Pascal's bet

Blaise Pascal

Paris Pascal is an argument proposed by the mathematician and philosopher Blaise Pascal to demonstrate the rationality of religious faith. The text of the argument is a fragment of the thoughts contained in section VIII “It is wiser to believe than not to believe what the Christian religion teaches” of the posthumous work Thoughts on Religion and Other Objects ( Fr. Pensées sur la religion et sur quelques autres sujets , in translations into Russian the name is often shortened to “Thoughts”), written in 1657 - 1658 .

The essence of reasoning

Pascal reasoned:

God is or not. Which side are we leaning to? Reason cannot solve anything here. Endless chaos separates us. At the edge of this infinity, a game is being played whose outcome is unknown. What will you bet on?

What do you bet on - religion or atheism ? In order to find the answer, Pascal suggested that the chances of the existence or absence of God are approximately equal, or at least that the probability of the existence of God is greater than zero. Then two options are possible:

Living without faith is extremely dangerous, since the possible "loss" in the case of the existence of God is infinitely great - eternal torment . If God does not exist, then the price of “gain” is low - unbelief gives us nothing and requires nothing from us. (Atheistic choice only allows you to save money, time and effort in the performance of religious rites and so on.)
Living according to the canons of faith is not dangerous, although a little more difficult because of fasting , all kinds of restrictions, rituals and the associated costs of time and money. The cost of “losing” in the absence of God is low — the cost of rituals and efforts for a righteous life. But the possible "gain" in the event of the existence of God is infinitely great - soul salvation , eternal life .

In his other works, Pascal considered the strategies of card games from a mathematical point of view. These works were based on an analysis of the probabilities of events in order to select the optimal bet size. To compare and select options for actions (events) that occur with different probabilities, you need to multiply the possible prize (win, bonus, result) by the probability of this event. The obtained values can be compared for different events and compared with costs (rates). The game theory , developed in the 20th century, is based on these ideas.

To make a decision in favor of one of the proposed options, Pascal used the same reasoning. What is the assessment of the options under consideration?

When we multiply, even if there is a high probability that there is no God, by a small value of the prize, we get a value less than the “small value” of the prize, that is, small and always final.
When you multiply any non-zero, even very small, probability that God will show mercy to a person for his virtuous behavior, an infinitely large value of the prize results in an infinitely large value.

Pascal concludes that the second option is preferable, that it’s silly to grab at finite values if you can get infinite ones:

What are you risking making such a choice? You will become a faithful, honest, humble, grateful, good-doing person, capable of sincere, true friendship. Yes, of course, base pleasures — fame, voluptuousness — will be ordered for you — but will you not receive anything in return? I tell you, you will win a lot even in this life, and with every step along the chosen path, the gain will be more and more doubtless for you and the more worthless you have set against the undoubted and infinite without sacrificing anything.

Analysis in terms of decision theory

Decision theory considers Pascal's bet to be a decision in the face of uncertainty. To make the best decision, you need to determine the value matrix (the resulting gains and costs).

Pascal did not consider the option “Do not believe”, provided that God exists. From his description it should be assumed that the gain from unbelief in the case of the existence of God is a finite number, although without specifying its sign ^[1] . For example, we can assume that this gain is zero or even a positive value. Denote this number as $f_{0}$ ${\ displaystyle f_ {0}}$ .

	God exists	God does not exist
Believe	$+\infty$ ${\ displaystyle + \ infty}$ (infinite prize value)	$-1$ ${\ displaystyle -1}$ (some final costs)
Do not believe	$f_{0}$ ${\ displaystyle f_ {0}}$ (some final value)	$+1$ ${\ displaystyle +1}$ (some final savings)

If we denote the probability that God exists, as $p$ ${\ displaystyle p}$ , then the probability that there is no God is equal $1-p$ ${\ displaystyle 1-p}$ .

To make the optimal decision (the most rational of the available choices), you should analyze each line of behavior and find the mathematical expectation of the corresponding gain. To this end, each element of the matrix (value) should be multiplied by the probability of an event in the corresponding column, and then the sum of such products should be found for the corresponding choice (that is, for each row). You should choose the option for which the amount received, that is, the mathematical expectation, is greater. Probability is estimated by a number from 0 to 1, but Pascal considered the option “ greater than 0, but less than 1 ”.

