Faith and Wagering: Two Types of Decision-Making Under Uncertainty (Column 661)

A few weeks ago a question from Yehuda reached my Q&A section (here), including a quotation from William James:

Skepticism, then, is not abstention from choice; it is a choice of a certain kind of risk. “Better to risk the loss of truth than the chance of error”—this is precisely the position of the denier of faith. He bets quite as actively as the believer; he wagers on all the other horses save the hypothesis of faith, just as the believer wagers on the hypothesis of faith against all the other horses. He who preaches that skepticism is our duty until ‘sufficient evidence’ for religion is found is, as against the faith-hypothesis, simply telling us that it is wiser and better to yield to our fear that it is false than to our hope that it might be true. This is not intellect standing over and against all feelings; it is only intellect ruled by a single feeling. And what justifies this feeling’s claim to supreme wisdom? Of all possible errors, what proof is there that the error born of hope is graver than the error born of fear?

Yehuda asked what I think of this passage. In my reply I wrote to him as follows:

He draws a comparison between the skeptic and the believer and sees both as wagering. That is of course a mistake. The believer is not a gambler; he arrives at a conclusion (even if not certain). The skeptic who has no position may arguably be wagering. But even then one must discuss what it is proper to wager on in his situation. This brings us to Pascal’s wager, and I have already written my opinion about it on the site (search here). In fact, if a person believes on the basis of a wager, he is not truly a believer.

Afterwards I thought it appropriate to clarify more sharply the distinction I drew here between wagering and decision-making under uncertainty. That is what I wish to do in this column. It is a very subtle distinction, and I suggest reading the column and reflecting on the claims. I am not sure I managed to express it well in words.

Pascal’s Wager

In column 408 (and also in the Q&A, for example here, and more) I discussed Pascal’s wager. His claim was that if two possibilities stand before us—there is a God or there is not—it is in our interest to believe in Him and observe the commandments because of differences in expected value. If I believe and keep commandments, then on the assumption that God exists I gain infinite reward, and if He does not exist nothing happens—just a minor loss of time and a few pleasures that were forbidden to me. But if I do not believe and do not keep commandments, then on the assumption that God exists I will receive a dreadful eternal punishment, and if He does not exist I have gained those aforementioned pleasures. The conclusion is that even if the probability that God exists is very small, the expected value of observing the commandments and believing in Him is far greater than the alternative’s expected value.

I noted there that although Pascal is one of the fathers of probability theory, it is surprising to discover a probabilistic error here (or an error in the interpretation and meaning of probability), but I will not enter into that now. Here I wish to focus on another point. First, in that column I argued against this reasoning that it grounds observance of commandments but not faith, and I wondered what value there is to such observance. My assumption was that expected-value considerations cannot bring a person to believe, only to observe commandments. In the Q&A here I was told that in the original Pascal speaks also about faith and not only the religious cultus. As noted, that is very odd, since such a wager cannot ground faith. I will clarify this further.

Suppose a person confronts two possibilities—that God exists and that He does not—and he reaches a conclusion about the probability of each. If the probability of one is significantly higher than the other, it is reasonable to adopt the one with the higher probability. In such a case he indeed reaches the conclusion that he believes in God. But Pascal’s wager makes a different calculation: it does not compare the probabilities of the two possibilities (that God exists or not); rather, it computes expected value. For the sake of discussion Pascal is even prepared to assume that the probability of God’s existence is very low, and nonetheless the expected value of belief and observance is higher than the alternative, and therefore one should believe and observe. But now ask the person whether he believes in God. His answer should be no, since the probability of His existence is very low. One cannot use expected-value considerations to ground faith. They can ground only behavior. The most such a person can conclude is that apparently there is no God, and nonetheless he will observe commandments in order to gain a higher average payoff. I do not intend to dispute what Pascal meant (I did not read him in the original), but if this was his intention, then there is yet another mistake in addition to those I listed in that column.

