Pascal's Wager: This is an argument proposed by Pascal in his 'Pensées' in favor of God's existence, which he construes as if it concerns a case similar to a simple bet on heads or tails, as follows:
|God does not exist
||God does exist
|God does not exist
|God does exist
(4) + ∞
These are then the four possibilities for one's bets, and according to Pascal one has the following pay-offs for these cases - and I will discuss these supposed pay-offs below:
(1) In fact God does not exist, and one bets God does not exist:
One gains a finite sum of happiness a, in one's finite life.
(2) In fact God does not exist, and one bets God does exist:
One is mistaken and one looses nothing.
(3) In fact God does exist, and one bets God does not exist:
One is mistaken and the reward or punishment depends on God, if He cares.
(4) In fact God does exist, and one bets God does exist:
Heavenly jackpot: Infinite bliss, blessings, and rewards - infinite pie in the sky.
Pascal's argument is, then, that one should bet God exists, and live accordingly, because if He does one will be rewarded with infinite bliss, and if He doesn't exist, one looses nothing (or, more realistically: one leads a miserable life keeping ones religious vows while gaining nothing after one dies, except happiness for the worms that eat one's dead body), whereas if one bets that God does not exist, at best one gains the finite pleasures of not living a boring life according to religious vows, while one risks God's possibly infinite displeasure if He turns out to be there after all.
And one can see this in the table: There is more to gain if only one bets God does exist, for a+0 < ?+∞ anyhow.
There is much one can object here, such as that this kind of wager does not do honour to God, as it seems it all is to be decided by self-interest, or that it would have been nicer of God if He had provided better evidence for His existence, so that one needed not to bet or act on faith, on the line 'credo quia absurdum' (I believe it because it is absurd - Tertullian).
My own basic objections are three, a minor and technical one, a major logical one, and a major epistemological one.
The first, minor and technical objection is that one can not reckon with infinities in this way, and that it is even quite possible that there are no infinities other than unlimited collections.
So suppose that this is waived, and that, instead, one takes it that the pay-off if one is right is large, but finite, which avoids the objection just made, apart from the possible incredibility of pie in the sky as promised by religions.
Thus Pascal's case - in fact: his promise, or belief - comes down to the statement that, whatever is the case, the outcomes of the possible bets are such that the possible gains and losses if God does not exist are (much) smaller than if He does Exist.
The second objection is that one has to live or at least believe according to some religion if one bets that God exists, and that there are many religions, all of which oppose and contradict each other.
What if one decides to bet on Catholicism, and God exists but is a Calvinist, or a Mohammedan, or a congregation of Hinduistic Gods, or something quite unimaginable but divine that looks down with severe contempt on humans who bet on His existence, instead of following their own finite capacities for reasoning?
In any of these cases - God exists, but not quite as you think, and with quite different rewards and punishments than those you or Pascal so fondly attributed to him, and indeed possibly precisely the opposites of those you imagine, just like many sincere Catholics have believed Calvinists were bound for hell, and many sincere Calvinists believed the same about Catholics - the losses may very well more than balance the possible gains of betting on His existence and living or believing according to a false teaching about Him.
And please note the opinions of a Catholic saint and a Protestant leading theologian on the subject of rewards and punishments:
"That the saints may enjoy their beatitude and the grace of God more abundantly they are permitted to see the punishment of the damned in hell." (St. Thomas Aquino, Summa Theologica)
"The sight of hell's torments will exalt the happiness of the saints forever." (Jonathan Edwards)
Thus, any promises Pascal's God may offer if He exists may be balanced or offset by any threats any other God may have in store for those who believe in Pascal's God (or indeed for betting in this way about divine existence).
So one can not rationally reckon in terms of this sort of imaginary pay-offs based on nothing but arbitrary assumptions or groundless faith - or indeed, if one wishes to believe according to Tertullian ('credo quia absurdum') it is better to do so directly, and without specious arguments, and also without moving oneself by hopes for great and divine rewards.
The third objection, which is epistemological, is in line with the previous objection, but objects to the pay-offs, also apart from God's existence: One should not consider whatever pay-offs one can imagine to justify one's bets, if what one is concerned with is the truth of something, for such pay-offs may make the less probable seem better, whereas what matters cognitively is not the rewards one may get if one is right, but the probability that one is right.
All of this is not to say probability theory is completely useless, but it is to say it should be used well, and the least that should be done is to reformulate the possibilities not in the form 'God exists' but in the form 'my God exists' - which immediately draws attention to the fact that there may be your God as well, and there may be different schemes of rewards and punishments for either, and this invites questions about evidence, both for your God, my God, and the diverse imagined rewards and punishments, and so on.
As I indicate in the lemma Religion my own hypothesis about God is that He does not exist, and as I indicated in my third objection, I hold that it is immaterial and confusing to consider the promises of pie in the sky as relevant for the probability of the truth of His existence, just as one should not consider it logically relevant for the probability of the truth of the possibility that one has cancer that one would be so very happy if this possibility were false.