Pascal's Wager

Blaise Pascal

Pascal's Wager is an argument in apologetic philosophy devised by the seventeenth-century French philosopher, mathematician and physicist Blaise Pascal (1623–62).[1] It posits that humans all bet with their lives either that God exists or not. Given the possibility that God actually does exist and assuming an infinite gain or loss associated with belief or unbelief in said God (as represented by an eternity in heaven or hell), a rational person should live as though God exists and seek to believe in God. If God does not actually exist, such a person will have only a finite loss (some pleasures, luxury, etc.).[2]

Pascal formulated the wager within a Christian framework. The wager was set out in section 233 of Pascal's posthumously published Pensées ("Thoughts"). These previously unpublished notes were assembled to form an incomplete treatise on Christian apologetics.

Historically, Pascal's Wager was groundbreaking because it charted new territory in probability theory, marked the first formal use of decision theory and anticipated future philosophies such as existentialism, pragmatism and voluntarism.[3]

The wager

The wager uses the following logic (excerpts from Pensées, part III, §233):

  1. God is, or God is not. Reason cannot decide between the two alternatives.
  2. A Game is being played... where heads or tails will turn up.
  3. You must wager (it is not optional).
  4. Let us weigh the gain and the loss in wagering that God is. Let us estimate these two chances. If you gain, you gain all; if you lose, you lose nothing.
  5. Wager, then, without hesitation that He is. (...) There is here an infinity of an infinitely happy life to gain, a chance of gain against a finite number of chances of loss, and what you stake is finite. And so our proposition is of infinite force, when there is the finite to stake in a game where there are equal risks of gain and of loss, and the infinite to gain.
  6. But some cannot believe. They should then 'at least learn your inability to believe...' and 'Endeavour then to convince' themselves.
Category Quotation(s)
Uncertainty in all This is what I see, and what troubles me. I look on all sides, and everywhere I see nothing but obscurity. Nature offers me nothing that is not a matter of doubt and disquiet.[4]
Uncertainty in Man's purpose For after all what is man in nature? A nothing in relation to infinity, all in relation to nothing, a central point between nothing and all and infinitely far from understanding either.[5]
Uncertainty in reason There is nothing so conformable to reason as this disavowal of reason.[6]
Uncertainty in science There no doubt exist natural laws, but once this fine reason of ours was corrupted, it corrupted everything.[7]
Uncertainty in religion If I saw no signs of a divinity, I would fix myself in denial. If I saw everywhere the marks of a Creator, I would repose peacefully in faith. But seeing too much to deny Him, and too little to assure me, I am in a pitiful state, and I would wish a hundred times that if a god sustains nature it would reveal Him without ambiguity.[4]

We understand nothing of the works of God unless we take it as a principle that He wishes to blind some and to enlighten others.[8]

Uncertainty in skepticism It is not certain that everything is uncertain.[9]

Pascal asks the reader to analyze the position of mankind, this crisis of existence and lack of complete understanding. While Mankind can discern a great deal through reason, it is also hopelessly removed from knowing everything through it. He describes Mankind as a finite being trapped within an incomprehensible infinity, thrust into being from non-being for a brief life only to go out again, with no explanation whatsoever of "Why?" or "What?" or "How?" The finite nature of our being constrains reason with respect to every form of knowledge. Now, assuming that reason alone cannot determine whether or not God exists, the ontological question is reduced to a coin toss. However, making a choice to live as though God exists or does not exist is unavoidable even if the ontological question is inconclusive. In Pascal's assessment, participation in this Wager is not optional because Mankind is already thrust into existence. So even if God's existence cannot be independently confirmed or denied, nevertheless the Wager is necessary and the possible scenarios must be considered and decided upon pragmatically.

Explanation

The wager is described in Pensées this way:

If there is a God, He is infinitely incomprehensible, since, having neither parts nor limits, He has no affinity to us. We are then incapable of knowing either what He is or if He is....

..."God is, or He is not." But to which side shall we incline? Reason 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. What will you wager? According to reason, you can do neither the one thing nor the other; according to reason, you can defend neither of the propositions.

Do not, then, reprove for error those who have made a choice; for you know nothing about it. "No, but I blame them for having made, not this choice, but a choice; for again both he who chooses heads and he who chooses tails are equally at fault, they are both in the wrong. The true course is not to wager at all."

