Hume’s Problem of Induction is Probably Not All That Problematic
(Philosophy of Science)
Hume’s problem of induction is widely considered to be unsolvable, despite many attempts by philosophers to solve it (Henderson, 2020). In the following essay, I will describe Hume’s (1739; 1748) problem of induction and how it demonstrates that using previous experience is insufficient as a means of providing justification for inductive inferences. And that, because science is based on inductive inferences, the problem of induction would seemingly present scientific knowledge as belief rather than knowledge (Salmon, 1966). Despite this, I will argue that induction is still the most reasonable method we have for accumulating knowledge (Salmon, 1974), and that it is pragmatic for us to do so (Reichenbach, 1938; 1949). I will also argue that probability theory and statistical techniques are implemented by scientists to mitigate the problem of induction and provide some measure of confidence in inductive inferences (Dale, 1999). I will also argue that Popper’s (1935) theory of falsification provided scientists with a tool to partially avoid induction through the means of deductively eliminating hypotheses. And that Schurz’s (2019) meta-induction technique has given us a promising new way to accept a hypothesis that is based on its optimality. Thus, even though the problem of induction is not solved, and our scientific knowledge based on induction is always fallible, it is still prudent for us to engage in both induction and the practice of science.
David Hume’s (1739; 1748) problem of induction was originally laid out in A Treatise of Human Nature and again in An Enquiry Concerning Human Understanding. Hume talks about there being two forms of arguments, demonstrative and probable. There have been many interpretations of what Hume meant by these distinctions, along with many interpretations of what Hume meant by his overall argument (Henderson, 2020). So, for this purpose of this essay, I will use Wesley Salmon’s (1966) interpretation, where demonstrative arguments have been interpreted as valid deductive reasoning, and probable arguments have been interpreted as inductive reasoning (an overview of deductive and inductive reasoning can be seen in Figure 1). However, there are also different interpretations of what inductive reasoning is, where it can be interpreted in a wide or a narrow sense (Schurz, 2019). The wide sense encompasses all arguments that are not deductive, including abduction and inference to the best explanation. Whereas the narrow sense refers primarily to inductive predictions (i.e., observations transferred to future instances) and inductive generalization (i.e., observations transferred to all possible instances). This essay will use the term inductive reasoning in the narrow (or sometimes called Humean) sense of the term.
“…because science is based on inductive reasoning, Hume’s problem of induction is a significant problem, since it seemingly renders science as belief rather than knowledge.”
Hume’s (1739; 1748) problem of induction is concerned with how it is we can justify inductive reasoning. Where deductive reasoning guarantees the truth of the conclusion (given that the premises are true and the argument is valid), inductive reasoning extrapolates from what has been observed to make predictions or generalizations about the unobserved (Salmon, 1966). But, in doing so, it is assuming that the natural world around us is uniform and will behave similarly in the future to how it has in the past. Unfortunately, attempts to justify induction and this uniformity assumption leads to arguments that are either invalid or are circular in the way that the conclusion is presumed by the premise (see Figure 2 for examples). This results in both the assumption of the uniformity of nature and inductive reasoning as being rationally unjustified. And because science is based on inductive reasoning, Hume’s problem of induction is a significant problem, since it seemingly renders science as belief rather than knowledge. Bertrand Russell famously wrote that if there is no solution to Hume’s problem, then “there is no intellectual difference between sanity and insanity” (Russell, 1946). Because of this, it is considered one of the greatest problems in philosophy, and many philosophers have attempted to create solutions to the problem (Henderson, 2020). I will now discuss four responses to Hume’s problem of induction: Reichenbach’s (1938; 1949) best alternative response, the probability response, Popper’s (1935) falsification response, and Schurz’s (2019) meta-induction response.
Best Alternative Response
“This argument presents inductive reasoning and assuming the uniformity of nature as the most reasonable option we have out of the possible alternatives.”
Hans Reichenbach (1938; 1949) believed Hume’s problem of induction to be unsolvable, yet he provided a weak form of justification for induction by arguing that we have pragmatic grounds for engaging in inductive reasoning. Reichenbach used the example of a fisherman going to fish in an unexplored part of the sea where it is unknown whether he will catch fish or not. According to Hume’s problem of induction, to cast his net there would be unjustified since it would be based on inductive reasoning. Yet, Reichenbach argues that the fisherman should cast his net because “it is preferable to try even in uncertainty than not to try and be certain of getting nothing” (Reichenbach, 1938). More recently, Wesley Salmon (1974) has created a reconstruction of Reichenbach’s argument (summarized in the utility matrix shown in Figure 3). It presents two possible world states (i.e., uniform or nonuniform) and two possible methods of predicting the future (i.e., inductive reasoning or guessing). If it turns out that nature is nonuniform, then regardless of whether predictions are based on simply guessing or inductive reasoning, they will likely fail. However, if it does turn out that nature is indeed uniform, then predictions based on guessing could bring success or failure, but predictions based on inductive reasoning are more likely to be successful. This argument presents inductive reasoning and assuming the uniformity of nature as the most reasonable option we have out of the possible alternatives.
