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Popper (negativly) solved the problem of induction by showing that there is no class of sentences (analytic/synthetic, a priori/a posteriori) in which a principle of induction can be phrased without invoking an infinite regress or admitting synthetic a priori statements. He then presented a different methodology (fallibilism) which helps us do the work the principle of induction should have done in the first place (or something like that).

Quine and others did show, or rather argued, that the distinctions a priori/a posterior and synthetic/analytic are bad, superficial etc...

Given that one buys into the arguments against both distinctions, do the arguments against the principle of induction lose strength (since they seem to rely on them)?


Edit: Poppers solution of the problem of induction

In one of the first chapters of 'The Logic of Scientific Discovery' Popper shows that it is impossible to formulate a principle of induction. His method is to look at each category of statements and show that no principle of induction can be formulated.

Analytic a priori: If the principle of induction would be in this category, there wouldnt be a problem of induction. There is a problem of induction, therefore it can't be in this category.

Analytic a posteriori: For Popper not possible.

Synthetic a priori: Popper believes that this is not possible, and argues that Kant was wrong in supposing there are such statements.

Synthetic a posteriori: Every principle of induction would require an higher-order principle of induction. Popper dislikes infinite regresses, therefore it cannot be such statement.

As he has shown now, in neither category is it possible to formulate a principle of induction, therefore there is no such thing. I was wondering, if one abandones these distinctions, how much is left of Poppers argument.

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First, what is described in the question as Popper's solution, is not his solution, but his formulation of the problem. That is, the problem of induction. In essence, Popper fully accepted David Hume's presentation of the problem of induction. Yet Popper rejected Hume's psychological solution to the problem, and offered a solution of his own, involving the method of refutation.

I approached the problem of induction through Hume. Hume, I felt, was perfectly right in pointing out that induction cannot be logically justified...

I found Hume's refutation of inductive inference clear and conclusive. But I felt completely dissatisfied with his psychological explanation of induction in terms of custom or habit.
(Popper, "Conjectures and Refutations")

Second, Quine's criticism of the analytic / synthetic distinction does not seem to me to touch Popper's formulation. Quine's argument amounts to the result, that a statement that seems synthetically true today may seem analytically true tomorrow, and false next week. But since Popper's formulation covers all the analytic / synthetic combinations, it wouldn't matter for the argument if the principle of induction changes between analytic and synthetic, as Quine allows.

Therefore, to the extent that Popper's formulation was valid before, it is still valid.

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I don't think Popper ever used all that synthetic a priori, posteriori stuff. As a result, you whole objection is based on an interpretation of Popper that doesn't have anything to do with what he really wrote. As a previous answer pointed out your misreading is irrelevant anyway. Popper rejected the idea that arguments of any kind provide justification: they do not show an idea is good or true or whatever. Any argument assumes premises and rules of inference that may be wrong and as a result the conclusion of an argument may also be wrong. The value of the conclusion is that it provides us with guesses to test, not that the conclusion is justified. Whether the argument is labelled this way or that way has nothing to do with the problem. See "Realism and the Aim of Science" Chapter I.

Now, some of the objections in other answers should be addressed:

Why is it rational to act on the basis of a decision informed by the best tested and corroborated theory, to apply it to new situations, to decide to use it as basis for practical action? Corroboration says absolutely nothing about the future performance of a theory. In what sense, therefore, is the decision to act a rational one? The reply of Popper is that since it is the best theory, what could be more rational than acting on such a theory, than holding a “pragmatic belief in the results of science”. This reply is not entirely satisfactory. For under the circumstances, the rational thing to do is not to act at all. If our best theory provides us with no clue as to the prospect of achieving our goals, then it cannot sufficiently motivate us to act. For our best theory to guide us in our actions, its past success should give us some reason for its future success.

This objection assumes that using a theory has to be justified and Popper rejected that demand. Decisions to use a theory or not can be decided by guessing and criticism just as scientific theories are created in the first place. An untested theory may be rejected as a basis for action because it is untested and you don't want the first test to be one where a load of people might die if you're wrong. It is better to test an idea in a safe environment.

