Popper on probability

Question

I cannot understand how and why Popper rejects the idea of a theory being "probably true".

If a theory can be -more likely- than another (as he states), it means that a theory is more likely to be true than false than the other; that it has higher probability to be true than false, thus that it is "probably true". I am not questioning the idea that the probability of an event to happen isn't changed by the number of experiment that you do, but the idea for which "there is no way to tell if a theory is probably true or not".

The first time I thought about this I resolved in this way: "Popper tends to criticize, with Bacon, our tendency to demand regularity from nature. Therefore, he might have thought that induction by probability works only because we think that the laws of nature stays the same over time." Fair enough, if it was not for a small problem.

For this same "Baconian" reason, even Popper resolution, which is deduction. If we have no reasons to believe that nature's laws will stay the same, we cannot say that a theory that was before corroborated or refuted will today or tomorrow still stay that way. Maybe one day we will discover that laws have changed and gravity is no more.

Is it a problem of precision? For example, for one that has always seen only white swan, how is it probably true that "[all] swans are white"? 80%? 90%? 99.9999%? And if he sees a new white swan, does that number get higher? And if he sees 1 black swan, (supposing he saw 99 white swans before), does that become a 99% chance of swan being white? Of course not, because if it was so before seing that black swan it had to be 99/99, which mean 100% sure that all swans are white.

You can solve this problem probably only when it is a linguistic problem, not a real problem pf probability. You may say for example: "it has high probability" or "it is more probable than", but still there is the question: how can it be?

This is my big problem with Popper for now. It's not about the probability of an event to occur, nor about the level of probability should be optimal or desirable for the scientific discovery.

I want to understand why a theory doesn't get more probably true when it is tested and succeed. Thanks for all the answers and sorry for my writing skills...

Answer 1

For Popper, either a theory reflects 'reality' (scare quotes to note how ill-defined that word is), or it does not. There is no sense talking about whether a theory is probably true; either the theory performs as expected (in which case it it true as far as we know), or the theory fails to perform as expected (in which case it is false, and we have to revise or replace it). There's no middle ground.

For example, let's say I choose one person from US at random — call this Person X — and I make a theory that Person X is a black male. We can refer to statistics here and note that nationwide blacks make up about 12.8% of the population and men make up about 49.2%, so the probability that we have randomly selected a black male is around 6.2%. Fine. But the fact is, either Person X is a black male or s'he isn't; that is not probabilistic at all.

Our choice may be probabilistic, but the thing chosen is not probabilistic at all. We need to keep this distinction in mind.

If we see ninety-nine white swans and one black one, we usually want to keep the general-universal that all swans are white — because it's a useful rule — and then puzzle over what's wrong with that one swan who isn't. But that is not how Popper sees it; one black swan should trash the theory outright. From Popper's view using probabilistic assessments compounds errors, allowing us to keep theories that have been demonstrably negated, through a nearly magical hand-waving act of dismissing disconfirming evidence on statistical grounds.

Popper's theories are dated, and have some serious difficulties dealing with exceptions and anomalies. No one in the philosophy of science really uses his theories any more, except for their historical value to the field (though he does have a cult following outside the discipline). I wouldn't sweat this too much.

Answer 2

Popper tends to criticize, with Bacon, our tendency to demand regularity from nature. Therefore, he might have thought that induction by probability works only because we think that the laws of nature stays the same over time.

This is not a correct account of Popper's views. Popper's position is that induction is impossible ("The Logic of Scientific Discovery" [LScD] Section 1):

According to a widely accepted view—to be opposed in this book — the empirical sciences can be characterized by the fact that they use ‘inductive methods’, as they are called.

In Section 6 of LScD he repeats this point:

Now in my view there is no such thing as induction. Thus inference to theories, from singular statements which are ‘verified by experience’ (whatever that may mean), is logically inadmissible.

Popper's criticises inductivism because it claims that ideas should be justified, but inductivism can't be justified (LScD, Section 1):

That inconsistencies may easily arise in connection with the principle of induction should have been clear from the work of Hume; also, that they can be avoided, if at all, only with difficulty. For the principle of induction must be a universal statement in its turn. Thus if we try to regard its truth as known from experience, then the very same problems which occasioned its introduction will arise all over again. To justify it, we should have to employ inductive inferences; and to justify these we should have to assume an inductive principle of a higher order; and so on. Thus the attempt to base the principle of induction on experience breaks down, since it must lead to an infinite regress.

Popper claims that scientific theories are created by guessing and may be eliminated by experimental testing.

