I feel the need to recycle a post I put on the Risk Analysis listserv over
two years ago:
Ian Hacking, part III. My favorite chapter in The Emergence of Probability
has to be Chapter 8, entitled 'The Great Decision'. The subject of this
chapter is Pascal's wager on the belief in God. Here are a few excerpts:
>>>IH. The wager is not easy to understand. Logicians have dismissed it.
They have been mistaken. Pascal's pages contain three distinct arguments.
Each is valid. Each has the form of a decision-theoretic argument of a
sort properly classified and characterized only in this century. Although
Pascal did not state his underlying principles, it seems clear that he did
know what he was doing. The reasoning was novel, but the popularity of
[Pascal's writing] made it a familiar fact that games of chance could serve
as models for other problems about form of decision under uncertainty.
>>>IH. The argument is directed at the sort of person who, not being
convinced of the proofs of religion, and still less by the arguments of
atheists, remains suspended between a state of faith and one of unbelief.
This assertion is extremely important. A decision problem requires an
exhaustive partition of possibilities. It is taken as a premise of the
argument that either there is no God, or else there is a God whose
characteristics are correctly reported by the Church. The God of the
Muslims, for example, is not admitted as a possibility. It is a corollary
that Pascal's argument is good for any decision problem with the same
formal structure. "An Imam could reason just as well this way', as Diderot
remarked. Pascal's partition of states of affairs may be out of place
today, but this is one thought from a book of thoughts. The other thoughts
contain other reasons bearing on this partition. There are also other
arguments directed directed at other special sorts of person, for example,
the arguments directed at Orthodox Jews whose partition, of course, is not
at all like that of the Parisian man about town.
>>>IH. '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 or disproof 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 he is not
willing to play the game. But in our case, by the mere fact of living, we
are engaged in play. We either believe in God, or we do not.
>>>IH. The three arguments are all valid. None are convincing. All rely
on dubious premises. The arguments are worthless as apologetics today, for
no present agnostic who understood the arguments would ever be moved to
accept all the premises. The most dubious of the premises is the
partition, with its concomitant assigment of utilities. God is (and belief
in Him brings salvation), or, God is not (and non-believers who have heard
the Word are damned). It is no criticism of Pascal that he assumes this
partition: he is directing his argument at his fellows who accept it.
>>>IH. Throughout this analysis I have freely used the word 'probability'.
It is not used by Pascal. All the technical terminology is that of the
theory of gaming, of chances and hazards and coins. No one today would
want to say that a chance set-up like a coin or loaded die determined
whether or not God should exist. We would express the argument in terms of
some idea of subjective or personal probability, saying, for example, that
no matter how slender our degree of belief in the existence of God, it is
not 0. Pascal does not speak of a quantitative measure of degree of
belief. He is saying that we are in the same epistemological position as
someone who is gambling about a coin whose aleatory properties are unknown.
>>>IH. Whatever its value as an apologetic, Pascal's logic remained.
Games were seen to serve as models of all sorts of decision under
uncertainty. Voltaire was too late when in 1728 he said of [Pascal's
wager], "This article seems a little indecent and peurile: the idea of a
game, and of loss and gain, does not befit the gravity of the subject."
My spin (Clark Carrington):
>>>CDC. Hacking refers to Pascal's form of reasoning as probability logic.
It is also known as probabilistic logic or logical probability. It is the
only form of reasoning that anyone has come up with which suitable for
addressing model uncertainty. Or to put it another way, model uncertainty
is another synonym for probability logic. And by 'mode'l or 'theory', we
may mean anything. Reality is just a theory. There are alternative models
of God. If we are going to construct a risk assessment that will settle a
dispute or provide a convincing argument for a particular course of action,
the partition of alternative theories must include all those entertained by
those party to the argument. All models are wrong, but some are more
useful than others.
>>>CDC. There is, of course, the matter of assigning probabilities.
Pascal adopted the principle of indifference, suggesting that the
probability of God vs. non-God are equal: Acting on the belief in God may
be recommended on the greater utility (according to Pascal) of believing in
God - Pascal suggests that if there is not God then it does no harm to
believe that there is. Hacking thinks the principle of indifference is
'monstrous', but offers no alternative. If one is going to make use of
probability logic, assigning probabilties is, rather directly, going to
become the issue.
>>>CDC. Other than adopting the principle of indifference, Pascal didn't
consider quantitative measures. Hacking acknowledged the possibility, but
didn't discuss it. Hacking dismissed Bacon and Cohen's conceptions of
induction as having "little use for probabilistic reasoning". Perhaps he
was too quick. The theory weights ("weight-of-the-evidence") spoken of by
Bacon, Whewell, Keynes, and Cohen are perhaps what are needed if we are to
say one theory is more probable than another. But the probability doesn't
just depend on the weight. It also depends on the other theories to which
it is compared.
>>>CDC. In current practice, the use of probability logic is intimately
tied to the employment of expert opinion. This need not be. If we can
agree on how to calculate weights, then anyone's theory should be fair
game. The earth is mounted on the back of a giant tortoise? Low weight,
but sure, why not? There is a practical matter. There is a limit to how
many theories one can consider. Each additional theory raises the cost of
the analysis. But with a computer, the cost isn't that much.