We calculate the products of interest to us and compose a matrix of results of the following form:

	God exists	God does not exist	Expected value
Believe	$+\infty$ ${\ displaystyle + \ infty}$	$(p-1)$ ${\ displaystyle (p-1)}$	$+\infty$ ${\ displaystyle + \ infty}$
Do not believe	$p\cdot f_{0}$ ${\ displaystyle p \ cdot f_ {0}}$	$(1-p)$ ${\ displaystyle (1-p)}$	$p\cdot f_{0}+(1-p)$ ${\ displaystyle p \ cdot f_ {0} + (1-p)}$

It's obvious that

$(1-p)\cdot (-1)=(p-1)$ ${\ displaystyle (1-p) \ cdot (-1) = (p-1)}$

and

$+\infty +(p-1)=+\infty$ ${\ displaystyle + \ infty + (p-1) = + \ infty}$

In any case, the mathematical expectation of winning for the option (line of behavior) "Believe" is positive and infinitely large. On the other hand, the mathematical expectation of winning for the Do Not Believe option (line of behavior) is always a finite quantity, even a large one. Pascal concludes that Rational is the rational choice.

Generalizations

It is obvious that the result is independent of probability $p$ ${\ displaystyle p}$ . What matters is that

${\displaystyle 0<semantics><mrow class=$ ${\ displaystyle 0 <p}$

and the gains in the absence of God are finite. That is, the specific values of the gain in the absence of God, −1 and +1, can be replaced by some finite $f_{1}$ ${\ displaystyle f_ {1}}$ and $f_{2}$ ${\ displaystyle f_ {2}}$ and consider the following payoff matrix:

	God exists	God does not exist
Believe	$+\infty$ ${\ displaystyle + \ infty}$ (infinite prize value)	$f_{1}$ ${\ displaystyle f_ {1}}$ (some final costs)
Do not believe	$f_{0}$ ${\ displaystyle f_ {0}}$ (some final value)	$f_{2}$ ${\ displaystyle f_ {2}}$ (some final savings)

This change will not affect the conclusions, we will again compare $+\infty$ ${\ displaystyle + \ infty}$ and some final value, which is equal to $p\cdot f_{0}+(1-p)\cdot f_{2}$ ${\ displaystyle p \ cdot f_ {0} + (1-p) \ cdot f_ {2}}$ .

The assumption of gaining from unbelief in the event of the existence of God may well be tightened. For example, we can assume that this gain is equal to $-\infty$ ${\ displaystyle - \ infty}$ (infinitely large punishment). With this assumption, we will compare $+\infty$ ${\ displaystyle + \ infty}$ (expected gain in case of faith in God) and $-\infty$ ${\ displaystyle - \ infty}$ (expected gain in case of disbelief). A rational choice will naturally be the same.

Criticism

Religious grades

Pascal's bet is referenced or analyzed by many religious books and publications. An example is the Internet portal “The Orthodox Encyclopedia“ The ABC of Faith ”” ^[2] (not to be confused with the “Orthodox Encyclopedia” ). The editors of the portal cited the opinion of S. L. Frank from the book “God is with us”, which critically considers the logical course of Pascal’s bet as having a strange and blasphemous error. The author rightly points out the moments of failure of the bet as some kind of spiritual evidence, asking the question: “What religious value does such a motivated determination to believe have?” Frank points out that internal foundations are necessary for faith, and the probabilistic approach in Pascal's reasoning has the features of "spiritual ugliness." At the same time, S. L. Frank finds another meaning in the “bet”: “In it you can catch a completely different idea, namely, that, having gone first“ at random ”along the path of faith, then we gain on it an experimental confirmation of its truth” , and for this thought there are reasons in Pascal's words: “ and with every step along the chosen path, there will undoubtedly be a gain for you .”

The Russian religious philosopher B. P. Vysheslavtsev considered Pascal's bet a paradoxical, and even comic, ^[3] version of the solution to the question of the appropriateness of faith.