Popper doesn't criticise our demand for regularity. He claims it is an indispensable methodological rule for science because scientific theories are created by conjecture and criticism, not by induction (LScD, Section 79):

I shall therefore take up as relevant only one of the points of this argument—the reference to the so-called ‘principle of the uniformity of nature’. This principle, it seems to me, expresses in a very superficial way an important methodological rule, and one which might be derived, with advantage, precisely from a consideration of the non-verifiability of theories.

Let us suppose that the sun will not rise tomorrow (and that we shall nevertheless continue to live, and also to pursue our scientific interests). Should such a thing occur, science would have to try to explain it, i.e. to derive it from laws. Existing theories would presumably require to be drastically revised. But the revised theories would not merely have to account for the new state of affairs: our older experiences would also have to be derivable from them. From the methodological point of view one sees that the principle of the uniformity of nature is here replaced by the postulate of the invariance of natural laws, with respect to both space and time. I think, therefore, that it would be a mistake to assert that natural regularities do not change. (This would be a kind of statement that can neither be argued against nor argued for.) What we should say is, rather, that it is part of our definition of natural laws if we postulate that they are to be invariant with respect to space and time; and also if we postulate that they are to have no exceptions.

You then continue:

For this same "Baconian" reason, even Popper resolution, which is deduction. If we have no reasons to believe that nature's laws will stay the same, we cannot say that a theory that was before corroborated or refuted will today or tomorrow still stay that way. Maybe one day we will discover that laws have changed and gravity is no more.

This makes no sense. A theory is not a collection of individual statements (LScD, Section 25):

For we can utter no scientific statement that does not go far beyond what can be known with certainty ‘on the basis of immediate experience’. (This fact may be referred to as the ‘transcendence inherent in any description’.) Every description uses universal names (or symbols, or ideas); every statement has the character of a theory, of a hypothesis. The statement, ‘Here is a glass of water’ cannot be verified by any observational experience. The reason is that the universals which appear in it cannot be correlated with any specific sense-experience. (An ‘immediate experience’ is only once ‘immediately given’; it is unique.) By the word ‘glass’, for example, we denote physical bodies which exhibit a certain law-like behaviour, and the same holds for the word ‘water’. Universals cannot be reduced to classes of experiences; they cannot be ‘constituted’.

A theory gives rise to a rule governing what events can happen, so if one event takes place that breaks the rule, then the rule is false. For similar reasons the following statement is wrong:

Is it a problem of precision? For example, for one that has always seen only white swan, how is it probably true that "[all] swans are white"? 80%? 90%? 99.9999%? And if he sees a new white swan, does that number get higher? And if he sees 1 black swan, (supposing he saw 99 white swans before), does that become a 99% chance of swan being white? Of course not, because if it was so before seing that black swan it had to be 99/99, which mean 100% sure that all swans are white.

Now, to take care of the issue of anomalies raised in one of the answers to your question. An observation is a guess about what happened in some particular region of space and time and the causes of that event. You can be wrong about what was happening in a particular region, e.g. - you might have magnetic fields in a region that futzes with an attempt to observe atoms without a magnetic field, say. So if you do an observation and it appears to contradict your theory, then that observation may be refuted by coming up with an independently testable guess about what was happening in that case. And if you did a single observation that appeared to refute a theory and you can't reproduce it then you might have been wrong about how the experiment works and you may decide the observation rather than the theory. See Section 8 and Chapter 5 of LScD for more on this issue.

If you are interested in understanding Popper's ideas better, then I recommend "The Fabric of Reality" Chapters 3 and 7 and "The Beginning of Infinity" by David Deutsch. This site also has a list of Popper readings:

https://fallibleideas.com/books#popper

And you can discuss Popper with people who are actually interested in understanding his work:

http://fallibleideas.com/discussion-info

Answer 3

I want to understand why a theory doesn't get more probably true when it is tested and succeed.

I recommend that you read Colin Howson, Hume's problem: Induction and the justification of belief (Clarendon Press, Oxford 2000). Howson focuses on David Hume and induction, but discusses Karl Popper in many places. I have lost the source of this abstract, but here is what the book is about:

Hume was correct when he wrote that no future event can be predicted from simple enumeration of observations, without also assuming the truth of the prediction itself. Attempts to justify induction, such as the "No-Miracles" argument, are unsuccessful because they assume what they set out to prove. Bayes created a system that estimates the likelihood of a future event by comparing the chance of a hypothesis being true against the weight of the evidence in its favor. This method does not justify induction; it does not justify reliance on estimates of future activity based on past events. However, Bayes's system makes such estimates precise.