The Problem of Induction

In critical rationalism, popper on 01/07/2011 at 1:45 pm

One of the aims of science, perhaps its most fundamental aim, is knowledge — not of past events — but of future events. Scientists want to ‘read the book of nature’, to borrow a phrase from Bacon. Think of the laws of nature as being general truths, or as they’re known in predicate logic, universal statements (“for all x, y”). So the question, to rephrase David Byrne is, how do I get there? How can scientists grasp hold of the laws of nature?

The popular answer is by ‘inductive inference’, called by Aristotle “the passage from individuals to universals.” [1] Inductive inference usually takes the following form: “This bacon is crispy. That bacon is crispy. … Therefore, all bacon is crispy.”

Scientists may infer that a universal statement holds when they have observed a sufficient number of instances of the law. Corroborating scientific theories through repeated testing is the key. And then there was Sextu-

And then there was Hume. [2]

Hume’s problem of induction is quite simple: you can’t get there from here. His argument is the following:

(1) General laws are formulated by inductive inference.

(2) Inductive inference is unjustified.

(3) If an inference is unjustified, it cannot count as knowledge. Therefore, we cannot have knowledge of general laws.

A common characteristic of all deductive logics is that the conclusion of a valid inference is implicit in the premises; or, the conclusion cannot have a greater logical content than the conjunction of the premises. Since there is no content-increasing inferences in logic, the problem of logical inference from existential statements to universal statements is unsolvable.

The proof runs as follows. Let the statements “This is a piece of bacon and it is crispy” (Ba), “This is a second piece of bacon and it is crispy” (Bb) … up to Br be true. Assume that s has not yet been tested for the property of crispiness.

If Ba⋀Bc⋀Bd⋀ … ⋀Br ⇒ Bs were analytic, then Bs could not be false, should Ba, Bc, Bd, … Br all be true. This logical inference would provide all the inductive inference scientists need. But should Ba, Bc, Bd, … Br be all true, Bs may still be false, so the if-then statement would also be false. Hence it is not an analytic truth, for counterexamples are conceivable. There is no logical inference to be made from past experience to future experiences.

As Karl Popper said, “…any conclusion drawn in this way may always turn out to be false: no matter how many instances of white swans we have observed, this does not justify the conclusion that all swans are white.” [3] No finite number of existential statements — such as “This is a piece of bacon and it is crispy” — implies a universal statement such as “All bacon is crispy”.

Popper again:

I formulate Hume’s logical problem of induction as follows:

(L1) Can the claim that an explanatory universal theory is true be justified by ’empirical reasons’; that is, by assuming the truth of certain test statements or observation statements (which, it may be said, are ‘based on experience’)?

My answer to the problem is the same as Hume’s: No, it cannot; no number of true statements would justify the claim that an explanatory universal theory is true.

There is a second logical problem, L2, which is a generalization of L1. It is obtained from L1 merely by replacing the words ‘is true’ by the words ‘is true or that it is false’:

(L2) Can the claim that an explanatory universal theory is true or that it is false be justified by ’empirical reasons’; that is, by assuming the truth of certain test statements or observation statements (which, it may be said, are ‘based on experience’)?

To this problem, my answer is positive: Yes, the assumption of the truth of test statements sometimes allows us to justify the claim that an explanatory universal theory is false. [3]

(A false test statement, thanks to the modus tollens rule of inference, reveals the falsity of the universal theory. However, his use of the word “justified” needs to be rejected.)

The naive solution to the problem of induction is to posit a uniformity of nature. That is, assert a priori that the future will be like the past. This looks like a perfectly valid solution. This argument employs the very inference we’re trying to justify: on all previous occasions the future has resembled the past. Therefore, the future will always resemble the past. Statements about the present and past are predictive statements about the future if nature is uniform in respect to this scientific theory, but scientists do not know whether or not nature is uniform in respect to this scientific theory.

This solution moves the problem of induction one step back. Since the principle of induction is not synthetic, and since there remains no method of producing predictions from past evidence that will guarantee their truth, inductive use or success justifies induction. This leads directly to an infinite regress through further and further principles of induction.

