The Problem of Induction

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Science has the problem of induction: No matter how much evidence we have for a conclusion, the conclusion could still conceivably be false. The best we can say is that it is ``unlikely'' that our conclusion is false when we are using inductive reasoning. Here's an example: suppose there is a barrel filled with 100 apples and the first apple I pull out off the top is very rotten. Few would wager from this single apple that we know all the apples in the barrel are rotten. However, small amounts of evidence need not always be weak. A biologist might be willing on the basis of this one apple to wager that all of the apples are likely to be rotten, if other information were provided like what temperature the apples were stored, and for how long, because of her general knowledge of bacteria and their ability to spread rapidly. If we have some world view or paradigm (a framework of a general consensus of belief of how the world works), we can do a lot of hypothetical work with just a few observations.

But without anything else to go on, concluding that all the apples are rotten from a single positive case is a very weak inductive inference.To make the inductive inference stronger, more apples need to be sampled. If I pull out 4 more apples off the top and all of them are also rotten, we'd now have a better basis for concluding that all the apples are rotten. This is called induction by enumeration. In general, the more positive cases in favor of a hypothesis, the stronger the hypothesis is. But how about the apples at the bottom? A stronger case could be made by choosing a representative sample---a sample that matches in characteristics the total population of things under investigation. In the case of the barrel of apples, a representative sample could be gathered by selecting one from the top, one from the very bottom, one from each side of the barrel, and one from the middle. If all five are rotten, this would strengthen the hypothesis considerably. A small representative sample is much stronger logically than is a large unrepresentative one. Five representative apples are better than 20 just off the top.

If you found that another 45 were rotten, would you bet your life savings that all the remaining 50 were rotten? Probably not, since it is still possible that some, even many, of them are not rotten. If you found that another 49 were also rotten, would you bet your life savings that the last remaining apple was rotten? Most people would, but they'd still have a lot of anxiety as the last apple was pulled from the barrel because it was still possible that the hypothesis, ``all the apples are rotten,'' was false. Hypotheses can only be confirmed, not logically proven to be true. Understand that it is possible to deduce true conclusions (the 5th apple will be rotten) from premises that may be false (all the apples are rotten). Because we can deduce true predictions from a false theory, no matter how long a theory has been successful in making predictions, it cannot be known to be true absolutely. It could be found to be false tomorrow.

Critics of science often attempt to use this logical window to repudiate many scientific conclusions. They also often commit the logical fallacy of appealing to ignorance, arguing that because the theory cannot be proved absolutely true, it must be false. But absence of evidence for absolute proof is not evidence of absence of truth. Critics of science fail to recognize the positive aspect of this logical doubt. Without room for doubt, there would be no room for self-correction, and we would be left with a cluttered clash of irrefutable beliefs.

The appeal to ignorance has been used to argue for the existence of things like the Loch Ness monster, Bigfoot, space aliens in flying saucers, etc. by saying one cannot prove they do not exist. In this excerpt from an article by Steven Hales (select the link), he shows how one can prove a negative beyond reasonable doubt.

Summary

• No matter how much evidence we have for a conclusion, the conclusion could still conceivably be false.
• The more positive cases in favor of a hypothesis, the stronger the hypothesis is.
• The most logically sound samples are those that are representative of the entire set.
• It is possible to make true conclusions from false assumptions.
• A hypothesis can only be confirmed but it cannot be proven absolutely true.
• Even though a scientific hypothesis cannot be proven absolutely true, that does not mean that it must be false.

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last updated: January 28, 2011