reason which they always do. So knowing this, a mugger can exploit rationalists, by playing on their devotion to expected value.
But rationalists (and atheists) hardly ever seem to want to fall for these appeals. I don't know if real Pascal's Muggings happen (although scams do, and appeals to enormous but unlikely future goods occupy some minds), but I would bet someone has offered Pascal's Wager to an atheist, and the atheist was unimpressed. But what's wrong with the Wager and the offer in the Mugging? If you stick to the possibility and logic, they look pretty good. Why do we have an intuition against them?
I'm not sure this really explains the intuition, but I can think of one problem with Pascal's Wager, which is that you have to ask "which God?". Any conceivable thing has a nonzero chance of being real -- we can call that the Pascalian nonzero chance. When we have some more solid reason to believe in something, the chance of it being real is also nonzero, is somehow higher, and can be called a non-Pascalian nonzero chance. Well, if being aligned with God requires us to do certain things and hold certain attitudes, we want to know which God that is, to know what those behaviors and attitudes specifically ought to be. If any one deity can be considered to possibly exist simply because he is conceivable, then why not all the other conceivable ones? If we have to satisfy every conceivable deity, we may end up with all of their demands cancelling each other out. And if they don't, it will probably take too much of our time to figure out how they all add up and balance. Deities could be into all kinds of things, in all kinds of proportions, to all kinds of tolerances.
Perhaps a similar objection can be made to the Mugger. Yes, there is a nonzero chance that the mugger is both honest and possessing sufficient magical powers, but if that chance is exactly as high as the chance that anything conceivable can be true, then it is equally likely that some magical thing will happen to cancel out the good the mugger promises (a leprechaun interferes, perhaps). Then if I take the offer, I'm out $100 and I don't get anything back in the end.
What is the size of a Pascalian nonzero? I don't know if anyone has defined it. A solid 1% chance is definitely non-Pascalian. That's something you can work with. 0.1% or 0.01% are also okay. If you keep adding zeros, though, at some point you get Pascalian. Presumably in people's minds (or guts) there's some exact quantification of "Technically it's got a nonzero possibility because it's conceivable". But I'd be hesitant to mention what it is for me, perhaps because if I did the math I'd have to do something I didn't want to do. I would have been hoping the low number I'd assign to my Pascalian nonzero would make it effectively zero in all practical calculations, but some mugger could offer me a sufficiently high reward, and I'd feel rationally compelled to give him $100.