Yes; but you must wager. It is not optional. You are embarked. Which will you choose then? Let us see. Since you must choose, let us see which interests you least. You have two things to lose, the true and the good; and two things to stake, your reason and your will, your knowledge and your happiness; and your nature has two things to shun, error and misery. Your reason is no more shocked in choosing one rather than the other, since you must of necessity choose. This is one point settled. But your happiness? Let us weigh the gain and the loss in wagering that God is. Let us estimate these two chances. If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is.

"That is very fine. Yes, I must wager; but I may perhaps wager too much." Let us see. Since there is an equal risk of gain and of loss, if you had only to gain two lives, instead of one, you might still wager. But if there were three lives to gain, you would have to play (since you are under the necessity of playing), and you would be imprudent, when you are forced to play, not to chance your life to gain three at a game where there is an equal risk of loss and gain. But there is an eternity of life and happiness. And this being so, if there were an infinity of chances, of which one only would be for you, you would still be right in wagering one to win two, and you would act stupidly, being obliged to play, by refusing to stake one life against three at a game in which out of an infinity of chances there is one for you, if there were an infinity of an infinitely happy life to gain. But there is here an infinity of an infinitely happy life to gain, a chance of gain against a finite number of chances of loss, and what you stake is finite.[6]

Pascal begins by painting a situation where both the existence and non-existence of God are impossible to prove by human reason. So, supposing that reason cannot determine the truth between the two options, one must "wager" by weighing the possible consequences. Pascal’s assumption is that, when it comes to making the decision, no one can refuse to participate; withholding assent is impossible because we are already "embarked", effectively living out the choice.

We only have two things to stake, our "reason" and our "happiness". Pascal considers that if there is "equal risk of loss and gain" (i.e. a coin toss), then human reason is powerless to address the question of whether God exists. That being the case, then human reason can only decide the question according to possible resulting happiness of the decision, weighing the gain and loss in believing that God exists and likewise in believing that God does not exist.

He points out that if a wager was between the equal chance of gaining two lifetimes of happiness and gaining nothing, then a person would be a fool to bet on the latter. The same would go if it was three lifetimes of happiness versus nothing. He then argues that it is simply unconscionable by comparison to bet against an eternal life of happiness for the possibility of gaining nothing. The wise decision is to wager that God exists, since "If you gain, you gain all; if you lose, you lose nothing", meaning one can gain eternal life if God exists, but if not, one will be no worse off in death than if one had not believed. On the other hand, if you bet against God, win or lose, you either gain nothing or lose everything. You are either unavoidably annihilated (in which case, nothing matters one way or the other) or lose the opportunity of eternal happiness. In note 194, speaking about those who live apathetically betting against God, he sums up by remarking, "It is to the glory of religion to have for enemies men so unreasonable..."

Inability to believe

Pascal addressed the difficulty that 'reason' and 'rationality' pose to genuine belief by proposing that "acting as if [one] believed" could "cure [one] of unbelief":

But at least learn your inability to believe, since reason brings you to this, and yet you cannot believe. Endeavour then to convince yourself, not by increase of proofs of God, but by the abatement of your passions. You would like to attain faith, and do not know the way; you would like to cure yourself of unbelief, and ask the remedy for it. Learn of those who have been bound like you, and who now stake all their possessions. These are people who know the way which you would follow, and who are cured of an ill of which you would be cured. Follow the way by which they began; by acting as if they believed, taking the holy water, having masses said, etc. Even this will naturally make you believe, and deaden your acuteness.
—Blaise Pascal, Pensées Section III note 233, Translation by W. F. Trotter

Analysis with decision theory

The possibilities defined by Pascal's Wager can be thought of as a decision under uncertainty with the values of the following decision matrix.

God exists (G) God does not exist (¬G)
Belief (B) +∞ (infinite gain) −1 (finite loss)
Disbelief (¬B) −∞ (infinite loss) +1 (finite gain)

Given these values, the option of living as if God exists (B) dominates the option of living as if God does not exist (~B), as long as one assumes a positive probability that God exists. In other words, the expected value gained by choosing B is greater than or equal to that of choosing ~B.

