# How valid is assignment of probabilites when evidence is totally lacking, as in Pascal's Wager?

The SEP article discussing Pascal's Wager states,

Premise 1 presupposes that you should have a probability for God’s existence in the first place. However, perhaps you could rationally fail to assign it a probability—your probability that God exists could remain undefined. We cannot enter here into the thorny issues concerning the attribution of probabilities to agents. But there is some support for this response even in Pascal’s own text, again at the pivotal claim that “[r]eason can decide nothing here. There is an infinite chaos which separated us. A game is being played at the extremity of this infinite distance where heads or tails will turn up…” The thought could be that any probability assignment is inconsistent with a state of “epistemic nullity” (in Morris’ 1986 phrase): to assign a probability at all—even 1/2—to God’s existence is to feign having evidence that one in fact totally lacks. For unlike a coin that we know to be fair, this metaphorical ‘coin’ is ‘infinitely far’ from us, hence apparently completely unknown to us. Perhaps, then, rationality actually requires us to refrain from assigning a probability to God’s existence (in which case at least the Argument from Superdominance would apparently be valid). Or perhaps rationality does not require it, but at least permits it. Either way, the Wager would not even get off the ground.

Where can I find a source which goes into depth about the "thorny issues concerning the attribution of probabilities to agents"? This question seems to deal with similar issues, and it also seems that there is a consensus that rationality requires/permits a non-assignment of probability. Is this the general philosophical consensus? Are there any sources which specifically talk about this?

• There is a Pascal Wager-like argument in favour of infinite life/immortality, where evidence of infinite spans is either inconclusive and/or delegated to myth and legend. Is this an acceptable style of answer to your question? Jul 27, 2019 at 15:48
• Another strange example of a Pascal Wager-like argument is exploring outer space, contingent on the universe being infinite. Do you accept this kind of answer? When you get into the topic of infinite gain, you enter some very strange territory of philosophy. Jul 27, 2019 at 15:49
• I'm not sure about the probabilistic implications of Kant. Do you want a citation of Kant? Jul 27, 2019 at 16:16
• @TautologicalRevelations I'm sure that some philosophers, such as Kant, reject this sort of probability. What I'm looking for is what the philosophical community as a whole thinks. Is this an overwhelmingly popular idea, or is there still debate?
– Josh
Aug 5, 2019 at 4:03
• Fair enough. :) :D Let's work towards giving you best answers possible. Aug 7, 2019 at 2:28

There are plenty of issues with subjective probability assignments to degrees of belief discussed e.g. in SEP's Subjective Probability Theory. I will only address the one outlined in the OP. For a book length treatment see Fundamental Uncertainty: Rationality and Plausible Reasoning volume edited by Marzetti and Brandolini.

The idea of distinguishing probabilistically quantifiable risk and unknown uncertainty goes back to Chicago school's economist Knight (his Risk, Uncertainty and Profit (1921)), and is termed Knightian uncertainty. Here is Knight:

"Uncertainty must be taken in a sense radically distinct from the familiar notion of Risk, from which it has never been properly separated... The essential fact is that 'risk' means in some cases a quantity susceptible of measurement, while at other times it is something distinctly not of this character; and there are far-reaching and crucial differences in the bearings of the phenomena depending on which of the two is really present and operating... It will appear that a measurable uncertainty, or 'risk' proper, as we shall use the term, is so far different from an unmeasurable one that it is not in effect an uncertainty at all."

Keynes echos Knight in The General Theory of Employment (1937), and explicitly rejects assigning probabilities under uncertainty:

"By `uncertain' knowledge, let me explain, I do not mean merely to distinguish what is known for certain from what is only probable. The game of roulette is not subject, in this sense, to uncertainty...The sense in which I am using the term is that in which the prospect of a European war is uncertain, or the price of copper and the rate of interest twenty years hence... About these matters there is no scientific basis on which to form any calculable probability whatever. We simply do not know."

From a technical point of view, we certainly can not have a probability space for events with unknown structure of outcomes. There is plenty of that in Pascal's setup, e.g. unknown alternative gods and their costs and rewards, see What fallacy in Pascal's Wager allows replacing God with the devil? Epistemological objections include overconfidence in models (Gray), insensitivity to robustness of evidence (Kyburg), suppression of dependency of events, etc. But, while the rejection of uncertain subjective probabilities is certainly a mainstream view, there is no consensus on it. After reviewing the above objections, Sinick concludes on Less Wrong:

"While some people have said that subjective probabilities of arbitrary events are not meaningful, there are definitions that make the notion of subjective probability meaningful, though arguably only as an intervals rather than as numbers. Using intervals rather than numbers addresses some of the objections that have been raised. A large part of the debate about whether one should assign subjective probabilities to arbitrary events is perhaps best conceptualized as a debate about how large the probability intervals that one assigns should be... The ways in which assigning subjective probabilities can be bad for one's epistemology seem to fall under the broad heading "failing to incorporate all of one's knowledge when assigning a probability and then using it uncritically, or forgetting that the probability that you assign to an event does not fully capture your knowledge pertaining to the event." These issues can be at least partially mitigated by keeping them in mind.

The idea of Knightean uncertainty has been revived and reinvented multiple times, including recently as Rumsfeld's "unknown unknowns", and Taleb's black swans. Taleb also coined a term ludic fallacy for "basing studies of chance on the narrow world of games and dice", confusing the unstructured randomness in life with the structured randomness of games. But it is interesting that he finds a rational seed in the Pascal's Wager despite rejecting the probability assignments required for it to work formally:

"But the idea behind Pascal's wager has fundamental applications outside of theology. It stands the entire notion of knowledge on its head. It eliminates the need for us to understand the probabilities of a rare event (there are fundamental limits to our knowledge of these); rather, we can focus on the payoff and benefits of an event if it takes place. The probabilities of very rare events are not computable; the effect of an event on us is considerably easier to ascertain (the rarer the event, the fuzzier the odds). We can have a clear idea of the consequences of an event, even if we do not know how likely it is to occur. I don't know the odds of an earthquake, but I can imagine how San Francisco might be affected by one. This idea that in order to make a decision you need to focus on the consequences (which you can know) rather than the probability (which you can't know) is the central idea of uncertainty."

• Comments are not for extended discussion; this conversation has been moved to chat. Aug 9, 2019 at 11:41
• I know this question is almost a month old by now, but I revisited it and I'm wondering what exactly you mean by "the rejection of uncertain subjective probabilities is certainly a mainstream view." What exactly does uncertain subjective probabilities mean? If, for example, you have some evidence to point towards one hypothesis, but this evidence is not conclusive, can a probability be assigned? If so, why draw the line there; why not make assignment under equal/no evidence possible? If not, how do we encode partial evidence into probability?
– Josh
Aug 21, 2019 at 4:23
• @Josh After moderators move comments to chat it is expected that all further discussion is posted there. I just meant the Knight-Keynes position. Aug 22, 2019 at 5:24