This is one distinction between a conclusion about faith and a wager. It rests on the fact that expected-value considerations can constitute a wager but cannot bring me to a conclusion about what I believe. But my response to James’s statements touched another aspect of the issue. There I dealt with the situation in which I compute the probabilities of the two possibilities before me (that God exists or not), and because the probability that God exists is, for me, sufficiently high, I decide on that basis to believe in Him. My claim there was that even in such a case it is important to distinguish between wagering and believing.

Faith and Wagering: Sharpening the Difficulty

I will preface by saying that there is no claim on earth about which we can have certainty. As human beings, the possibility of error always lies at our door. Hence one might say that every claim we adopt and every piece of knowledge we have is a wager. Under uncertainty we compare the probabilities of two possibilities and decide in favor of the higher probability. So when can we say that some knowledge we have is not a wager? Note that James’s remarks were specifically about faith, whereas here we see that the same could be said about any claim and any item of knowledge in any field. Is every non-certain claim truly a wager? Is every non-certain claim “belief”? And therefore—are the two the same?

First we must distinguish among several different situations. Suppose there are two mutually exclusive possibilities (see the previous column) and we must choose to “believe” one of them. There are cases in which a probabilistic calculation can be made and a numerical chance assigned to each. In such a case it is reasonable to choose the one with the higher chance. If the situation is even, it is reasonable not to decide (to remain in doubt). But in most cases we have no way to make a probabilistic calculation. In such a case we attribute to each possibility some plausibility. Plausibility is not a number representing chance but a degree of confidence I have that the possibility is correct. One might call this “intuition” (see column 653). In such a case I choose the option that seems to me more plausible. On the face of it there is no difference here from the earlier situations, except that in the latter there is no defined number I attach to each option. Still, I should choose the one with the greater plausibility (and not probability).

In all these cases it seems the conclusion is the result of a wager, and it is hard to see a state of “faith” that can be distinguished from wagering. I am betting on the higher probability or plausibility. So when can we speak of knowledge or faith (which, as noted, are never certain) that are not wagers?

The Concept of “Knowledge” in Law and Beyond

In columns 226 and 228 I discussed statistical evidence in law. One of the examples I used to present the fundamental problem was the following (the example appears at the beginning of this judgment). Imagine a hundred prisoners in the prison yard. Ninety-nine of them assaulted a guard and killed him, but it is not known who was the one who did not participate. Now they stand trial, and each claims he is the one who did not take part. Seemingly there is a 99% chance he is lying, and such a chance should suffice to convict him. And yet, in standard legal thinking one does not convict on such a basis. Now imagine a case in which two witnesses come and testify that Reuven killed Shimon. In such a situation, we of course convict Reuven, although there is some chance the witnesses erred or are lying. For the sake of discussion, let that chance also be 1%, that is, here too there is a 99% chance Reuven murdered. In this case we do rely on the 99% and convict him. What is the difference between the two cases? Why, in the prison-yard case, is 99% not enough for us, whereas in the eyewitness case it is enough to convict?

A common explanation of the distinction is that in the case of testimony we have knowledge and on its basis we may convict, whereas in the prison-yard case we have no knowledge, and therefore there is no justification to convict anyone there. In those columns I questioned this explanation and asked why calling something “knowledge” should change anything. At bottom we must decide whether 99% confidence suffices for criminal conviction or not. Labels should not change anything. Indeed, in those two columns I proposed several explanations for this distinction, which I will not revisit here. The reason is that, in the legal context, it does not much matter whether we have knowledge or not, but what the chance is that we are right. But that does not mean that the distinction between situations in which we treat our stance as “knowledge” and others is empty (mere semantics). In other contexts it is indeed possible to accept such a distinction even without the explanations I offered there.