However, several flaws have been identified in Reichenbach’s argument (Henderson, 2020). The most notable of which is Gerhard Schurz (2008; 2019) pointing out that people believe in many other non-inductive methods beyond simply guessing, such as intuition and various supernatural abilities. Schurz argues that it is possible to conceive of other possible nonuniform worlds where these other methods are more successful than inductive reasoning (this is what the “[?]” in Figure 3 refers to). Another notable flaw identified by Marc Lange (2011) is that the pragmatic justification assumes that there are no costs associated with engaging in an action that results from inductive reasoning, and if there were, it is no longer clear whether the action would be the most reasonable one to take. Indeed, I would argue that scientists are often confronted with many possible hypotheses and courses of action to take, and that despite Reichenbach’s (1938; 1949) argument that inductive reasoning is pragmatic to use, it does not help scientists choose out of those many options available to them. Despite these limitations, I believe that Reichenbach has given us a convincing argument that presents both inductive inferences and scientific practices that are based on them as reasonable and pragmatic to engage in.
Probability Response
“Since inductive inferences are only probable, probability theory would seem like a plausible way of responding to the uncertainty that probable situations face.”
Many great philosophers, including Rudolf Carnap and Hans Reichenbach, have attempted to solve Hume’s problem of induction using probability theory (Ladyman, 2002). Since inductive inferences are only probable, probability theory would seem like a plausible way of responding to the uncertainty that probable situations face. Thomas Bayes was the creator of one of the first and most important means of dealing with the problem of induction using probability, in what may have been a direct response to Hume (Henderson, 2020). Bayes (1764) developed what is now known as Bayes’ theorem and is still used to this day. Pierre-Simon Laplace expanded upon Bayes’ work and created the famous “rule of succession” (Laplace, 1814). Their work combines to become what is now known as the Bayes-Laplace argument (Henderson, 2020). It relies on using Bayes’ theorem to invert the probability distribution to go from the sampling distribution to the population distribution. Although, this method is based on several assumptions and has been subjected to much criticism. A similar approach, called the combinatorial approach, was originally proposed by Donald Williams (1947) and was then later advanced by David Stove (1986). It states that the inference from sampling distribution to the population distribution can be based on statistical syllogism. It is where if a certain percentage of observations (within an item classification) have a specific trait within a sample, you can then infer that the specific trait will match their populations with that certain percent of probability. Both Williams and Stove claimed that they had provided a solution to Hume’s problem of induction, and their work is now known as the Williams-Stove argument (Henderson, 2020). Although, it too is based on assumptions and has also been subjected to much criticism.
Scientists now use statistical methods that are based on an amalgamation of many of these probability developments (Dale, 1999). These methods all make assumptions and face various minor problems (Henderson, 2020), but there are two main problems that all theories of probability face within the context of Hume’s problem of induction. Firstly, they rely on the underlying assumption of the uniformity of nature (Schurz, 2019). Secondly, in cases of inductive generalization, induction is being projected out to an infinity of possible cases, and when divided by no matter how much evidence that is in support of that induction, the probability of them becomes zero (Chalmers, 2013). This second problem means that using probability is primarily only relevant in the context of inductive predictions or where there are clearly defined population parameters. That is, they are less relevant when it comes to inductive generalization, where observations are being generalized to all possible instances. Therefore, scientists simply need to be well-trained in statistics and to recognize the limitations of probability theory, and be made aware that it is not a full solution to Hume’s problem. And the first problem becomes not so problematic if you combine probability theory with Reichenbach’s (1938) argument that we have pragmatic grounds for both the assumption of the uniformity of nature and engaging in inductive reasoning.
“…Thomas Young’s (1804) double slit experiment…provides evidence that at what may be the most fundamental level of reality, nature isn’t either uniform or nonuniform, it is probabilistic.”
It is also possible that Hume’s own argument presenting nature as either uniform or nonuniform is a false dichotomy, for there are many ways in which we don’t expect the future to resemble the past. If we can assume that there is continuity between the levels of reality (e.g., quantum level, atomic level, cellular level, etc.), then Thomas Young’s (1804) double slit experiment (illustrated in Figure 5) — along with the many contemporary modifications of it in quantum physics — provides evidence that at what may be the most fundamental level of reality, nature isn’t either uniform or nonuniform, it is probabilistic. Though, of course, this itself is an inductive claim that is based on the future continuing to resemble the past.
