What If God Was

Pascal’s Wager

Along with Pierre de Fermat, Blaise Pascal is often credited with the initial development of mathematical probability in the 17^th century.  But Hacking (1975) also credits him with introduction of the first great model uncertainty problem:

‘God is, or he is not’ is Pascal’s expression of his partition. ‘Which way should we incline? Reason cannot answer.’  That is, there is no valid proof of God’s existence.  Instead we adopt the following model:

A game is on at the other end of an infinite distance, and heads or tails is going to turn up. Which way will you bet?

The model is then reinforced.  When reason cannot answer, a sensible man can say that he will not play the game.  But in our case, by the mere fact of eating living, we are engaged in play.  We either believe in God, or we do not.

So, there we have it:  A two-node probability tree:

Pascal then goes on to use the wager to argue that, just in case there is a heaven and a hell, it is better to behave as if God exists.  But as Hacking (1975) notes, there is a problem:

The argument is valid.  The premises are dubious, if not patently false.  Few non-believers now can suppose that Pascal’s partition exhausts the possibilities.  If we allow just one further alternative, namely the thesis of some fundamentalist sects, That Jehovah damns all those who toy with ‘holy water and sacraments’, then the Catholic strategy no longer dominates.  That is, there is one possible state of affairs, in which strategy does not have the best pay-off.

This raises what are perhaps the two most crucial issues for the employment of a probability tree as a form of argument.  First, the nodes of the tree must at least be plausible.  Second, the nodes included must exhaust the range of alternatives.  Or, to put the two together, a credible probability tree must encompass the range of plausible alternatives.

Pascal did not refer to his wager as a form of probability, and he did not appear to think of degrees of belief that are in some way quantifiable.  The nodes of his probability tree were like the sides of a coin or dice, where each is equiprobable. 

The Probability of Causes

David Hume’s first work, A Treatise of Human Nature (1739) did not sell well.  It was most assuredly a bit long-winded.  His more concisely worded second work, An Enquiry concerning Human Understanding (1748), did much better.  While the Treatise distinguished between the Probability of Chance and the Probability of Causes, the Essay provides the better foundation for understanding the vast difference between the two:

ALL the objects of human reason or enquiry may naturally be divided into two kinds, to wit, Relations of Ideas, and Matters of Fact. Of the first kind are the sciences of Geometry, Algebra, and Arithmetic; and in short, every affirmation which is either intuitively or demonstratively certain. That the square of the hypothenuse is equal to the square of the two sides, is a proposition which expresses a relation between these figures. That three times five is equal to the half of thirty, expresses a relation between these numbers. Propositions of this kind are discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe. Though there never were a circle or triangle in nature, the truths demonstrated by Euclid would for ever retain their certainty and evidence.

Matters of fact, which are the second objects of human reason, are not ascertained in the same manner; nor is our evidence of their truth, however great, of a like nature with the foregoing. The contrary of every matter of fact is still possible; because it can never imply a contradiction, and is conceived by the mind with the same facility and distinctness, as if ever so conformable to reality. That the sun will not rise to-morrow is no less intelligible a proposition, and implies no more contradiction than the affirmation, that it will rise. We should in vain, therefore, attempt to demonstrate its falsehood. Were it demonstratively false, it would imply a contradiction, and could never be distinctly conceived by the mind.

The 'probability of chance', alluded to in the Treatise, falls in the first category of deductive reasoning.  It corresponds to Pascal’s mathematical probability and what is now generally referred to as statistical probability.  On the other hand, the 'probability of causes' falls in the second category of inductive reasoning, and it sounds suspiciously like Pascal’s Wager.  This is especially evident from this passage from the Treatise:

We may observe, that there is no great as not to allow of a contrary possibility; because otherwise ‘twou’d cease to be a probability, and wou’d become a certainty. That a probability of causes, which is most extensive, and which we at present examine, depends on a contrariety of experiments; and ‘tis evident that an experiment in the past proves at least a possibility for the future.

Hume is clearly thinking of probability as a degree of belief involving competing theories.  Not only that, he clearly has an idea about how degrees of belief are strengthened:

As the habit, which produces the association, arises from the frequent conjunction of objects, it must arrive at its perfection by degrees, and must acquire new force from each instance, that falls under our observation.

So, Hume thought association was evidence of causation.  That idea just might stick.

References

Hacking I (1975).  The Great Decision.  In: The Emergence of Probability. Cambridge: Cambridge University Press.

Hume. D. (1739). A Treatise of Human Nature, EC Mossner (ed.), London: Penguin Books. 

Hume, D. (1748). Enquiry Concerning Human Understanding, A. Flew (ed.), La Salle: Open Court Press.  

Official Post Soundtrack

Osborne, J (1995).  One of Us.  In: Relish, Track 6.

Post Note

Thesis Post #1  A main purpose of this blog is to get some public review, so I welcome any suggestions for improving it.  At some point, I will assemble the theses into a hyperlinked web document.  I figure there will be about 95.

I am starting with this thesis because the two paragraph Hume quote explains my handle of riskylogic; I freely admit that I am arguing from dubious premises.