Problem of induction

The problem of induction is the philosophical question of whether inductive reasoning is valid. That is, what is the justification for either:

# generalizing about the properties of a class of objects based on some number of observations of particular instances of that class (for example, the inference that "all swans we have seen are white, and therefore all swans are white," before the discovery of black swans) or
# presupposing that a sequence of events in the future will occur as it always has in the past (for example, that the laws of physics will hold as they have always been observed to hold).

The problem calls into question all empirical claims made in everyday life or through the scientific method. Although the problem dates back to the Pyrrhonism of ancient philosophy, David Hume introduced it in the mid-18th century, with the most notable response provided by Karl Popper two centuries later.

Ancient origins

Pyrrhonian skeptic Sextus Empiricus first questioned induction, reasoning that a universal rule could not be established from an incomplete set of particular instances. He wrote [Sextus Empiricus. "Outlines of Pyrrhonism" trans. R.G. Bury (Loeb edn) (London: W. Heinemann, 1933), p. 283.] :

The focus upon the gap between the premises and conclusion present in the above passage appears different from Hume's focus upon the circular reasoning of induction. However, Weintraub claims in The Philosophical Quarterly [Weintraub, R. (1995). What was Hume's Contribution to the Problem of Induction? The Philosophical Quarterly 45(181):460-470] that although Sextus' approach to the problem appears different, Hume's approach was actually an application of another argument raised by Sextus [Sextus Empiricus. "Against the Logicians" trans. R.G. Bury (Loeb edn) (London: W. Heinemann, 1935) p. 179] :Although the criterion argument applies to both deduction and induction, Weintraub believes that Sextus' argument "is precisely the strategy Hume invokes against induction: it cannot be justified, because the purported justification, being inductive, is circular." She concludes that "Hume's most important legacy is the supposition that the justification of induction is not analogous to that of deduction." She ends with a discussion of Hume's implicit sanction of the validity of deduction, which Hume describes as intuitive in a manner analogous to modern foundationalism.

Formulation of the problem

In inductive reasoning, one makes a series of observations and infers a new claim based on them. For instance, from a series of observations that at sea-level (approximately 14psi) samples of water freeze at 0°C (32°F), it seems valid to infer that the next sample of water will do the same, or, in general, at sea-level water freezes at 0°C. That the next sample of water freezes under those conditions merely adds to the series of observations. First, it is not certain, regardless of the number of observations, that water always freezes at 0°C at sea-level. To be certain, it must be known that the law of nature is immutable. Second, the observations themselves do not establish the validity of inductive reasoning, except inductively. In other words, observations that inductive reasoning has worked in the past do not ensure that it will always work. This second problem is the problem of induction.

David Hume

David Hume described the problem in "An Enquiry concerning Human Understanding", §4, based on his epistemological framework. Here, "reason" refers to deductive reasoning and "induction" refers to inductive reasoning.

First, Hume ponders the discovery of causal relations, which form the basis for what he refers to as "matters of fact." He argues that causal relations are found not by reason, but by induction. This is because for any cause, multiple effects are conceivable, and the actual effect cannot be determined by reasoning about the cause; instead, one must observe occurrences of the causal relation to discover that it holds. For example, when one thinks of "a billiard ball moving in a straight line toward another," ["Enquiry", §4.1.] one can conceive that the first ball bounces back with the second ball remaining at rest, the first ball stops and the second ball moves, or the first ball jumps over the second, etc. There is no reason to conclude any of these possibilities over the others. Only through previous observation can it be predicted, inductively, what will actually happen with the balls. In general, it is not necessary that causal relation in the future resemble causal relations in the past, as it is always conceivable otherwise; for Hume, this is because the negation of the claim does not lead to a contradiction.

Next, Hume ponders the justification of induction. If all matters of fact are based on causal relations, and all causal relations are found by induction, then induction must be shown to be valid somehow. He uses the fact that induction assumes a valid connection between the proposition "I have found that such an object has always been attended with such an effect" and the proposition "I foresee that other objects which are in appearance similar will be attended with similar effects." ["Enquiry", §4.2.] One connects these two propositions not by reason, but by induction. This claim is supported by the same reasoning as that for causal relations above, and by the observation that even rationally inexperienced or inferior people can infer, for example, that touching fire causes pain. Hume challenges other philosophers to come up with a (deductive) reason for the connection. If he is right, then the justification of induction can be only inductive. But this begs the question; as induction is based on an assumption of the connection, it cannot itself explain the connection.

In this way, the problem of induction is not only concerned with the uncertainty of conclusions derived by induction, but doubts the very principle through which those uncertain conclusions are derived.

Interpretations and proposed explanations

Hume

Although induction is not made by reason, Hume observes that we nonetheless perform it and improve from it. He proposes a descriptive explanation for the nature of induction in §5 of the "Enquiry", titled "Skeptical solution of these doubts". It is by custom or habit that one draws the inductive connection described above, and "without the influence of custom we would be entirely ignorant of every matter of fact beyond what is immediately present to the memory and senses." ["Enquiry", §5.1.] The result of custom is belief, which is instinctual and much stronger than imagination alone. ["Enquiry", §5.2.]

Rather than unproductive radical skepticism about everything, Hume said that he was actually advocating a practical skepticism based on common sense, wherein the inevitability of induction is accepted. Someone who insists on reason for certainty might, for instance, starve to death, as they would not infer the benefits of food based on previous observations of nutrition.