A second solution is to retreat to assigning a probability about future events. Inductive inferences don’t always work, but they’re very likely to lead to true conclusions. “All x are y” should be replaced with “Probably, all x are y.” This leads to the tens of thousands of modern forms of Bayesianism.

First, it must be understood that universal statements refer to an open class of instances: it covers an infinite number of possible states of affairs. No matter the finite number of inductive inferences collected, such as the statement “all bacon that up to this time have been observed are crispy,” the universal statement will always have a greater logical content.

Even if it was valid to retreat to probability, reasoning from a finite number of observed cases would lead to a zero probability of universal statements. Increasing the content of a theory decreases its probability: due to the infinite number of test cases, any finite number of existential statements cannot increase the probability of a theory with so much content.

If we were to weaken inductive inference to the point where induction gives only “Probably, the next x will be y,” Hume’s argument still stands. For instance, if Popper’s formulation of the problem of induction were instead “Can the claim that a [predictive statement] is true be justified by ’empirical reasons’; that is, by assuming the truth of certain test statements or observation statements (which, it may be said, are ‘based on experience’)?”, again the answer is ‘No’; Hume’s problem of induction is not restricted to conclusions about the truth of predictions, nor is it restricted to universal statements. As demonstrated previously, the real power of Hume’s argument is that it extends to any claim that transcends present experience.

A third solution is to give up on inductive inference while still agreeing with Hume’s skeptical conclusion: we cannot have knowledge of general laws the same way as in a priori disciplines, thus we can have no knowledge of any general laws. The problem of induction is in fact a special case of the general problem of justification. Any attempt to give a sufficient reason to justify a statement or value judgement leads us for logical reasons into a trilemma that Sextus Empiricus saw thousands of years ago.

If we dogmatically break off the justification, all foundationalist systems as put forward by Kuhn, Wittgenstein and Carnap leads to the hermeneutic tradition of Habermas, Derrida, Foucault, &c.

A.J. Ayer produced a fourth solution. He dissolves the problem by demonstrating that it is a pseudo-problem, not a genuine problem. His reasoning is as follows.

There are only two possible types of meaningful justification of induction, and both are impossible to acquire. The first means that induction must be true by definition. But this is a mistake since no factual conclusions can be derived from definitions. Statements about definitions only tell us about the use of the word.

The second type would be to verify its truth. But this would be to use induction to justify induction. This is not acceptable, since it assumes that induction is reliable when discussing the reliability of induction. Ayer concludes that no meaningful solution is possible, and thus, the problem of induction is then a pseudo-problem. Induction is then valid not through deductive logic, but valid through inductive logic.

‘Reality’ is only accessible to us in terms of how we understand and interpret it. If there is no foundation to construct a system of knowledge,then the best we can do is the therapeutic task of ‘deconstructing’ our purported systems of knowledge by showing the arbitrary interpretations upon they are based.


(1) Aristotle Topica, I, XII, 105 a 13 [My original translation in my notes was “the progress from particulars to universals”; I felt this online edition was rather poor, but it is the only online available copy I could find.]


[W]hen they propose to establish the universal from the particulars by means of induction, they will effect this by a review either of all or of some of the particular instances. But if they review some, the induction will be insecure, since some of the particulars omitted in the induction may contravene the universal; while if they are to review all, they will be toiling at the impossible, since the particulars and infinite and indefinite. Thus on both grounds, as I think, the consequence of that induction is invalidated. (Sextus Empiricus, Outlines of Pyrrhonism, II, xv, 204.)

(3) Karl Popper, Logic of Scientific Discovery, p. 4.

(4) ibid., Objective Knowledge: An Evolutionary Approach, p. 7.

  1. Sometimes the word “justified” is just used to mean “is logically entailed by.” I suspect that is what Popper meant. Unfortunately, the stronger meaning of “justified” often gets used interchangeably, and so I prefer to avoid using the term altogether.

    • I feel the same way: words carry a great deal of mental baggage along with them. For example, if someone wants to call Spinoza’s god ‘God,’ fine by me, but it just ends up confusing others. ‘Falsification’? I try not to use it often much the same reason. Heck, even the word ‘reason’ is too loaded.

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