The presence of other religions and gods

Although the bet is quite abstract, Pascal himself regarded it as an argument in favor of Christianity , and not religion in general or any religion other than Christianity. In the book of Pascal, the word "religion", as you can see, is a synonym for Christianity , and the bet itself is considered in section VIII "It is wiser to believe than not to believe what the Christian religion teaches."

The French philosopher- educator Denis Diderot put forward one of the first objections to Pascal's argument. This objection is not so much against religion, but against taking Pascal's bet as an argument in favor of a particular religion. In 1770, Didro remarked that even though Pascal had spoken out in favor of Christianity, a Muslim could have made the same argument in favor of Islam: ^[4]

LIX. Pascal said: “If your religion is false, you risk nothing, considering it true; if it is true, you risk everything, considering it false. ” Some imam could say the same as Pascal.
- Addition to the "Philosophical Thoughts", or various objections to the works of various theologians

William James , an American representative of the philosophy of pragmatism , in 1897 expressed a similar idea ^[5] - “Bet” cannot serve as proof of the existence of God, since it justifies faith in any religion that promises an eternal afterlife ^[6] . Pascal implied a choice in favor of Christianity. But the bet does not justify which of the many religions and cults are truly true. If there is no criterion for choosing a religion, then you can make a mistake when choosing: what to believe, who to pray for. And faith in a false religion can produce the same end result as disbelief in any of the religions.

James’s thoughts echo the well-known phrase that “you cannot go to Paradise of one religion without going to Hell of all others”. For more correct conditions and conclusions of the Pascal bet, it is necessary to introduce into the condition a listing of additional real options and their consequences. In addition to believing in a certain version of the Christian God and unbelief of an atheist, there is also the possibility of faith in other gods, for example, in Zeus , or the choice of another religion. If Islam is recognized as the third and equal option, then the Christian faith can lead to endless loss: Islam can turn out to be the “right” religion, and hell awaits the Christian, like any “unfaithful” one.

Richard Dawkins also points out that, apparently, it is more profitable not to believe at all than to believe in the wrong god ^[7] - in itself, the number of gods and goddesses on whom you can bet is already refuting the logic of Pascal's argument.

Endless Prize Value

For Pascal, the value of the prize was obvious and was based on biblical instructions: “without faith, it is impossible to please God; for it is necessary that he who comes to God believes that He is, and gives to those who seek Him ”( Heb. 11: 6 ),“ whoever believes and is baptized will be saved; but he who does not believe will be condemned ”( Mark 16:16 ). But in order to perceive the Bible as evidence or a basis for action, one must already be a believer, and this contradicts Pascal's original intention to convince an unbeliever that it is more reasonable to believe. ^[eight]

Even assuming the reward of believers, there is no guarantee that the prize will be of infinitely great value. Thus, the conditions of the bet do not guarantee that the believer is indeed in a better position than the unbeliever.

If we proceed from the dogmas of Christianity itself, the assertion that the reaction of the Christian God is known in advance directly contradicts the biblical assertion that it is impossible to know the divine purpose (“O abyss of wealth and wisdom and knowledge of God! How incomprehensible are His destinies and His paths are unexplored!”; Rome 11:33 ).

Choice of Faith

Richard Dawkins notes that it is impossible to believe or not to believe in something of one’s choice. ^[7]

Pascal's answer to the problem of the impossibility of faith by choice is as follows ^[9] :

"So; but my hands are tied, and I am unable to speak; I am forced to play and imprisoned; they don’t let me out, but I am created so that I can’t believe it. What am I to do now? ”
It's right. But understand, at least, that you are powerless to believe, because, despite the coercion on the part of the mind, you still cannot believe. Try to convince yourself not by obtaining new evidence of the existence of God, but by diminishing your passions. You want to achieve faith, but you do not know the path to it; If you want to recover from unbelief, ask for medicine. Ask about it from those who, like you, were connected, and now sacrifice all their wealth. These people know the path that you intend to go, and have recovered from an illness that you want to get rid of. Start as they started; and they started by doing everything as they had already believed: drinking holy water, ordering masses, etc. This will teach you to believe and humble your mind.
- Blaise Pascal

According to the famous philosopher John Leslie Mackey , this means that a person must transform his will by performing religious rituals until his will really becomes susceptible to faith ^[10] .

