Pascal's Wager is one of the most famous arguments in the world, although it's usually not understood correctly. It's perhaps not surprising it is garbled; the argument is found in the Pensees, which are just notes for a work that was never fully written, and it is spread across multiple notes. In addition, Pascal's Wager is often confused with a similar (and derivative) wager argument, that found in the Port-Royal Logic. (The most obvious distinguishing characteristic is that wager arguments derived from the Port-Royal Logic include hell or infinite misery in the wager. This is a mutation not found in the original wager, and other wager arguments derived directly from Pascal's do not have it.) To simplify matters greatly, the actual wager argument is an argument against a particular kind of position, namely, that we should not believe that God exists on the ground that there is no way to know. It is not, contrary to a common popular presentation of it, an argument that God exists; it is an argument that we do in fact have reasons to believe that God exists (practical reasons), and that people making this claim are motivated not by rational considerations but by passions. We can roughly characterize the whole thing as having three stages:
(1) Even if we assume that there are no theoretical reasons for believing that God exists, we nonetheless have practical reasons for doing so. This is because we re motivated in inquiry not just by truth and error but also by happiness and misery. If we wager that God exists and are right, we can gain complete happiness; if we wager that God exists and are wrong, we do not thereby lose what we did not have anyway. Therefore wager that God exists. Ah, but the interlocutor says, gain and loss are not the only things to consider in a wager; we might wager too much. Well, the wagerer says, the happiness we could gain is infinite happiness, and we don't have a corresponding loss if we are wrong, so we can stake anything that would be reasonable to wager in return for infinite happiness. But what we stake is always going to be finite, compared to that infinite gain. Therefore wager that God exists. Ah, says the interlocutor, but it's not just gain, loss, and stake that we need to consider; we also need to consider certainty, and the uncertainty is infinite. But, the wagerer notes, the beginning position makes the uncertainty the same on both sides, so our risk is the same, either way, and therefore can't affect the wager. Therefore wager that God exists.
(2) But, says the interlocutor, even if this were the better wager, this is not how you gamble safely; you try to see the 'inside tip'. Well, says the wagerer, this gets us into the theoretical reasons that people actually give, that were simply dismissed out of hand by the original position.
(3) But these reasons are inadequate actually to make me believe, says the interlocutor; I think believing on the basis of them would require me to stultify myself. Well then, says the wagerer, we should start with the fact that we now know that it's not lack of reasons but psychological inability that is creating the inability to believe. But we are not mere slaves to our passions. Since we now recognize that there are reasons to believe, we know that the reason you don't believe is fear, worry, etc., arising from how your passions respond to the idea. The thing to do then is to stop overthinking it. Act on your practical reasons to believe; stop worrying about being stupid by getting out and doing the things that can reshape your passions.
All this is very rough, of course. But it gives us enough of the idea, and we can call (1) the Calculation Step, (2) the Inquiry Step, and (3) the Practice Step of Pascal's Wager. The Port-Royal version has a slightly different Calculation Step, and its Inquiry and Practice Steps are mostly implicit (although the Practice Step is at least strongly implied simply from structuring it as a wager), but we can characterize it in much the same way.
However, this structure is quite general, and we can run wager arguments in other contexts besides the existence of God. Locke recognized this (Essay II.xxi); he adapts the Port-Royal Calculation Step to argue that "morality, established upon its true foundations, cannot but determine the choice in any one that will but consider". If we act on a well-founded morality, our potential gain is infinite, our stake only finite; if we do not act on it, our potential loss is infinite in comparison to what is only a finite savings in the stake. Locke also very clearly has the Inquiry and Practice Steps, although in a sense his structure opens the Calculation Step earlier in the chapter, then develops it in parallel with the Inquiry and Practice Steps, then sums up with the complete Calculation Step in section 72.
But we can have wager arguments that do not have to go out to infinite happiness or misery. Consider a position that we should suspend judgment on induction because we have no reason to think that the future will resemble the past. Someone could build a wager argument against this:
(1) Calculation Step: Even if we assume that we have no theoretical reasons to accept that the future will resemble the past, we can have practical reasons to do so. If we act on the belief that we can predict the future from the past, we can live a life, with all the good that comes with that, which is a gain; if we simply suspend judgment on it, it seems that we would have to suspend serious living, which would be a loss -- all our projects, our art and science and social relations and education and so forth, involve the belief that we can know something of the future from the past. But, even setting aside any goods that might come from these, all of these things (art and science and society and education and so forth) are goods themselves. So wager that we can learn for the future, from the past.
(2) Inquiry Step: Nonetheless, we do in fact have reasons to think that the future will resemble the past; even if they aren't regarded as probative, we definitely do have reasons. There is a coherence to the belief that the future will resemble the past; it hangs together well. There are mathematical and logical limitations to how different the future could be from the past, which implies that at least some minimal structure must endure. And if we assume this belief, we can live our life and while doing so look further into possible theoretical reasons for believing it.
(3) Practice Step: Thus the position with which we started seems to be a case of overthinking -- we are considering all possibilities, and because we can't sort all the possibilities, we give up. We are worried that we're missing something. So, given that we've seen that we have reasons to accept the view that the future will resemble the past, we know that our worry is just that, a passional response, and the thing to do is to stop being shut up in our own heads and live our lives, which will involve wagering that the future resembles the past. Yes, we might be wrong in particular cases, but we will have a life, and we can try to learn what we need to learn in order to be less wrong.
A great many other things could be defended in the same way from various kinds of analogous skepticisms: the reliability of our memory, the integrity of our reasoning, the existence of a world independent of us, the existence of other people, and so forth. In each case we might have to modify things here and there to fit the topic, but some kind of wager can be run for each. Again, in these wager arguments we do not establish the truth of what is defended; rather, we establish the reasonableness of it as a working assumption, and usually the unreasonableness (arising from overthinking the matter) of not taking it as a working assumption, by noting that we have practical reasons for it, that given these we can accept or at least investigate the theoretical reasons for it even if they aren't always probative, and that any inability to accept is not a based on the reasons but on the passions, which are often better handled by training rather than arguing.