Falsification Response
“Popper proposed his theory of falsification as a means of side-stepping the problem of induction, where he posited that the justification of a hypothesis can be achieved using deduction alone.”
Karl Popper (1935) also believed that Hume’s problem of induction is unsolvable, but he argued that it did not matter because science does not need to use inductive inferences. Popper proposed his theory of falsification as a means of side-stepping the problem of induction, where he posited that the justification of a hypothesis can be achieved using deduction alone. To do this, one must propose a hypothesis with testable predictions, conduct observations, and then compare those predictions with the actual observations. If there is a contradiction between them, then Popper believed the hypothesis to be falsified. If there is no contradiction, then Popper believed the hypothesis to be corroborated. Popper’s (1983; 1979) idea of corroboration is where one is basing future actions and further predictions based on the past success of hypotheses that have yet to be falsified. But Peter Singer (1974) argued that Popper has simply obscured the role that induction plays in science by hiding it with falsification. Indeed, there are several problems and limitations to Popper’s solution, where it cannot be applied to forms of science that use statistical hypotheses (Schurz, 2019). But the biggest problem is that — despite Popper arguing that it does not involve induction — the step of corroboration is a form of inductive inference (Salmon, 1981; Schurz, 2019). This is because the act of favoring one theory over another based on its past success is what Alan Musgrave (2002) refers to as epistemic induction and what Gerhard Schurz (2019) refers to as meta-induction, which I will discuss in the next paragraph. Despite these limitations, Popper’s (1935) theory of falsification has provided scientists with a valuable tool to — at least partially — circumvent Hume’s problem of induction, by providing a deductive method for eliminating hypotheses. The remaining problem is finding a rationally justified means for accepting a hypothesis.
Meta-Induction Response
“…because it is a successful justification at the level of meta-induction, it enables one to choose the most predictively successful method and then use it at the level of object-induction. Thus, giving one a rationally justified means of accepting a form of induction based on its optimality.”
The title of Gerhard Schurz’s (2019) book makes the bold claim that he has solved Hume’s problem of induction. Although, it is clarified in the text that he means that in a weak sense, where he has provided a non-circular justification of the optimality of induction. This is contrasted with a reliability justification of induction, which would be a justification of induction in a strong sense, if it were possible. Optimality is defined as having greater predictive success than other competing methods. He further clarifies that his justification is also only applicable at the level of meta-induction, and not at the level of object-induction. Object-induction is defined as at the level of events that have been observed to occur. Whereas meta-induction is defined as at the level of an aggregation of competing methods of prediction. Schurz states that Reichenbach’s best alternative response was an optimality justification of induction at the level of object-induction, and that is why it failed. Whereas his optimality justification of induction is at the level of meta-induction, and that is why it is successful. Furthermore, because it is a successful justification at the level of meta-induction, it enables one to choose the most predictively successful method and then use it at the level of object-induction. Thus, giving one a rationally justified means of accepting a form of induction based on its optimality.
Schurz’s (2019) approach is a variation of Nicolò Cesa-Bianchi and Gábor Lugosi’s (2006) regret-based learning strategy, which he calls attractivity-weighted meta-induction (wMI). This method predicts a weighted average of the predictions of other non-meta-inductive methods, using their attractivities as weights. The attractivity of a non-meta-inductive method is the difference between the non-meta-inductive method’s success rate and the success rate of wMI. Schurz claims that the wMI method is optimal in the long-run (i.e., in the short run it may suffer) for all possible worlds (i.e., it does not assume the uniformity of nature) and with a finite set of prediction methods. However, Arnold Eckhart (2010) argues that this finite (as opposed to infinite) set of prediction methods is a limitation of Schurz’s method, and it is therefore not a complete solution to Hume’s problem. Schurz (2019) concedes to Eckhart’s point, yet he argues that it is a sufficient justification for all practical purposes. Schurz then explains that his wMI solution can be applied to cognitive science, social epistemology, cultural evolution, and any other epistemic problem where certain conditions are met. These conditions include clearly defined goals and environmental feedback regarding the extent to which the goals have been reached. I would argue that these conditions also present a limitation to Schurz’s solution, since they limit its applicability. Despite this, it is still a compelling method of dealing with Hume’s problem of induction, which scientists could utilize (within the appropriate conditions) to further circumvent it.