God can learn about the insincerity of faith

Richard Dawkins believes that Pascal’s bet can only be an argument in favor of pretending to be a believer (the god you supposedly believe in may turn out to be omniscient and figure out your tricks). ^[7]

Faith May Not Have a Reward

Richard Dawkins expresses doubts about the validity of the assertion that the best way to appease God is to believe in him (can it be assumed that God would just as willingly reward kindness, generosity, sincerity, and / or a purposeful search for truth) ^[7] .

The historian and philosopher Richard Carrier argues ^[11] :

Suppose that there is a certain god who watches over us and decides which of the souls of the dead to take to heaven, and this god really wants only good people to inhabit the heavens. He will probably only choose from those who consciously made significant efforts to learn the truth. [...] That is, if they really care about doing the right things and evading the wrong, then it necessarily follows from this that they should really be concerned about knowing the right and wrong. Since for such cognition it is necessary to learn many fundamental facts about the universe (for example, about the presence of God), it necessarily follows from such that these people should really be bothered by the constant search, verification and confirmation of the fact that their beliefs about such things are more likely all true. Therefore, only such people can prove to be sufficiently virtuous and trustworthy in order to earn a place in heaven, if God does not want to fill the heavens with careless, irresponsible and unreliable people. [...] Those who are truly interested in doing right things and evading wrong ones should sincerely be interested in the truth of certain statements, including the statement “God exists,” and should consider this issue with the same responsibility and attention as any another ethical issue.
Original text
Suppose there is a god who is watching us and choosing which souls of the deceased to bring to heaven, and this god really does want only the morally good to populate heaven. He will probably select from only those who made a significant and responsible effort to discover the truth. [...] That is, if they have a significant and trustworthy concern for doing right and avoiding wrong, it follows necessarily that they must have a significant and trustworthy concern for knowing right and wrong. Since this knowledge requires knowledge about many fundamental facts of the universe (such as whether there is a god), it follows necessarily that such people must have a significant and trustworthy concern for always seeking out, testing, and confirming that their beliefs about such things are probably correct. Therefore, only such people can be sufficient moral and trustworthy to deserve a place in heaven - unless god wishes to fill heaven with the morally lazy, irresponsible, or untrustworthy. [...] If anyone is sincerely interested in doing right and avoiding wrong, they must be sincerely interested in whether certain claims are true, including "God exists", and must treat this matter with as much responsibility and concern as any other moral question.
- Richard Carrier

Justification for other statements

Arguing in the spirit of Pascal's bet, one can get other results ^[12] :

For example, it is easy to show that faith in a strict (petty vengeful, angry, etc.) rather than a super-merciful god is preferable. That is, Orthodox Judaism is preferable to Hasidic or reformist (and especially Christianity!), While the Old Believers are preferable to Nikonianism, etc. How? But very simple. Fearing a strict god, we risk nothing even before the merciful. But the low threshold of fear and prohibitions before the merciful is extremely dangerous if it turns out to be strict!
- Khilkevich Y. M.

In addition, in the same way, any superstition can be justified ^[12] .

Based on such reasoning, we can conclude that Pascal's bet is not the final proof of anything. It does not evaluate the truth or falsity of a statement. An assumption is made of the possibility of an event, even with minimal probability. Rather, it is an explanation of the dominant line of behavior in a situation with uncertain conditions .