In fact, according to decision theory, the only value that matters in the above matrix is the +∞ (infinitely positive). Any matrix of the following type (where f1, f2, and f3 are all finite positive or negative numbers) results in (B) as being the only rational decision.[10]

God exists (G) God does not exist (¬G)
Belief (B) +∞ f1
Disbelief (¬B) f2 f3

Criticism

Criticism of Pascal's Wager began in his own day, and came from both atheists, who question the 'benefits' of a deity whose 'realm' is beyond reason, and the religiously orthodox, who primarily take issue with the wager's deistic and agnostic language. It is criticized for not proving God's existence, encouragement of false belief and the problem of which religion and which God should be worshipped.[11]

Nature as not a proof of the existence of God

Voltaire (another prominent French writer of the Enlightenment), a generation after Pascal, rejected the idea that the wager was "proof of God" as "indecent and childish", adding, "the interest I have to believe a thing is no proof that such a thing exists".[12] Pascal, however, did not advance the wager as a proof of God's existence but rather as a necessary pragmatic decision which is "impossible to avoid" for any living person.[13] He argued that abstaining from making a wager is not an option and that "reason is incapable of divining the truth"; thus, a decision of whether or not to believe in the existence of God must be made by "considering the consequences of each possibility".

Honestly judged, however, Voltaire's critique concerns not the nature of the Pascalian wager as proof of God's existence, but the contention that the very belief Pascal tried to promote is not convincing. Voltaire hints at the fact that Pascal, as a Catholic Jansenist, believed that only a small, and already predestined, portion of humanity would eventually be saved by God.

In this context Voltaire explained that no matter how far someone is tempted with rewards in order to believe in Christian salvation, the result will be at best a faint belief.[14] Pascal, in his Pensees, agrees with this, not stating that people can choose to believe (and therefore make a safe wager), but rather that some can not believe.

As Étienne Souriau explained, in order to accept Pascal's argument, the bettor needs to be certain that God seriously intends to honour the bet; he says that the wager assumes that God also accepts the bet, which is not proved; Pascal's bettor is here like the fool who seeing a leaf floating on a river's waters and quivering at some point, for few seconds, between the two sides of a stone, says: "I bet a million with Rothschild that it takes finally the left path." And, effectively, the leaf passed on the left side of the stone, but unfortunately for the fool Rothschild never said "I bet".[15]

Argument from inconsistent revelations

Since there have been many religions throughout history, and therefore many conceptions of God (or gods), some assert that all of them need to be factored into the wager, in an argument known as the argument from inconsistent revelations. This, its proponents argue, would lead to a high probability of believing in "the wrong god", which, they claim, eliminates the mathematical advantage Pascal claimed with his Wager. Denis Diderot, a contemporary of Voltaire, concisely expressed this opinion when asked about the wager, saying "an Imam could reason the same way".[16] J. L. Mackie notes that "the church within which alone salvation is to be found is not necessarily the Church of Rome, but perhaps that of the Anabaptists or members of The Church of Jesus Christ of Latter-day Saints or the Muslim Sunnis or the worshipers of Kali or of Odin."[17]

Another version of this objection argues that for every religion that promulgates rules, there exists another religion that has rules of the opposite kind. If a certain action leads one closer to salvation in the former religion, it leads one further away from it in the latter. Therefore, the expected value of following a certain religion could be negative. Or, one could also argue that there are an infinite number of mutually exclusive religions (which is a subset of the set of all possible religions), and that the probability of any one of them being true is zero; therefore, the expected value of following a certain religion is zero.

Pascal considers this type of objection briefly in the notes compiled into the Pensées, and dismisses it as obviously wrong and disingenuous:[18]

What say [the unbelievers] then? "Do we not see," say they, "that the brutes live and die like men, and Turks like Christians? They have their ceremonies, their prophets, their doctors, their saints, their monks, like us," etc. If you care but little to know the truth, that is enough to leave you in repose. But if you desire with all your heart to know it, it is not enough; look at it in detail. That would be sufficient for a question in philosophy; but not here, where everything is at stake. And yet, after a superficial reflection of this kind, we go to amuse ourselves, etc. Let us inquire of this same religion whether it does not give a reason for this obscurity; perhaps it will teach it to us.[19]

This short but densely packed passage, which alludes to numerous themes discussed elsewhere in the Pensées, has given rise to many pages of scholarly analysis.

Pascal says that unbelievers who rest content with the many-religions objection are people whose scepticism has seduced them into a fatal "repose". If they were really bent on knowing the truth, they would be persuaded to examine "in detail" whether Christianity is like any other religion, but they just cannot be bothered.[20] Their objection might be sufficient were the subject concerned merely some "question in philosophy", but not "here, where everything is at stake". In "a matter where they themselves, their eternity, their all are concerned",[19] they can manage no better than "a superficial reflection" ("une reflexion légère") and, thinking they have scored a point by asking a leading question, they go off to amuse themselves.[21]

As Pascal scholars observe, Pascal regarded the many-religions objection as a rhetorical ploy, a "trap"[22] that he had no intention of falling into. If, however, any who raised it were sincere, they would want to examine the matter "in detail". In that case, they could get some pointers by turning to his chapter on "other religions".

