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First off, in order to understand Hume's argument, it is important to understand the difference between deduction and induction. Deduction is a method for arriving at knowledge of specific things from knowledge of general things. For example, if it is known that all of Jupiter's moons have a smaller radius than Jupiter, one can deduce that Ganymede (Jupiter's largest moon) has a smaller radius than Jupiter. The specific case follows directly from the general.
Induction, on the other hand, deals with coming to general conclusions based on knowledge of specific cases. For example, if someone observes the sun rising and setting every (approximately) twenty-four hours every day they have been alive, and if that person talks to older people (or reads books written by people who lived long ago) that made the same observations, the inductive method leads that person to the conclusion that the sun will rise tomorrow, or at least that it is very, very likely (since the sun could unexpectedly go nova).
Another example of induction would be the distribution of how close to a target bullets fired from a certain gun will land. Say a gun is set up so as to neglect temperature change, wind and other such outside factors and then is fired many times toward a target. One might observe that ninety percent of the bullets fired (out of a large number of bullets) landed within two inches of the target. Performing more experiments, the number that comes within two inches may fluctuate slightly to between eighty-eight and ninety-two percent, but will always stay within two percent of ninety. Induction would state the general conclusion that around ninety percent of bullets fired from the gun (be they in the future, or hypothetical shots that never actually take place) will land within two inches of the target. Thus, induction does not have to deal only with observations in the one-hundred percent category, but can deal with probabilities as well, and the small fluctuations that these entail.
Back in the eighteenth century David Hume pointed out a huge problem with this method, known as the 'Problem of Induction'. He argued that there is simply no reason to accept general conclusions based on observations of specific instances, because the general conclusions are based around a number of unobserved (or even just possible) cases. He bases his argument on the idea that there are only two possible ways to justify induction and that both of them are insufficient. These two ways are demonstrative reasoning (relying only on thought), and experimental reasoning (relying on experience).
According to Hume, demonstrative reasoning is only justified if accepting the premises of an argument and rejecting the conclusion leads to a contradiction. However, there is often no contradiction associated with accepting premises but rejecting their inductive conclusion - for example, it is possible the sun will go nova tomorrow despite the conclusion of induction that it will rise.
The only other way of justifying induction would rely on experimental reasoning. Since the conclusion itself cannot be observed (because it deals not only with indefinitely many future instances and unobserved past instances, but also merely hypothetical ones), the only way to justify induction this way is by observing that in the past the conclusions of inductive true premises have turned out to be almost always true, and concluding from that the general notion that if inductive premises are true, their conclusion is also highly likely to be true.
There are two problems with this justification for induction. Firstly, the observed cases where conclusions of true premises were observed true are not actually observations of the conclusion, but only more specific instances - in other words, there are still indefinitely many more future, unobserved or merely possible cases where the conclusion could still turn out false. Secondly, the argument is circular in that it uses the inductive method to justify the inductive method. It relies on an inductive jump from (alleged) past cases of observed true conclusions following from true premises to the conclusion that all such cases would happen that way. Such a jump is not justified because the inductive method it uses is what is in question in the first place. Therefore Hume concludes that since neither of the two possible justifiers of induction - demonstrative reasoning and experimental reasoning - are adequate, induction cannot be justified.
Failed Attempts to Escape the Problem
In response to Hume's dilemma, most philosophers have conceded his main idea and then attempted to justify inductive reasoning in other ways that do not contradict that idea. The two most common such arguments are the 'pragmatic vindication' of induction, and the 'ordinary language' justification.
The pragmatic vindication states that where any relevant truth is to be found, the inductive method can theoretically find it. The inductive method suggests that after taking a number of specific observations, their result should be tentatively accepted as representing the general situation until new observations come in that force revision of the result. The pragmatist in this case asserts that as time goes on and further results continue to come on, the total result will approach a limiting value (the true general situation) if any such limiting value exists. Since for any situation there may not be any such limiting value (as with the inductive conclusion that a certain proportion of people will buy trucks instead of cars next year based on true observed premises about the proportion of people who bought trucks in the past), they agree with Hume that inductive reasoning cannot be justified as leading to true propositions, but only that the inductive method is vindicated because it succeeds wherever success is possible, wherever the limiting value does exist.