Notes

↑ Alan Hájek , Stanford Encyclopedia of Philosophy
↑ Semyon Frank, “God is with us”, excerpts from the chapter “Faith as a religious experience” // Internet portal “Orthodox Encyclopedia“ ABC of Faith ””.
↑ B.P. Vysheslavtsev . Chapter XII. Pascal // "The Eternal in Russian Philosophy" . - 1. - New York: them. Chekhov, 1955 .-- 302 p. “We find a peculiar solution to the dilemma in the argument that is known as Pascal’s bet. The argument can be considered paradoxical, even comic. ”
↑ History of the philosophy of Giovanni Reale and Dario Anticeri (neopr.) . www.krotov.info. Date of treatment August 3, 2016. Archived on August 3, 2016.
↑ James W. The Will to Believe: and Other Essays in Popular Philosophy (1897) - “As well might the Mahdi write to us, saying,“ I am the Expected One whom God has created in his effulgence. You shall be infinitely happy if you confess me; otherwise you shall be cut off from the light of the sun. Weigh, then, your infinite gain if I am genuine against your finite sacrifice if I am not! “His logic would be that of Pascal; but he would vainly use it on us, for the hypothesis he offers us is dead. No tendency to act on it exists in us to any degree. "
↑ Encyclopedia of Philosophy
↑ ¹ ² ³ ⁴ Chapter 3. Evidence of the existence of God / God as an illusion // Richard Dawkins (neopr.) . scisne.net. Date of treatment August 2, 2016.
↑ Andre Comte-Sponville. Philosophical Dictionary . - Litres, 2015-03-30. - 1949 p. - ISBN 9785457745698 .
↑ Blaise Pascal - “Thoughts” (neopr.) . bookwu.net. Date of treatment February 4, 2018.
↑ JL MacKie. The Miracle of Theism: Arguments for and Against the Existence of God . - Clarendon Press, 1982-01-01. - 284 p. - ISBN 9780198246824 .
↑ Richard Carrier, 2002: The End of Pascal's Wager: Only Nontheists go to Heaven
↑ ¹ ² Khilkevich Ya. M. Paris Pascal as a log paradox

Links

Blaise Pascal's “Thoughts” (inaccessible link from 11/05/2013 [2307 days]) (HTML, breakdown of book sections)
Blaise Pascal "Thoughts" (simplified translation)
Pascal's Wager from Internet Encyclopedia of Philosophy
Boris Ponomarev. Pragmatic Pascal Betting Approach

more in Russian philosophy "| link = http://www.kirsoft.com.ru/freedom/KSNews_125.htm | edition = 1 | place = New York | publisher = Chekhov | year = 1955 | pages = 302 | isbn =}} "We find a peculiar solution to the dilemma in the argument known as Pascal's bet. The argument can be considered paradoxical, even comic."

[1] Alan Hájek , Stanford Encyclopedia of Philosophy

[2] Semyon Frank, “God is with us”, excerpts from the chapter “Faith as a religious experience” // Internet portal “Orthodox Encyclopedia“ ABC of Faith ””.

[3] B.P. Vysheslavtsev . Chapter XII. Pascal // "The Eternal in Russian Philosophy" . - 1. - New York: them. Chekhov, 1955 .-- 302 p. “We find a peculiar solution to the dilemma in the argument that is known as Pascal’s bet. The argument can be considered paradoxical, even comic. ”

[4] History of the philosophy of Giovanni Reale and Dario Anticeri (neopr.) . www.krotov.info. Date of treatment August 3, 2016. Archived on August 3, 2016.

[james-5] James W. The Will to Believe: and Other Essays in Popular Philosophy (1897) - “As well might the Mahdi write to us, saying,“ I am the Expected One whom God has created in his effulgence. You shall be infinitely happy if you confess me; otherwise you shall be cut off from the light of the sun. Weigh, then, your infinite gain if I am genuine against your finite sacrifice if I am not! “His logic would be that of Pascal; but he would vainly use it on us, for the hypothesis he offers us is dead. No tendency to act on it exists in us to any degree. "

[6] Encyclopedia of Philosophy

[:0-7] ¹ ² ³ ⁴ Chapter 3. Evidence of the existence of God / God as an illusion // Richard Dawkins (neopr.) . scisne.net. Date of treatment August 2, 2016.

[8] Andre Comte-Sponville. Philosophical Dictionary . - Litres, 2015-03-30. - 1949 p. - ISBN 9785457745698 .

[9] Blaise Pascal - “Thoughts” (neopr.) . bookwu.net. Date of treatment February 4, 2018.

[10] JL MacKie. The Miracle of Theism: Arguments for and Against the Existence of God . - Clarendon Press, 1982-01-01. - 284 p. - ISBN 9780198246824 .

[11] Richard Carrier, 2002: The End of Pascal's Wager: Only Nontheists go to Heaven

[X-12] ¹ ² Khilkevich Ya. M. Paris Pascal as a log paradox