As David Wetsel notes, Pascal's treatment of the pagan religions is brisk: "As far as Pascal is concerned, the demise of the pagan religions of antiquity speaks for itself. Those pagan religions which still exist in the New World, in India, and in Africa are not even worth a second glance. They are obviously the work of superstition and ignorance and have nothing in them which might interest 'les gens habiles' ('clever men')[23]"[24] Islam warrants more attention, being distinguished from paganism (which for Pascal presumably includes all the other non-Christian religions) by its claim to be a revealed religion. Nevertheless, Pascal concludes that the religion founded by Mohammed can on several counts be shown to be devoid of divine authority, and that therefore, as a path to the knowledge of God, it is as much a dead end as paganism.[25] Judaism, in view of its close links to Christianity, he deals with elsewhere.[26]

The many-religions objection is taken more seriously by some later apologists of the wager, who argue that, of the rival options, only those awarding infinite happiness affect the wager's dominance. In the opinion of these apologists "finite, semi-blissful promises such as Kali's or Odin's" therefore drop out of consideration.[27] Also, the infinite bliss that the rival conception of God offers has to be mutually exclusive. If Christ's promise of bliss can be attained concurrently with Jehovah's and Allah's (all three being identified as the God of Abraham), there is no conflict in the decision matrix in the case where the cost of believing in the wrong conception of God is neutral (limbo/purgatory/spiritual death), although this would be countered with an infinite cost in the case where not believing in the correct conception of God results in punishment (hell).[27]

Furthermore, ecumenical interpretations of the Wager[28] argue that it could even be suggested that believing in a generic God, or a god by the wrong name, is acceptable so long as that conception of God has similar essential characteristics of the conception of God considered in Pascal's Wager (perhaps the God of Aristotle). Proponents of this line of reasoning suggest that either all of the conceptions of God or gods throughout history truly boil down to just a small set of "genuine options",[29] or that if Pascal's Wager can simply bring a person to believe in "generic theism" it has done its job.[30]

Argument from inauthentic belief

Some critics argue that Pascal's Wager, for those who cannot believe, suggests feigning belief to gain eternal reward. This would be dishonest and immoral. In addition, it is absurd to think that God, being just and omniscient, would not see through this deceptive strategy on the part of the "believer", thus nullifying the benefits of the wager.[11]

Since these criticisms are concerned not with the validity of the wager itself, but with its possible aftermath — namely that a person who has been convinced of the overwhelming odds in favor of belief might still find himself unable to sincerely believe — they are tangential to the thrust of the wager. What such critics are objecting to is Pascal's subsequent advice to an unbeliever who, having concluded that the only rational way to wager is in favor of God's existence, points out, reasonably enough, that this by no means makes him a believer. This hypothetical unbeliever complains, "I am so made that I cannot believe. What would you have me do?"[31] Pascal, far from suggesting that God can be deceived by outward show, says that God does not regard it at all: "God looks only at what is inward."[32] For a person who is already convinced of the odds of the wager but cannot seem to put his heart into the belief, he offers practical advice.

Explicitly addressing the question of inability to believe, Pascal argues that if the wager is valid, the inability to believe is irrational, and therefore must be caused by feelings: "your inability to believe, because reason compels you to [believe] and yet you cannot, [comes] from your passions." This inability, therefore, can be overcome by diminishing these irrational sentiments: "Learn from those who were bound like you. . . . Follow the way by which they began: that is by doing everything as if they believed, by taking holy water, by having Masses said, etc. Naturally, even this will make you believe and will dull you. —'But this is what I am afraid of.' —And why? What have you to lose?"[33]

Whether or not one can genuinely choose to believe a particular story based on the appeal to possible consequences, is trivialized. The urgency Pascal expressed that one must choose, and quickly, is not convincingly an argument to choose at all. The wager requires a correct choice (lose or win) among two scenarios that are part of the religious story itself. One must choose first to be convinced in the dichotomy of two initial propositions, of what "losing" or "winning" mean, as potential outcomes. While obedience and professing one's belief results in reward, skepticism or inability to believe result in punishment, both of these propositions are foregone conclusions. The wager is based on these two assumed-to-be established conditions and their outcomes. From there, the gambler's choice degrades into a coercion on the basis of self-interest, swayed by an appeal to fear. The god in the scenario is Pascal's personal conception of how the creator is or behaves and so the wager is congruent with the jealous and angry God of the New Testament. Pascal's idea of God would include what is predictable behavior. He describes God's feelings as your primary concern, when stated plainly. What might upset him or not, will determine (your) outcome.