In column 228 I presented the argument of Enoch and colleagues, who attempt to explain the concept of “knowledge” in non-legal contexts (and then, as I explained there—not rightly, in my view—reapply it to legal contexts). Consider a person who bought a lottery ticket whose chance of winning is 1 in a million. The drawing was held and the winning numbers were selected. In fact, the reality is that the person did not win, but meanwhile I (a third party—but it could be him himself) still do not know what numbers came up. Can I now say that I know he did not win the lottery? The chance he won is 1 in a million, and seemingly one may indeed say with confidence that he did not win. Moreover, as a matter of fact he truly did not win—that is, the claim would be true (it matches the state of affairs in the world). And yet, in such a situation it is difficult to say, “I know he did not win.” I can say, “I surmise he did not win,” or “I believe he did not win,” and the like. But it is not correct, in such a state, to say, “I know he did not win” (the fact that in reality he did not win is irrelevant, because I do not know that). By contrast, consider a person who bought a lottery ticket whose chance of winning is higher, say 1 in a thousand. Now the drawing is held and the winning numbers are published in the newspaper. Again, suppose as a matter of fact he did not win, and the numbers in the paper are not his. There is, however, some chance the paper erred, or there is a misprint. Let us say for the sake of discussion that this chance is also 1 in a thousand. If so, the person again is in a state where the chance he won is 1 in a million (the product of the chances), but in this case it is correct for him to say, “I know I did not win” (of course not with full certainty). According to those authors, although in both cases the chance he won is 1 in a million, in the first case he cannot say he knows, and in the second he can.

This distinction obviously recalls the prison-yard example, but here we are not dealing with rules of evidence. Even there, one cannot say “I know Reuven took part in killing the guard,” as opposed to the case of two witnesses, where I will indeed allow myself to say, “I know Reuven killed Shimon.” Note that the chance that I am right is identical in both cases. Enoch and colleagues hang this distinction on the philosophical concept of “knowledge.”

Back to Wager and Faith

The conclusion from all the above is that there are situations in which I make a “wager” between two possibilities and my conclusion can be considered knowledge I possess—and to these I will refer as faith. In contrast, there are cases that are wagers but whose conclusion is not knowledge—and these, for me, are wagers. We saw that the difference between the two is not necessarily probabilistic, but there is a strong intuitive sense that it exists and is correct. As we saw with Pascal, a wager is a consideration that concerns primarily one’s mode of conduct and not the plane of beliefs. On the plane of beliefs we are supposed to use considerations that yield a conclusion I adopt as such, and not merely as a basis for conduct. One could say that wagering is a pragmatic consideration, whereas beliefs are grounded in substantive, intrinsic considerations. On pragmatism and its relation to intrinsic considerations, see columns 588, 501, and 480. Regarding James’s faith on the planes of emotion and experience, see column 630.

I have still not offered here an explanation of the difference between these two types of considerations. I only tried to help you find this distinction within yourselves as well. We still need to elucidate and clarify it. Before doing so, let us look at the matter from another angle.

Faith and Plausibility

Above I distinguished between plausibility and probability. I explained that we use probabilities when we must choose among several possibilities before us, and computing the chances gives us a tool to do that. In the terminology I am using here, by definition probability belongs to the domain of wagering. We have no substantive stance regarding any of those possibilities, and yet we must choose one of them. Probability is the tool that helps us do so. Plausibility, by contrast, is a different kind of consideration. Above I explained that we resort to it mainly when we have no way to carry out a quantitative calculation of probabilities. Here I will add another, more important point. There are cases in which I do have a stance regarding some claim, and that stance is grounded in its plausibility relative to the alternatives. Intuitively it seems more correct to me. This is not probability but plausibility, for here we are dealing with a stance regarding the content of the option itself. A consideration of plausibility is not a wager, since its outcome is a stance that adopts one option as true (even if not with certainty). Recall: a wager is a pragmatic consideration for decision-making; its outcome is not the adoption of the stance itself.