The main problem with the pragmatist view is that without an infinite number of observations, we have no way to know if we are approaching the limiting value. Thus, at no actual point can we be justified in accepting that the value observed so far is indicative of any true limiting value that may exist. As such, the pragmatic approach fails to yield any meaningful or useful justification for induction, but leaves us right back where we started.
The ordinary language justification takes a different approach. It points out that in ordinary linguistic usage, in other words common sense, someone who accepts an inductive conclusion (like the conclusion that the sun will rise tomorrow) is considered reasonable, while someone who denies that conclusion is considered unreasonable by common sense. Thus, the alleged 'problem' of induction does not actually exist, but comes from a misunderstanding of the clear dichotomy between deductive and inductive reasoning.
Proponents of the 'problem' of induction, according to ordinary language philosophy, make the mistake of trying to apply deductive standards to inductive reasoning, when in fact the two types of reasoning are different and have different standards. In deduction, conclusions must follow conclusively from the premises, but that is not necessary in induction. According to the ordinary language philosopher, this fact does not make induction any less reasonable, however, just a different type of reason.
The obvious problem with the view of the ordinary language philosopher is that induction defined this way seems useless; we are after a reasoning that leads to likelihood that the conclusions are true, which this type of induction does not do. The problem of induction is focused on the fact that an induction that does not lead to likelihood of truth means that we have no reason to accept much of science and human knowledge (which is based around inductive reasoning) as true. This problem is not addressed by either of the two arguments given, by pragmatists or ordinary language philosophers.
BonJour's Response to Hume
The previous approaches were based on accepting Hume's basic conclusions, and trying to show that induction is still useful. Since they have failed, it seems that the best approach to take instead is to attack Hume's conclusions about the failure of his two types of reasoning (demonstrative and experimental), either by showing that one of them can justify induction, or that there is a workable third way to justify induction. There is no way around the circularity of experimental reasoning, so the only possibility is something not based on experience. Hume is undeniably right that no contradiction exists in accepting inductive premises but rejecting the conclusion, so a justification of induction cannot be based simply on Hume's traditional definition of demonstrative reasoning, where accepting premises and rejecting their conclusion leads to a contradiction.
Laurence BonJour proposes a unique justification for induction without relying on Hume's classification of demonstrative reasoning as the only a priori reasoning. He accepts that in cases where there is no convergence on a limiting value, there is absolutely no reason to accept the standard inductive conclusion as true.
However, where there is an apparent convergence (perhaps with small fluctuations, but necessarily converging in an overall sense), such a convergence needs to be explained. The only two likely explanations are that the apparent convergence is a result of chance or that there indeed exists such a value upon which the observed results are converging. If the results persist, the possibility that they are based on chance becomes extremely low and gets lower the longer it remains, because so many other possible chance results exist which would not lead to any apparent convergence.
The possibility of an apparent convergence early on is not surprising, but such a possibility shrinks exponentially lower the longer it remains after successive observations. So the only explanation that remains for such convergences is that they do in fact approach a real value. The best explanation for the fact that they approach such a value is that there is some regularity in nature which a series of observations tends to reflect with enough observations to neglect possible illusions of such a reflection based upon chance. This explanation means that the limit value seen in convergent results is approximately the true value, which means that the inductive conclusive of the standard premises (those results) justify accepting the standard inductive conclusion as true.
One objection to this explanation is that it seems possible that there is a regularity in nature that produces a true proportional value of how a general thing is, but that the convergence of specific results does not actually represent that value. If the convergence does not represent the actual value, then the only possible explanations for it are chance (which has been ruled out above) or a systematic skewing of the results.
Such a systematic skewing could only be caused by something regular, such as the way the experiments are set up that take the observational results; or perhaps the act of observing itself causes the skewing. However, these possibilities do not discount the justification for inductive reasoning, because they deal only with the results of such skewed experiments, whereas inductive reasoning deals with the real value of things - what would happen without any such outside skewing. Thus, in cases where there is a convergence of results upon a limiting value, and only in such cases, there appears to be justification for accepting the standard inductive conclusion that follows from those results.