Pascal's Wager is often compared to Anselm's Ontological Argument for the Existence of God, where a subtle insertion of a foregone conclusion, is presumed true and valid while the remaining propositions follow effortlessly. That which humans can capably conceive, is an a priori announcement that humans can't conceive of much. Worse, Pascal goes beyond Anselm to claim personal knowledge of his god, and what that god expects, while presenting the appearance of a hypothetical scenario.

Some other critics have objected to Pascal's Wager on the grounds that he wrongly assumes what type of epistemic character God would likely value in his rational creatures if he existed. More specifically, Richard Carrier has objected by positing an alternative conception of God that prefers his creatures to be honest inquirers and disapproves of thoughtless or feigned belief:

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. . .Therefore, only such people can be sufficiently moral and trustworthy to deserve a place in heaven — unless God wishes to fill heaven with the morally lazy, irresponsible, or untrustworthy.

The End of Pascal's Wager: Only Nontheists Go to Heaven[34]

However, as noted above, nowhere in the establishment of the wager does Pascal appeal to feigned belief; God, being omniscient, would not succumb to such trickery and unwittingly reward the disingenuous. Rather, in the passage following the establishment of the wager, Pascal addresses a hypothetical person who has already weighed the rationality of believing in God through the wager and is convinced by it, but remains unable to sincerely believe. Again, as noted above, Pascal offers this person a way to escape the irrational sentiment that compels him to withhold belief in God after the validity of the wager has been rationally conceded. This way consists of applying oneself to spiritual discipline, study, and community.

In practical terms, therefore, this "alternative" scenario of God valuing rational belief and honest inquiry which is offered by Carrier and other critics is actually not very different from Pascal's own formulation of the scenario. Indeed, Pascal is unabashed in his criticism of people who are apathetic about considering the issue of whether God exists. In note 194, he retorts: "This carelessness in a matter which concerns themselves, their eternity, their all, moves me more to anger than pity; it astonishes and shocks me; it is to me monstrous." Far from glorifying blind irrationality, one of the chief aims of Pascal's arguments in the Pensées was to shake people out of what he saw as their ignorant complacency so that they could rationally approach this most crucial existential matter. Pascal says in note 225: "Atheism shows strength of mind, but only to a certain degree." Unbelievers who persistently endeavor in an honest, rational effort to search for the truth are commended by Pascal, to the exclusion of those who are merely dismissive.

A different angle on this concern is that it is a non-sequitur relationship between points 1 and 5. The truth of the "God is/God is not" binary exclusion seems unchallengeable. However, it cannot be drawn from this that there must of necessity be a consequence to that reality unless we make a presumption. Pascal postulates that there is infinite bliss or infinite loss attached to the gamble which we are compelled to make. But there is nothing given to support this as a natural sequitur, nor even that this "God-is" has the capacity to effect any future outcome at all, other than this "suppose there is a god who is watching us and choosing" presumption. And that is unproven.

