Thinking and Deciding

HYPOTHESES IN SCIENCE 167
how the planetary motions could be explained more simply by assuming that the
earth and the planets revolved around the sun, no crucial experiment or potential observation
was known that could be used to demonstrate which of the two theories
was superior. The Copernican theory won out, according to Lakatos, because this
theory was more progressive as a program for research; one could ask and answer
more questions within it than within the Ptolemaic theory. When Kepler assumed the
Copernican theory was true, for example, he was able to show that planets moved in
elliptical orbits, and Newton, in turn, showed how these ellipses could be explained
by the inverse-square law of gravity.
For Lakatos, then, a theory is useful if it generates valuable research. Lakatos’s
view implies that theories cannot be compared directly with one another, to determine
their closeness to truth or their probability. We can compare theories only in
hindsight, after we have seen what research they generated. Again, this view is not
useful advice to practicing scientists, unless they can foresee the future.
We could, however, use probability theory (as we do with other hypotheses) to
evaluate the effect of experimental results on hypotheses. By asking about the effect
of various results on our hypotheses before we do an experiment, we can determine
the value of the experiment before we do it. To find how the probability of a scientific
hypothesis changes when we obtain an experimental result, we would assign a prior
probability to each hypothesis, p(Hi) (where i takes a different value for each hypothesis).
Observations and experimental results would constitute the data, the Dj s
(with j taking different values for different results). After assigning a likelihood to
each datum given each hypothesis, p(Dj |Hi), we could then use Bayes’s theorem to
calculate the posterior probability of the hypothesis, p(Hi|Dj ). The data cannot, in
science, be expected to raise the probability of a hypothesis precisely to one or lower
it precisely to zero. Other things being equal, however, a good scientific experiment
would be one with potential outcomes that have a high probability, given some
hypothesis, and a low probability given others. If we obtain such a result, our probabilities
for the various hypotheses would change greatly, and we would, therefore,
have learned a lot.
Probability theory would constitute part of the normative theory of science. The
other part of the normative theory would be the theory of decision making, which
is explored in Part III. 4 The theory is normative but not necessarily prescriptive.
Although scientists can, and sometimes do, calculate probabilities as specified by
this theory, they can probably do nearly as well by following certain prescriptive
rules of thumb, which we shall discuss shortly.
This normative theory can help us to understand why the prescriptive advice of
Popper and Platt, on how to formulate and test scientific hypotheses, is good advice
when it is possible to follow it. For Popper, a good result (D, datum) is one that has
a probability of 1, given the hypothesis being tested (H), and a very low probability
4 This view — that the normative theory of science consists of probability theory and decision theory
— is advocated by Horwich, 1982, especially pp. 1–15, 51–63, and 100–136. I expressed my own support
for it earlier in Rationality and Intelligence, 1985a, ch. 4. Its first clear expression seems to be in Savage
(1954, ch. 6).