Lakatos's objection that it is not possible to conclusively prove a theory wrong completely ignores what Popper actually said about how we should respond to a failed test. He said we can propose any fix we like to a failed test if it is independently testable, not that a theory should be rejected immediately in the face of an objection. See chapter 5 of Logic of Scientific Discovery, especially section 29.

Martin Gardner didn't understand Popper's ideas at all. Popper pointed out that the only tests that matter are those that have some prospect of refuting a theory. And only those that actually refute a theory result in us learning a lot. Whether scientists feel good about failing a test or intend to refute their theories is irrelevant.

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After Lukas editing his question, and by his comments too, I realize it will be difficult to make him see what is accessory and what is the core in the criticism of Popper. I leave to others to do so. It is repeated too by many that this site is not a forum for debate. Therefore now I edit my answer and I will stick only to the main core question:

"Is Poppers Solution to the Problem of Induction still valid?"

The issue here will be the main criticism of the Popper's solution to the problem of induction.

Popper said that induction is not justifiable. That a theory has been corroborated in the past "says nothing whatever about future performance." Popper said that it is possible to avoid assuming that the future will, or probably will, be like the past, and this is why he has claimed to have solved the problem of induction. We do not have to make the assumption, he tells us, if we proceed by formulating conjectures and attempting to falsify them. He says that, as a basis for action, we should prefer "the best-tested theory." Popper never adequately defined the notion of severity for tests, a concept on which much depended, since the more severe the test a theory passed, the better its corroboration. This can only mean the theory that has survived refutation in the past; but why, since Popper says that past corroboration has nothing to do with future performance, is it rational to prefer this? Corroboration is not another term for confirmation since it does not involve any notion of inductive support for a theory. Theories remain as unsupported hypotheses or conjectures forever. Corroboration is not a measure of verisimilitude. Saying that the better corroborated theory is also the one that is closer to truth would be no more than a guess.

Why is it rational to act on the basis of a decision informed by the best tested and corroborated theory, to apply it to new situations, to decide to use it as basis for practical action? Corroboration says absolutely nothing about the future performance of a theory. In what sense, therefore, is the decision to act a rational one? The reply of Popper is that since it is the best theory, what could be more rational than acting on such a theory, than holding a “pragmatic belief in the results of science”. This reply is not entirely satisfactory. For under the circumstances, the rational thing to do is not to act at all. If our best theory provides us with no clue as to the prospect of achieving our goals, then it cannot sufficiently motivate us to act. For our best theory to guide us in our actions, its past success should give us some reason for its future success. In short, Popper must allow for inductivism. The concept of corroboration cannot explain why it is rational for scientists to base their future predictions on the best corroborated theory. To do this, it is inevitable for them to accept some kind of principle of induction. Without the inductive assumption, the fact that a theory was refuted yesterday is quite irrelevant to its truth-status today. Corroboration is also uncertain and can never be quantified by degree of probability.

Wesley Salmon in his paper “Rational Prediction” focuses attention on the practical case in which one must decide on a course of action on the basis of a theory. Salmon asks how one is to choose between alternative theories which make conflicting predictions as a basis on which to act. According to Popper, the action should be based on the most highly corroborated of the competing theories. But this suggests that corroboration has inductive force. For while corroboration relates to a theory’s past success in surviving tests, if it is to serve as a basis for future action then past survival of tests must be of relevance to what will take place in the future. But if corroboration is to be taken into account in determining a future course of action, this amounts to an inductive inference from past success in surviving tests to the likely continuation of such success into the future. Again, it therefore appears that Popper’s falsificationist philosophy of science rests at base on an assumption that is inductive in nature.