Colin Howson

Colin Howson interpreted Hume to say that an inductive inference must be backed not only by observations, but also by an independent "inductive assumption." [Howson, 2.] Howson combined this idea with Frank P. Ramsey's view on probabilistic reasoning to conclude that "there is a genuine logic of induction which exhibits inductive reasoning as logically quite sound given suitable premisses, but does not justify those premisses." [Howson, 4.] In this sense, the strength of inductive reasoning is comparable to that of deductive reasoning. [Howson, 2.]

C.S. Lewis

C.S. Lewis, a 20th century popular theologian, argues in "Miracles" that the past-future variety of problem of induction can be easily solved by presupposing the existence of a consistent Creator who would create a consistent Universe. Such a Creator allows us to stipulate that the Universe works according to consistent rules, since the Creator would not create a Universe that was so contrary to his own nature. Thus, we can assume that once we have learned one of the Creator's rules, it will continue to hold in the future.

Lewis further argues that God claims in the Bible to be exactly such a consistent Creator.

Karl Popper

Karl Popper, a philosopher of science, sought to resolve the problem of induction [cite book
title = The Logic of Scientific Discovery
author = Karl Popper
year = 1959
accessdate = 2007-12-27
pages = Ch. 1
quote = ...the theory to be developed in the following pages stands directly opposed to all attempts to operate with the ideas of inductive logic.] [cite news
url = http://www.abc.net.au/rn/science/ss/stories/s75303.htm
title = A Portrait of Sir Karl Popper
author = Alan Saunders
work = The Science Show
publisher = Radio National
date = 2000-01-15
accessdate = 2007-12-27] ` in the context of the scientific method.He argued that science does not rely on induction, but exclusively on deduction, by making modus tollens the centerpiece of his theory. [cite news
url = http://www.accessmylibrary.com/coms2/summary_0286-26166758_ITM
title = Probability, Econometrics and Truth: The Methodology of Econometrics (book review)
author = Aris Spanos
work = Journal of the American Statistical Association
date = 2002-09-01
accessdate = 2007-12-28] Knowledge is gradually advanced as tests are made and failures are accounted for. [cite news
url = http://www.atimes.com/atimes/Middle_East/HL16Ak01.html
title = Civilization is at stake
author = John Dowd
work = Asia Times Online
date = 2006-12-16
accessdate = 2007-12-28]

Wesley C. Salmon critiques Popper's falsifiability by arguing that in using corroborated theories, induction is being used. Salmon stated, "Modus tollens without corroboration is empty; modus tollens with corroboration is induction." [cite book
title = The Foundations of Scientific Inference
author = Wesley C. Salmon
year = 1967
pages = 26
accessdate = 2007-12-27]

Nelson Goodman's New Problem of Induction

Nelson Goodman presented a different description of the problem of induction in the article "The New Problem of Induction" (1966). Goodman proposed a new predicate, "grue". Something is grue if it has been observed to be green before a given time t, or if it has been observed to be blue thereafter. The "new" problem of induction is, since all emeralds we have ever seen are both green and grue, why do we suppose that after time t we will find green but not grue emeralds? The standard scientific response is to invoke Occam's razor.

Goodman, however, points out that the predicate "grue" only appears more complex than the predicate "green" because we have defined grue in terms of blue and green. If we had always been brought up to think in terms of "grue" and "bleen" (where bleen is blue before time t, or green thereafter), we would intuitively consider "green" to be a crazy and complicated predicate. Goodman believed that which scientific hypotheses we favour depend on which predicates are "entrenched" in our language.

W.V.O. Quine offers the most practicable solution to the problem by making the metaphysical claim that only predicates which identify a "natural kind" (i.e. a real property of real things) can be legitimately used in a scientific hypothesis.

Mathematical Induction

The problem of induction exists within the philosophy of science, but not within the philosophy of mathematics, which may seem puzzling. However, scientific induction and mathematical induction are very different. Scientific induction relies on generalizing a set of observations in a non-deductive manner, while mathematical induction relies on axioms to make deductive arguments. For example, proving general theorems about the natural numbers requires appealing to the axiom of induction. The epistemic question of why mathematician are allowed to use these arguments is sidestepped by making induction a part of the systems they are investigating. To over-simplify the issue, it could be said that mathematics replaces the question "Why should we use inductive arguments at all?" with "What conclusions do we reach if we allow certain kinds of inductive arguments?"

Notes

References

* cite book
url = http://18th.eserver.org/hume-enquiry.html#4
title = An Enquiry concerning Human Understanding
author = David Hume
year = 1910
origyear = 1748
publisher = P.F. Collier & Son
accessdate = 2007-12-27
* cite book
url = http://books.google.ca/books?id=pNfwrSTzz8cC
title = Hume's Problem: Induction and the Justification of Belief
author = Colin Howson
publisher = Oxford University Press
year = 2000
accessdate = 2008-01-07

External links

* [http://plato.stanford.edu/entries/induction-problem/ The Problem of Induction] - Stanford Encyclopedia of Philosophy
*
* [http://www.iep.utm.edu/h/humeepis.htm#H4 David Hume: Metaphysics and Epistemology] at the Internet Encyclopedia of Philosophy
* [http://www.geocities.com/ResearchTriangle/Facility/4118/dcs/hume/hume.html Probability and Hume's Inductive Scepticism] (1973) by David Stove
* [http://www.utilitarian.net/singer/by/19740502.htm Discovering Karl Popper] by Peter Singer
* [http://www.dspace.cam.ac.uk/bitstream/1810/3475/1/Inaugural.html The Warrant of Induction] by D. H. Mellor
* [http://journal.ilovephilosophy.com/Article/Hume-s-Problem-of-Induction/47 Hume and the Problem of Induction]