So I am trying to understand the critical rationalist arguments against the inductive and subjective interpretations of probability. I am not all that familiar with the matter, and so I have likely made some elementary mistakes — feedback is appreciated. Although the views expressed here are based on the arguments of Popper and Miller, I have only been exposed to their relevant work secondhand. I thank Kenneth Hopf for originally explaining the basic argument to me, though he shouldn’t be blamed for what are probably my errors.
A ROUGH HISTORY
Repeated failure to justify a principle of induction once caused much frustration among philosophers of science. If no finite set of evidence entailed the truth of any scientific hypothesis, then one could not be justified in believing that any such hypotheses were true. Science had a crisis of identity.
The enlightenment had elevated science to rival traditional political and religious authorities, but to fulfil this role science had to offer its own genus of justified true belief. Sense experience and the experimental method, buttressed with logic and mathematics, emerged as new authorities to challenge custom, might, and religious superstition. But the problem of induction threatened to undermine this edifice; it cast doubt upon the fundamental principles of the scientific method and challenged the integrity of practicing scientists.
Critics accused scientists of hypocrisy: their declarations required a leap of faith no less than religious dogma. Going beyond the evidence, the truth of neither could be guaranteed by rational authorities. But while the devout believer could then appeal to religious authorities for answers, the scientist’s decision to prefer one hypothesis over another was arbitrary. Further investigation revealed that even our most mundane beliefs could not be justified, e.g. other minds exist, the sun will rise tomorrow, and the future will resemble the past.
The problem of induction had been identified, but all attempts to discover a logical solution failed. Positing a special inductive rule of inference or principle of natural uniformity shifted the problem to justifying said principles, but all attempts to do that were demonstrably invalid or question begging.
The emerging field of mathematical probability suggested an alternative solution to the problem of induction: perhaps one could not be justified in believing that any hypothesis is true, but one could be justified in believing that some hypotheses are more probable than others. The theory of probability seemed to offer a way to justify preferences between competing hypotheses without the need to justify the belief that any hypothesis was actually true. Scientists could now answer charges of hypocrisy while remaining loyal to their rational authorities; the only concession was that establishing any hypothesis with absolute certainty would be impossible.
To make the transition from believing in the truth or falsity of hypotheses to believing in their probability, philosophers needed to figure out how to assign and interpret the probability of a proposition. In logic, one would normally assign truth-values to propositional variables, but truth and falsity were now just limiting cases represented by 1 and 0, and the full range of numbers between them could now be assigned.
Two rival interpretations of probability emerged to make sense of statements such as “the probability of P is .75.” The frequentists wanted to interpret probabilities similarly to the idea of truth: probabilities were descriptions of objective facts or, specifically, classes of events. The subjectivists wished to interpret probabilities as the subjective degree of confidence that one has (or should have) given the evidence. With the ascension of Bayes’s theorem and difficulties with the frequentist interpretation, the subjectivists eventually came to dominate.
During all this, the problem of induction was not abandoned; the inductive nature of science could be preserved if induction was probabilistic. While no finite set of evidence entails the truth of any scientific hypothesis, the future events predicted by a hypothesis may be more probable given supporting evidence of past events. The problem of induction had apparently been solved, albeit in a rather more pragmatic and qualified sense than originally desired.
The critical rationalist critique of induction has many facets; not all of the objections will be covered here. However, before moving onto the central argument of this post, one point is worth emphasising. Parted from the presuppositions which led to the problem of justifying beliefs with sense experience, induction just doesn’t do anything. For critical rationalists, induction has no useful role in the critical evaluation of rival conjectures; it is reduced to merely another invalid inference: a logical fallacy. The rejection of induction runs far deeper than just the observation that induction is invalid or question begging, and the following arguments should be understood in this context.
[EDIT: I have attempted to restate the following argument using a different approach here.][EDIT: Well, there was another argument. I accidently deleted it; I’ll rewrite it again soon.]
Let A and B be propositional variables. Suppose the probability of A is less than probability of A given B. In such a case, B is said to support or partially confirm A by probabilistic inference. For example, suppose
p(A) = .6
p(B) = .4
p(B|A) = .5
From Bayes’s theorem, one can derive
p(A|B) = .75
By subtracting p(A) from p(A|B), one can now calculate precisely how much A is supported by B:
p(A|B) – p(A) = .15
The result seems straightforward: A is not a logical consequence of B, but B increases the probability of A by 15 percent. B appears to be amplified by probabilistic inference. While the induction falls short of implying the truth of A, it nonetheless increases the probability of A being true. Therefore, some say, one should invest more confidence in the truth of A given B than one would have before accepting B.
The counterargument to this view depends on a simple logical equivalence:
A =||= (A v B) & (A v ~B)
It follows that both have the same probability. That is, anything that changes the probability of one side of the logical equivalence must equally change the probability of the other side. The point of decomposing the equivalence on the right is to reveal how the probability of A relates to the probabilities of A v B and A v ~B. In fact, the degree to which B supports A can be calculated by adding the degree to which B supports A v B and A v ~B, respectively:
p(A v B|B) – p(A v B) + p(A v ~B|B) – p(A v ~B) = .15
On the inductive view of probabilistic inference, B is amplified to imply that A is more probably true. This would would mean the logical consequences of A which are not also logical consequences of B should be more probable given B. However, since the probabilities of both A and B are greater than 0 and less than 1, it follows that
p(A v B|B) > p(A v B)
p(A v ~B|B) < p(A v ~B)
In other words, while B increases the probability of A v B, it actually reduces the probability of A v ~B. From this we get,
if p(A|B) > p(A), then |p(A v B|B) – p(A v B)| > |p(A v ~B|B) – p(A v ~B)|
That is, to the extent that B increases the probability of A, it does so by increasing the probability of A v B more than it decreases the probability of A v ~B. However, since A v B is a logical consequence of B to begin with, the increase in probability is a purely deductive inference.
The inductive view of probabilistic inference rests on the fallacy of decomposition, i.e. assuming that what is true for the whole must be true for its parts. Not only do logical consequences of A which are independent of B not increase in probability, they may actually decrease in probability. This concludes the refutation of inductive probability.
The subjective interpretation of probability might be retained even if probabilistic inference is not inductive. While it may be true that probabilistic inference cannot amplify B, it can still be used to select among alternative propositions. Probabilities can help us keep score and choose preferences without any presumption of induction. More importantly, probabilities can still be interpreted as the subjective degree of confidence that one has (or should have) in a proposition given some other proposition.
The critique of the subjective interpretation of probability is an extension of the previous argument against inductive probability. Consider that
p(A v ~B|B) < p(A v ~B), therefore p(~(A v ~B)|B) > p(~(A v ~B))
Since the probabilities of A v ~B and ~(A v ~B) must add to 1, if B decreases the probability of the A v ~B, then it must increase the probability of ~(A v ~B). Therefore, in the subjective interpretation, given B, one should have increased confidence in both A and ~(A v ~B), but that is a flat contradiction with a probability of 0.
While the probability of A v ~B remains greater than ~(A v ~B), what this argument demonstrates is that probabilities do not behave like subjective degrees confidence or belief. When one says that he is more confident in some proposition, he does not mean to suggest that he is also more confident in the contradiction of that proposition, but that is exactly where one may end up with in the subjective interpretation. The concept of subjective degrees of confidence or belief just does not have the same formal structure as the concept of probability, because the former is truly “inductive” in a way that the latter is not.