In this context we can cite R. A. I. Kook in Ein Ayah on Shabbat 30. The Gemara there addresses contradictions in Ecclesiastes:

It is written, “Do not answer a fool according to his folly,” and it is written, “Answer a fool according to his folly.” There is no contradiction: this one concerns matters of Torah, and that one concerns worldly matters. As in the case of the man who came before Rabbi: he said to him, “Your wife is my wife and your sons are my sons.” He said to him, “Would you like to drink a cup of wine?” He drank, and the matter dissipated. A man came before Rabbi Ḥiyya and said to him, “Your mother is my wife and you are my son.” He said to him, “Would you like to drink a cup of wine?” He drank, and the matter dissipated. Rabbi Ḥiyya said: May Rabbi’s prayer have helped him, that he not render his sons mamzerim. For when he prayed, he would say, “May it be Your will, Lord our God, that You save me today from brazen people and brazen women.”

In both cases a person comes before Rabbi or Rabbi Ḥiyya and attests about an event that (allegedly) occurred. To Rabbi he says that his wife committed adultery and his sons are mamzerim; to Rabbi Ḥiyya he says that his mother committed adultery and he himself is a mamzer. Both wave the fellow away with a straw and completely ignore his words.

R. Kook there (sections 106–108) explains that the lesson of this aggadah is the superiority of inner confidence and certainty over external evidence. A single witness came and said what he said. Seemingly they should have been concerned, for it is always possible that reality as we understand it is not so. Every person is sure he is the child of his parents, but there are situations in which that is not the case. The claim is that if I grasp reality in a certain way, and do so very clearly, a challenge to that on the strength of some juridic evidence must be very strong before I consider it. Contrary to what people think, not every situation for which I lack proof is categorized as a doubt. On the contrary, if the situation is clear to me, then to render it doubtful requires a good reason. This is the meaning of the statement in Kiddushin 80a:

R. Ḥiyya bar Abba said in the name of R. Yoḥanan: We administer lashes on the basis of presumptions (ḥazakot), we stone and burn on the basis of presumptions, yet we do not burn terumah on the basis of presumptions. We administer lashes on the basis of presumptions, as Rav Yehuda said in the name of Rav Yehuda: If a woman is presumed to be a niddah among her neighbors, her husband receives lashes for relations with her as a niddah. We stone and burn on the basis of presumptions, as Rabba bar Rav Huna said: A man and a woman, a boy and a girl, who were raised together in one household are stoned one on the other and burned one on the other [for forbidden relations]. R. Shimon ben Pazi said in the name of R. Yehoshua ben Levi in the name of Bar Kappara: There was an incident with a woman who came to Jerusalem with a child riding on her shoulders; she raised him, and later had relations with him, and they brought them to court and stoned them—not because he was certainly her son, but because he was “attached” to her [i.e., presumed so].

If a given state of affairs is publicly presumed to be true, then that is the state, and we treat it as a full fact. To be sure, it may always be that we are mistaken, but so long as no serious challenge has arisen, for us this is the clear truth. I believe I once brought here a case that occurred in our synagogue in Bnei Brak. An avrekh once gave a lecture in honor of his son’s bar mitzvah and claimed that the common practice of asking a child his age to know whether to count him for a minyan is not halakhically correct—who says he is believed? But in fact this is what people do, and in my opinion rightly so. If a child is presumed to be of age and it seems to us this is the case, then he is of age. It is not due to his credibility; rather, we do not need to resort to credibility considerations. This is not a case of doubt. Credibility and evidence become relevant only after we have decided that the matter is doubtful. But if the matter is not doubtful in our eyes, there is no need to resort to proofs and evidence. That is precisely what happened in the Gemara there in Shabbat. The challenge was grounded in some reasonable external juridic consideration: a single witness. But Rabbi and Rabbi Ḥiyya had deep inner confidence in their stance. They were presumed to be legitimate sons of their parents, and that is the state. That presumption overrides a challenge on the strength of some external rational consideration or juridic evidence. These begin to be relevant only after we have decided we are in a state of doubt. Juridic evidence operates on the probabilistic plane. But a public presumption (ḥazakah) of a state creates plausibility.

This very much recalls the distinction we drew above regarding statistical evidence in law. There too, when we have two witnesses, they render the state (for us) conclusive. Now we know this is the case, even though we do not have certainty and error is always possible. But a “majority” consideration (a statistical consideration), even if as reliable as two witnesses, does not create knowledge. The matter remains doubtful. A single witness does not create knowledge. But a presumption—i.e., a state that the public at large treats as such—is a state that we know. It is not a consideration that resolves a doubt; rather, here there is no doubt to begin with.