Variations

See also

Notes

  1. ↑ Connor, James A. (2006). Pascal's wager : the man who played dice with God. San Francisco: HarperSanFrancisco. p. 180-1. ISBN 9780060766917.
  2. ↑ "Blaise Pascal", Columbia History of Western Philosophy, page 353.
  3. ↑ Alan Hájek, Stanford Encyclopedia of Philosophy
  4. ↑ 4.0 4.1 Pensée #229
  5. ↑ Pensée #72
  6. ↑ 6.0 6.1 Pensée #272
  7. ↑ Pensée #294
  8. ↑ Pensée #565
  9. ↑ Pensée #387
  10. ↑ Alan Hájek, Stanford Encyclopedia of Philosophy
  11. ↑ 11.0 11.1 Dawkins, Richard (May 21, 2007). "Chapter 3: Arguments for God's existence". The God Delusion. Black Swan. pp. 130–132. ISBN 9780552773317.
  12. ↑ Remarques sur les Pensees de Pascal XI
  13. ↑ Durant, Will and Ariel (1965). The Age of Voltaire. p. 370.
  14. ↑ Vous me promettez l’empire du monde si je crois que vous avez raison: je souhaite alors, de tout mon coeur, que vous ayez raison; mais jusqu’à ce que vous me l’ayez prouvé, je ne puis vous croire. […] J’ai intérêt, sans doute, qu’il y ait un Dieu; mais si dans votre système Dieu n’est venu que pour si peu de personnes; si le petit nombre des élus est si effrayant; si je ne puis rien du tout par moi-même, dites-moi, je vous prie, quel intérêt j’ai à vous croire? N’ai-je pas un intérêt visible à être persuadé du contraire? De quel front osez-vous me montrer un bonheur infini, auquel d’un million d’hommes un seul à peine a droit d’aspirer?
  15. ↑ À vrai dire le célèbre pari de Pascal, ou plutôt le pari que Pascal propose au libertin n'est pas une option désintéressée mais un pari de joueur. Si le libertin joue «croix», parie que Dieu existe, il gagne (si Dieu existe) la vie éternelle et la béatitude infinie, et risque seulement de perdre les misérables plaisirs de sa vie actuelle. Cette mise ne compte pas au regard du gain possible qui est infini. Seulement, l'argument suppose que Dieu accepte le pari, que Dieu dit «je tiens». Sans quoi, nous dit Souriau, le libertin « est comme ce fou : il voit une feuille au fil de l'eau, hésiter entre deux côtés d'un caillou. Il dit : «je parie un million avec Rothschild qu'elle passera à droite». La feuille passe à droite et le fou dit : «j'ai gagné un million». Où est sa folie? Ce n'est pas que le million n'existe pas, c'est que Rothschild n'a pas dit : «je tiens». ». (Cf. l'admirable analyse du pari de Pascal in Souriau, L'ombre de Dieu, p. 47 sq.) - La Philosophie, Tome 2 (La Connaissance), Denis Huisman, André Vergez, Marabout 1994, pp.462-63
  16. ↑ Diderot, Denis (1875–77) [1746]. J. Assézar, ed. Pensées philosophiques, LIX, Volume 1 (in French). p. 167.
  17. ↑ Mackie, J. L. (1982). The Miracle of Theism, Oxford, pg. 203
  18. ↑ Wetsel, David (1994). Pascal and Disbelief: Catechesis and Conversion in the Pensées. Washington, D. C.: The Catholic University of America Press, p. 117. ISBN 0-8132-1328-2
  19. ↑ 19.0 19.1 Pensée #226
  20. ↑ Wetsel, Pascal and Disbelief, p. 370.
  21. ↑ Wetsel, Pascal and Disbelief, p. 238.
  22. ↑ Wetsel, Pascal and Disbelief, pp. 118 (quotation from Jean Mesnard), 236.
  23. ↑ Pensée #251
  24. ↑ Wetsel, Pascal and Disbelief, p. 181.
  25. ↑ Wetsel, Pascal and Disbelief, p. 182.
  26. ↑ Wetsel, Pascal and Disbelief, p. 180.
  27. ↑ 27.0 27.1 Alan Hájek, Stanford Encyclopedia of Philosophy
  28. ↑ For example: Jeff Jordan, Gambling on God: Essays on Pascal's Wager, 1994, Rowman & Littlefield.
  29. ↑ Paul Saka, The Internet Encyclopedia of Philosophy - Pascal's Wager
  30. ↑ Paul Saka, The Internet Encyclopedia of Philosophy - Pascal's Wager
  31. ↑ Pensée #233
  32. ↑ Pensée #904
  33. ↑ Pensée #233. Gérard Ferreyrolles, ed. Paris: Librairie Générale Française, 2000.
  34. ↑ The End of Pascal's Wager: Only Nontheists Go to Heaven
  35. ↑ Boyarin, Daniel (2009). Socrates & the fat rabbis. University of Chicago Press. p. 48. ISBN 0-226-06916-8.
  36. ↑ Weaver, John B. (2004). Plots of epiphany: prison-escape in Acts of the Apostles. Walter de Gruyter. p. 595. ISBN 978-3-11-018266-8.
  37. ↑ Aleksandrovich Florenskiĭ, Pavel (1997). The pillar and ground of the truth. Princeton University Press. p. 37. ISBN 0-691-03243-2.
  38. ↑ al-Juwayni A Guide to Conclusive Proofs for the Principles of Belief, 6
  39. ↑ Nicholas Ostler (2005). Empires of the Word, HarperCollins.

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