Problems with Induction Glenn Mason-Riseborough (29/8/1997) Introduction It is impossible to make an infinite (or even an extremely large) number of observations. Thus, to make universal statements of truth about our physical world we must make our finite observations meaningful and then generalise. Naive induction is a process of inferring general rules about a set of objects from particular instances in that set. It is distinguished from mathematical induction in that it is not deductively valid. In this essay, I will discuss the problem of naive induction—namely, that it cannot show absolute certainty. I will show why deductively valid arguments give absolute certainty, and hence why mathematical induction is seen as a valid form of argument, whereas arguments using naive induction are questioned. I will discuss some specific problems with naive induction and give examples that demonstrates its fallibility. I will also discuss some attempts at finding solutions to the problems raised, and comment on the importance naive induction has in everyday decision-making. Deductively Valid Arguments Ever since the Ancient Greeks (Eves 1963) we have prized deductive validity over other forms of argument because of its path to certain knowledge. Deductively valid arguments are arguments in which if the premises are true then the conclusion must also be true (i.e. it cannot be false). A valid argument makes no claims about the truth of the premises. It is an argument based entirely on the relationship between the premises and the conclusion. Mathematical induction is an example of such an argument. In general, mathematical induction has the following form: Let p(m), p(m+1), . . . be a sequence of propositions. If (B) p(m) is true and (I) p(k+1) is true whenever p(k) is true and m ? k, then all the propositions are true. B is called the basis step and I is the inductive step. B asserts that the first member, m of some sequence of propositions is true. I then asserts that if any member, k of that sequence is true then the next member, k+1 is true. Thus we know without a doubt that given these two conditions, all members of the sequence of propositions are true (i.e. the argument is deductively valid). On the other hand, arguments from naive induction still leave room for the possibility of counter-examples. The structure is as follows: Given some extremely large (or infinite) set ?, we can generate ? such that ? is a randomly selected sufficiently large finite subset of ?. If for all x, where , some proposition Px is true then Py is true for all . This is not a deductively valid argument because there is a possibility that there exists some y such that Py is false. We can see from these definitions that mathematical induction is a more powerful form of argument than naive induction. This is because we gain definite knowledge statements due to its deductive validity whereas naive induction only gives us a probabilistic answer. Naive Induction—Problems and Solutions The question we need to pose at this point is ‘what is wrong with a probabilistic answer?’ Surely if a result is probably correct, then it is meaningful. It could be argued that in many situations all we need is ‘almost’ certainty to satisfy claims. However, when we are looking for universal statements of truth, almost is not good enough—we need to be absolutely certain. This leads us to our next question; ‘is there a point in which the size of our sample set is sufficiently large that we can be guaranteed of absolute certainty?’ Logic dictates that we must answer ‘no’. This is because in a universal statement there are an infinite number of possible situations. Dividing by infinity (no matter what the numerator is) will always leave us with a probability of zero. In everyday situations however, we often generalise from samples, and different circumstances usually dictate different sample sizes. For example statistical surveys typically have a sample size of about 1000 subjects to achieve statistical significance. On the other hand, animals often need only one pairing of food and sickness before they avoid that particular food. (1) Perhaps if we were able to reduce our universal set to a size that can be finitely observed, then we could be absolutely certain of the answer. This could be done by selecting our samples from a sufficiently wide range of different situations, under many different conditions. Unfortunately however, there are an infinite number of different situations, and we can not know which ones are important in advance. The white swan argument below is an example which shows that however we select our sample, we will always have a selection bias which will leave us with an unrepresentative sample. White Swan Argument There was an assumption made in the past that all swans were white. This assumption was based on the argument using naive induction as follows: All swans that have been observed in the past have been white. Therefore, all swans are white. When Australia was explored, it was discovered that there exists some swans that are black. Our universal claim on the colour of swans, that is ‘all swans are white’ was found to be false, and had to be discarded. Our naively inductive argument had been based on a sample that was unrepresentative of the population. What we need is some guarantee that our sample is unbiased. Principle of Uniformity of Nature This principle was an attempt to introduce an additional premise to turn naive induction into a deductively valid argument. This principle states that the universe is consistent and thus any sample we used should be consistent with the rest of the population. Explicitly the deductive argument is: All observations we make will guarantee consistent results given consistent situations. Some observation, o logically implies some proposition p. Therefore, whenever we replicate observation o, proposition p will always be true. Unfortunately our argument supporting the principle is circular because it is itself a argument using naive induction, as shown below. All observations of the universe in the past have been consistent. Therefore, the universe is consistent. Hence, we are left with a situation in which we cannot make any universal truth claims about the world in which we live because we do not know if there is a counter-example beyond our field of vision. Practical solutions In the real world, animals need to make decisions based on reasoning using naive induction in order to survive. For example, this particular place (supermarket, tree, fishing hole, etc.) has been a food source in the past, therefore it is probable that there will be food when I go there again. Using this form of reasoning is simply a survival feature which (inductively) works because it has worked in the past. Behaviourism acknowledges this form of reasoning, and many behavioural laws of psychology assume that animal behaviour is based on reasoning using naive induction (e.g. classical conditioning). Conclusions Despite our best attempts to generate some certainties from naive induction, we must accept that conclusions based on arguments using naive induction are not guaranteed. Whatever we do, there is still the possibility that there is some counter- example out there that we have not yet found. Arguments using naive induction are not deductively valid, and even adding additional premises only results in circularity. Thus, we must conclude that naive induction cannot play a part in the discovery of objective truths about the universe. However, we also must accept that in some situations uncertainty is the only option, and that in order to survive in this world we must sometimes use naive induction to make decisions. References: Eves, H. (1963) A survey of Geometry, Volume One. Boston: Allyn and Bacon. Mazur, J. E. (1994) Learning and Behaviour (3rd edn.), Englewood Cliffs, NJ: Prentice-Hall. Endnots: 1 See for example Mazur (1994) for a discussion on taste-aversion learning