Faith and Intuition

In column 653 (and several other columns cited there) I discussed the meaning of intuition, and in particular its being a recognizing faculty, not merely a thinking one. At the end of that column I addressed the link between intuition and faith, and I explained that the faculty called “intuition” is a way of looking at the world (not by means of the senses, but with the “eyes of the intellect”). We can use that picture to clarify further the difference between faith and wagering.

When several options stand before me and I have faith that one of them is true, we saw that this is not a probabilistic consideration. It is a plausibility I attribute to that option. When two witnesses come before me and say Reuven murdered, that turns the fact into knowledge for me. But a majority consideration like the prisoners in the yard is a statistical consideration, not knowledge. The difference is that testimony or direct apprehension is an intuitive gaze at the thing. Probability or plausibility is an indication that I grasp that this thing is true. I have a stance regarding the specific prisoner, not a general statistical state of affairs that merely reflects upon him (as a member of a broader class). It is not a statement about a class to which he belongs, but a statement about him. Therefore, although the “plausibilities” in both cases are similar, in the case of plausibility and intuition this is knowledge; in the case of majority or probability it is not knowledge but a general consideration that radiates onto the case before me. Probability always relates to the individual under discussion as an individual member of a class, and the statements concern the class as a whole. Knowledge is always a claim that relates to the case or person before me. Even when I see something there remains a doubt lest my sight deceives me, but that sight contains information about the person before me. The possibility of error is an incidental annex. By contrast, a probabilistic (majority) consideration does not relate to the person or case before me. It is not knowledge about the person, but perhaps knowledge about the world.

Faith in God

To conclude I wish to return to faith in God. There are different ways to reach faith in Him. There are philosophical arguments of various kinds, each of course based on different premises. And there are immediate feelings, intuitions, that He exists. Sometimes the arguments generate that sense, and in that respect they are merely means that describe my intuition. I look at the world and understand that it has a Creator. I can describe this as a philosophical consideration, but that is only a form of description. By contrast, one who believes only by virtue of philosophical reasoning—this is an external faith, in fact some sort of wager. The first believer asserts that God exists; the second asserts that God must exist and only from there infers that He probably does exist.

This may be the meaning of the well-known story about the Jewish philosopher Hermann Cohen, who related that he met a simple Jew in the street and expounded to him his philosophical doctrine regarding faith in God. The Jew listened and finally asked him: “But where is the Holy One, blessed be He, in all this?” Hermann Cohen accepted the remark. There is something personal in a believer’s relation to his God, and philosophical arguments do not necessarily bring us there. Note: this is not a psychological or pragmatic distinction (a denigration of philosophical belief). It is an epistemic distinction. The simple Jew knows that God exists; the philosopher only knows that He must exist. That is not personal knowledge, for it does not deal with God. Such a faith deals with our world or the world of ideas and claims that within them something of this sort must exist. The dimension of encounter is absent, and therefore it is hard to speak here of knowledge that God exists. Faith in God is knowledge that He exists. An indication—however reliable—that He exists is not faith. The believer must know that God exists (though not with certainty), not wager on His existence, even if the wager is based on high probabilities.

This matter is delicate and tricky, and it may at first glance look like something existentialist, Heaven forbid. But it is not. It is a philosophical claim about our knowledge and its nature. A believing person is one who knows that God exists, not one who wagers on His existence or has decided that He must exist. Even if a person is not convinced of His existence, there is a difference between a person who is 90% convinced that He exists and a person who is 90% convinced that He must exist—even though, seemingly, the former follows from the latter. As long as the person has not actually made that inference, he is wagering but not a believer. To my judgment, in order for a person to be counted in a quorum he must be a believer, that is, one who knows that God exists. A wager—even at the same level of reliability (likelihood)—does not express knowledge.