Popper’s theory of method suggests that theories are to be rejected the moment they entail a false prediction. Lakatos denies that there are critical tests, in the Popperian sense, in science. Ruthless elimination of theories does not appear to be the norm in actual science. The point here is that the ‘falsification/corroboration’ disjunction offered by Popper is far too logically neat: non-corroboration is not necessarily falsification, and falsification of a high-level scientific theory is never brought about by an isolated observation or set of observations. Such theories are, it is now generally accepted, highly resistant to falsification. They are falsified, if at all, Lakatos argues, not by Popperian critical tests, but by research gradually grinding them to a halt. Popper's distinction falsifiability does not in the end do full justice to the fact that all high-level theories grow and live despite the existence of anomalies which are incompatible with the theories. The existence of such anomalies is not usually taken by the working scientist as an indication that the theory in question is false; on the contrary, he will usually, and necessarily, assume that the auxiliary hypotheses which are associated with the theory can be modified to incorporate, and explain, existing anomalies. Philosophers of science who hold that the actual practice of science is of relevance to the normative methodology of science will be little inclined to adhere to the Popperian picture in the face of historical evidence of anti-falsificationist practice in science.

To Martin Gardner, every falsification of a conjecture is simultaneously a confirmation of an opposite conjecture, and every conforming instance of a conjecture is a falsification of an opposite conjecture. If Popper bet on a certain horse to win a race, and the horse won, you would not expect him to shout, "Great! My horse failed to lose!". For Popper, the more tests for falsification a theory passes, the more it gains in "corroboration”. It's not so much that Popper disagreed with inductivists as that he restated their views in a bizarre and cumbersome terminology.

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    I feel like this is not the answer I am looking for, for the reasons that you seem to misunderstand me and that this may be very interesting, but offtopic to my question. I will edit my question to add further information in what I think is the connection between both distinctions and Poppers solution of the problem of induction – Lukas Aug 5 '13 at 20:53
  • @Lukas I would like to understand how you see that fallibilism enter in your question. And why, since Popper says that past corroboration has nothing to do with future performance, is it rational to prefer this? The more often a conjecture passes efforts to falsify it, Popper maintained, the greater becomes its "corroboration", although corroboration is also uncertain and can never be quantified by degree of probability. If so, how the scientific knowledge grow? How you see that every falsification of a conjecture is simultaneously a confirmation of an opposite conjecture – Annotations Aug 5 '13 at 21:54
  • This is not a question about Poppers fallibilism, this is not a question about growth of knowledge, not about corroboration...I really just see 2 options: Either my edit is wrong and Poppers argument is different from my presentation of it, then I need quotes, because im pretty sure I'm on the right track. Or it is correct, and then my question still is: Given that we buy into abandoning both distinctions, how much of poppers argument against the principle of induction is left? And neither your answer nor your comment are relevant to that question, if I am not mistaken. – Lukas Aug 6 '13 at 10:22
  • After you edited the question I edited the answer. – Annotations Aug 6 '13 at 18:34
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If knowledge is fallible then I see nothing terribly wrong with the principle of induction. Hume is just not that impressive since all he says is that induction does not fit any schema of deduction so it is not valid deductively. But it is still a valid type of reasoning. Popper's stuff becomes unworkable in the real world.

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    Hey there, welcome to philosophy and thanks for the answer! Is there any chance you could share a little bit more about why you find this answer persuasive? (What research could someone do to confirm it?) – Joseph Weissman Nov 30 '16 at 18:32
  • Hey Mark, but knowledge is on literally all sides considered to be factive: You can only know true things, in the real world and everywhere else (?). – Lukas Dec 1 '16 at 9:23
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I agree with Ricardo, picking a "best-tested" (in the Popperian sense, see Deborah Mayo for a definition) theory still involves induction because one assumes that it's best-tested status has not changed. A best-tested theory at time t1 could become a worst-tested theory in the future at time t2. One assumes stability and past evidence to support that assumption.

On a practical note, let's say i am checking some measurement for test-retest stability over a time interval T. I find the thing i am measuring as stable and now can use it in a double blind, randomized controlled trial that is run over a similar time interval. I may still keep a no-treatment control group as part of the design but my choice in using this measure is still purely inductive.

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