Sometimes in response to Kuhn and sometimes for independent reasons, philosophers tried to analyze particular instances of complex scientific discoveries, showing how the scientists involved appear to have followed identifiable methods and strategies. The most ambitious response to the empiricist orthodoxy tried to do exactly what was abandoned as hopeless—to wit, specify formal procedures for producing hypotheses in response to an available body of evidence. So, for example, the American philosopher Clark Glymour and his associates wrote computer programs to generate hypotheses in response to statistical evidence, hypotheses that often introduced new variables that did not themselves figure in the data. These programs were applied in various traditionally difficult areas of natural and social scientific research. Perhaps, then, logical empiricism was premature in writing off the context of discovery as beyond the range of philosophical analysis.
In contrast, logical empiricists worked vigorously on the problem of understanding scientific justification. Inspired by the thought that Frege, Russell, and Hilbert had given a completely precise specification of the conditions under which premises deductively imply a conclusion, philosophers of science hoped to offer a “logic of confirmation” that would identify, with equal precision, the conditions under which a body of evidence supported a scientific hypothesis. They recognized, of course, that a series of experimental reports on the expansion of metals under heat would not deductively imply the general conclusion that all metals expand when heated—for even if all the reports were correct, it would still be possible that the very next metal to be examined failed to expand under heat. Nonetheless, it seemed that a sufficiently large and sufficiently varied collection of reports would provide some support, even strong support, for the generalization. The philosophical task was to make precise this intuitive judgment about support.
During the 1940s, two prominent logical empiricists, Rudolf Carnap (1891–1970) and Carl Hempel (1905–97), made influential attempts to solve this problem. Carnap offered a valuable distinction between various versions of the question. The “qualitative” problem of confirmation seeks to specify the conditions under which a body of evidence E supports, to some degree, a hypothesis H. The “comparative” problem seeks to determine when one body of evidence E supports a hypothesis H more than a body of evidence E* supports a hypothesis H* (here E and E* might be the same, or H and H* might be the same). Finally, the “quantitative” problem seeks a function that assigns a numerical measure of the degree to which E supports H. The comparative problem attracted little attention, but Hempel attacked the qualitative problem while Carnap concentrated on the quantitative problem.
Rudolf Carnap, 1960.Courtesy of the University of California, Los Angeles
It would be natural to assume that the qualitative problem is the easier of the two, and even that it is quite straightforward. Many scientists (and philosophers) were attracted to the idea of hypothetico-deductivism, or the hypothetico-deductive method: scientific hypotheses are confirmed by deducing from them predictions about empirically determinable phenomena, and, when the predictions hold good, support accrues to the hypotheses from which those predictions derive. Hempel’s explorations revealed why so simple a view could not be maintained. An apparently innocuous point about support seems to be that, if E confirms H, then E confirms any statement that can be deduced from H. Suppose, then, that H deductively implies E, and E has been ascertained by observation or experiment. If H is now conjoined with any arbitrary statement, the resulting conjunction will also deductively imply E. Hypothetico-deductivism says that this conjunction is confirmed by the evidence. By the innocuous point, E confirms any deductive consequence of the conjunction. One such deductive consequence is the arbitrary statement. So one reaches the conclusion that E, which might be anything whatsoever, confirms any arbitrary statement.
To see how bad this is, consider one of the great predictive theories—for example, Newton’s account of the motions of the heavenly bodies. Hypothetico-deductivism looks promising in cases like this, precisely because Newton’s theory seems to yield many predictions that can be checked and found to be correct. But if one tacks on to Newtonian theory any doctrine one pleases—perhaps the claim that global warming is the result of the activities of elves at the North Pole—then the expanded theory will equally yield the old predictions. On the account of confirmation just offered, the predictions confirm the expanded theory and any statement that follows deductively from it, including the elfin warming theory.
Hempel’s work showed that this was only the start of the complexities of the problem of qualitative confirmation, and, although he and later philosophers made headway in addressing the difficulties, it seemed to many confirmation theorists that the quantitative problem was more tractable. Carnap’s own attempts to tackle that problem, carried out in the 1940s and ’50s, aimed to emulate the achievements of deductive logic. Carnap considered artificial systems whose expressive power falls dramatically short of the languages actually used in the practice of the sciences, and he hoped to define for any pair of statements in his restricted languages a function that would measure the degree to which the second supports the first. His painstaking research made it apparent that there were infinitely many functions (indeed, continuum many—a “larger” infinity corresponding to the size of the set of real numbers) satisfying the criteria he considered admissible. Despite the failure of the official project, however, he argued in detail for a connection between confirmation and probability, showing that, given certain apparently reasonable assumptions, the degree-of-confirmation function must satisfy the axioms of the probability calculus. Bayesian confirmation
That conclusion was extended in the most prominent contemporary approach to issues of confirmation, so-called Bayesianism, named for the English clergyman and mathematician Thomas Bayes (1702–61). The guiding thought of Bayesianism is that acquiring evidence modifies the probability rationally assigned to a hypothesis.
For a simple version of the thought, a hackneyed example will suffice. If one is asked what probability should be assigned to drawing the king of hearts from a standard deck of 52 cards, one would almost certainly answer 1/52. Suppose now that one obtains information to the effect that a face card (ace, king, queen, or jack) will be drawn; now the probability shifts from 1/52 to 1/16. If one learns that the card will be red, the probability increases to 1/8. Adding the information that the card is neither an ace nor a queen makes the probability 1/4. As the evidence comes in, one forms a probability that is conditional on the information one now has, and in this case the evidence drives the probability upward. (This need not have been the case: if one had learned that the card drawn was a jack, the probability of drawing the king of hearts would have plummeted to 0.)
Bayes is renowned for a theorem that explains an important relationship between conditional probabilities. If, at a particular stage in an inquiry, a scientist assigns a probability to the hypothesis H, Pr(H)—call this the prior probability of H—and assigns probabilities to the evidential reports conditionally on the truth of H, PrH(E), and conditionally on the falsehood of H, Pr−H(E), Bayes’s theorem gives a value for the probability of the hypothesis H conditionally on the evidence E by the formula PrE(H) = Pr(H)PrH(E)/[Pr(H)PrH(E) + Pr(−H)Pr−H(E)] .