In conclusion, despite Hume’s problem of induction not being completely solved, and scientific knowledge always being fallible, philosophers have given scientists a range of responses to deal with the problem. In this essay, I have argued that induction is the most reasonable method we have for accumulating knowledge (Salmon, 1974), and that there are convincing pragmatic reasons for us to engage in inductive reasoning (Reichenbach, 1938; 1949). I have also argued that when this pragmatism is combined with statistical probability, it gives scientists effective tools to deal with the uncertainty that is inherent in the inductive process. I have then argued that Popper’s (1935) theory of falsification gave us a valuable tool for eliminating hypotheses using deductive techniques. And that where falsification failed to provide an adequate justification for why one theory should be chosen over another, Schurz’s (2019) promising new meta-induction technique fills this gap. Even though all these techniques have their various limitations, and none provide a complete solution to Hume’s problem of induction on their own, scientists can and do combine a variety of these approaches. Doing so enables them to tackle the problem from a variety of perspectives which together establish a quite robust means of mitigating it. Given this, I think it would be hard to argue that Hume’s (1739; 1748) problem of induction means that our scientific knowledge is merely belief. In fact, I would argue that science is the single greatest tool we have for establishing any knowledge at all.
References
Bayes, T. (1764). An essay towards solving a problem in the doctrine of chances. Philosophical Transactions of the Royal Society of London, 53, 370–418. https://doi.org/10.1098/rstl.1763.0044
Cesa-Bianchi, N., & Lugosi, G. (2006). Prediction, learning, and games. Cambridge University Press.
Chalmers, A. (2013). What is this thing called science? (4th ed.). University of Queensland Press.
Dale, A. I. (1999). A history of inverse probability: From Thomas Bayes to Karl Pearson (2nd ed.). Springer.
Eckhart, A. (2010). Can the best-alternative justification solve Hume’s problem? On the limits of a promising approach. Philosophy of Science, 77(4), 584–593. https://doi.org/10.1086/656010
Henderson, L. (2020). The problem of induction. In E. N. Zalta, (Ed.) The Stanford Encyclopedia of Philosophy (Spring 2020 Edition). https://plato.stanford.edu/archives/spr2020/entries/induction-problem
Hume, D. (1739). A treatise of human nature. Oxford University Press.
Hume, D. (1748). An enquiry concerning human understanding. Oxford University Press.
Ladyman, J. (2002). Understanding philosophy of science. Routledge.
Lange, M. (2011). Hume and the problem of induction. In D. M. Gabbay, S. Hartmann, J. Woods (Eds.), Handbook of the history of logic (Vol. 10): Inductive Logic (pp. 43–91). North-Holland. https://doi.org/10.1016/B978-0-444-52936-7.50002-1
Laplace, P.-S. (1814). Essai philosophique sur les probabilités (A. I. Dale, Trans.). In G. J. Toomer (Ed., 1995). Pierre-Simon Laplace: Philosophical essay on probabilities (1st ed.). Springer.
Musgrave, A. (2002). Karl Poppers kritischer rationalismus. In J. M. Böhm, H. Holweg, & C. Hoock (Eds.), Karl Poppers kritischer rationalismus heute (pp. 25–42). Mohr Siebeck.
Popper, K. (1935). Logic der forschung. Verlag von Julius Springer.
Popper, K. (1979). Objective knowledge: An evolutionary approach (2nd ed.). Clarendon Press.
Popper, K. (1983). Realism and the aim of science. Hutchinson.
Reichenbach, H. (1938). Experience and prediction: An analysis of the foundations and structure of knowledge. University of California Press.
Reichenbach, H. (1949). The theory of probability: An inquiry into the logical and mathematical foundations of the calculus of probability. University of California Press.
Russell, B. (1946). A history of western philosophy. George Allen and Unwin Ltd.
Salmon, W. C. (1963). On vindicating induction. Philosophy of Science, 30(3), 252–261. https://doi.org/10.1086/287939
Salmon, W. C. (1966). The foundations of scientific inference. University of Pittsburgh Press.
Salmon, W. C. (1974). The pragmatic justification of induction. In R. Swinburne (Ed.), Justification of Induction, (pp. 85–97). Oxford University Press.
Schurz, G. (2008). The meta‐inductivist’s winning strategy in the prediction game: A new approach to Hume’s problem. Philosophy of Science, 75(3), 278–305. https://doi.org/10.1086/592550
Schurz, G. (2019). Hume’s problem solved: The optimality of meta-induction. The MIT Press.
Singer, P. (1974, May 2). Discovering Karl Popper. The New York Review. https://www.nybooks.com/articles/1974/05/02/discovering-karl-popper/
Stove, D. C. (1986). The rationality of induction. Clarendon Press.
Williams, D. C. (1947). The ground of induction. Harvard University Press.
Young, T. (1804). The bakerian lecture: Experiments and calculations relative to physical optics. Philosophical Transactions of the Royal Society of London, 94(1), 1–16. https://doi.org/10.1098/rstl.1804.0001
