anything about all or most guppies. Presumably the arguer does believe either Pl(a) or Pl(b), but if so, he does so on the basis of his previous experience with guppies and cat food. Somehow, the arguer must get from the proposition 'Every guppy I've had has died when I fed it cat food' to 'Most (or all) guppies die when fed cat food'. So far, in our discussions of arguments, we have said nothing to help us with this sort of inference.7

However, this kind of inference is very common, both in science and in ordinary life. We call it an inductive inference. This is our name for the case when you extrapolate from a sample of a total population of things either to something outside the sample, or to a generalisation about the population as a whole (so far, we have always gone the other way, either from a generalisation about a whole population to a claim about some smaller sample, or from the whole population to a claim about a single member of it). In the above case, the arguer needs to extrapolate directly from the sample of guppies they have experimented with to a proposition about the new guppy:

P1) Every observed guppy has died when fed cat food.

C1) My new guppy will die if I feed it cat food.

If we judge this inference to be inductively forceful - let as assume that it is - then, just as with any inductively forceful argument, we write 'Probably', in parentheses, before the conclusion:

P1) Every observed guppy has died when fed cat food. C1) (Probably) My new guppy will die if I feed it cat food.

7 As we pointed out, many probabilities can be based upon proportion. One of the tasks of statistical analysis is to show how this is so, even in cases that might initially seem to resist this strategy (see Note 2). Inductive inferences represent a class of cases where proportion certainly cannot be applied straightforwardly. If we know that, say, 74 per cent of the past 956 cases of A have been cases of B, we cannot without further ado assume that the probability that the next case of A will be a case of B is 74 per cent. The next case of A is not amongst the set of cases for which that proportion is known to have held. So it is not like the case of a card face down on the table, which is drawn from a set of cards for which the relevant proportion is known. In the inductive case we have to make the further assumption that probabilities taken from proportions of known cases can be transferred to unknown cases. Of course such inferences do often seem to be rationally justified; the question of why this is so - the 'Problem of Induction' - was first raised by David Hume in his Treatise of Human Nature (1 739), and remains an open problem.

Alternatively, the arguer could have obtained this conclusion by a slightly longer route. The arguer might have reached the conclusion by means of an extended argument, comprising an inductive inference to a generalisation about all guppies, together with a second, deductively valid inference from that generalisation to the conclusion concerning the arguer's new guppy:

P1) Every observed guppy has died when fed cat food. C1) All guppies die if fed cat food.

C2) My new guppy will die if I feed it cat food.

However, assuming the truth of PI, one could not assert the conclusion here with quite as much confidence as in the first case. We may assert C2 with as much, but no more, confidence than that with which we may assert CI. But since CI is a generalisation about all guppies, it stands a greater chance of being false than CI of the first guppy-argument - a statement about a single guppy. The reason is that the second argument depends upon a more ambitious extrapolation than the first, one that is not needed for the desired conclusion. Indeed, the inferences to CI in either case are fundamentally of the same type. They are both extrapolations from observed cases to unobserved cases. Each can be thought of as an extrapolation from a sample to a larger population of which the sample is a part. In the first case we can think of the larger population referred to as comprising the sample (the observed guppies) plus this one new guppy. In the second case, obviously, the larger population is simply the total population of guppies. Some inductive inferences are concerned to make a broad generalisation (all guppies), others are concerned only to extend to a few new cases, or to a single new case (the new guppy). But they are fundamentally the same type of inference. Thus we define:

To say that an inference is an inductive inference is to say: (a) it is not deductively valid; (b) its premise is a generalisation about a sample of a given population, and (c) its conclusion is a generalisation about the total population from which the sample is drawn.

Requirement (a) is to avoid counting the following sort of inference as inductive: 'Some black cats have no tails; therefore not all cats have tails'.

Inductive inferences frequently extrapolate from past to future. For example, if we infer, from Ireland's never having won the football World Cup, that probably it never will, our sample is all the World Cups that have so far been contested, and the total population is the set of all World Cups - past, present and future.

Sometimes the conclusion of an inductive inference is a more precise statistical generalisation such as '37 per cent'. For example, when a pollster finds that 37 per cent of a sample of the adult British population supports a given political party, the pollster might conclude that 37 per cent of the adult British population supports that party. How probable this conclusion is, however, will depend on how representative the sample is. This point requires further explanation.

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