Enumerative inferences

Imagine a turkey living happily on a turkey farm. Every morning the farmer brings corn for it to eat, and it doesn’t take much more than that to make a turkey happy. One morning the farmer approaches and the turkey (let’s suppose) thinks happily “Hooray, breakfast.” Supposing the turkey is reasoning at all, it’s reasoning non-deductively: On every morning up until now the farmer has brought corn, so today he’ll be bringing corn again. But alas, it’s Christmas morning, and the turkey is making a terrible mistake when he runs happily towards the farmer to get fed, because this time the farmer is bringing an axe instead.

The turkey was reasoning, if it was reasoning at all, pretty well: it was reasoning from true premises, and ones which gave it quite a lot of reason to believe its conclusion. Nevertheless, its conclusion was false. Moral: no matter how good your non-deductive argument, it’s still possible for your conclusion to be false.

Someone who is reasoning like this is taking a (large) number of cases of which they have experience, and inferring that a pattern which has occurred will continue to occur. They have, if you like, collected some data, and they are extrapolating from that data to formulate a conclusion. Inferences of this type are sometimes called “inductive inferences”. But because this is not the only type of induction, we have chosen to call these “enumerative inferences“. That is because a number of cases are collected, and, on the basis of that list of cases, a conclusion is reached about a new case.

Enumerative inferences are different from probabilistic arguments. Consider this probabilistic argument from above:

​P1. There are 75 black marbles in this bag and 25 white marbles.
P2. In my fist is a marble randomly selected from the bag.
                                                    
[Probably] C. The marble in my fist is black.

In this argument, the proportions of the contents of the bag are known, and, because this is a mathematical example, the degree of probability of the conclusion can be precisely calculated. (It is 75% likely that the marble in my fist is black.)

Now suppose I have a bag of marbles, and I know that there are 100 marbles in it. I know nothing about what colour the marbles are, however. I draw out the first 99 marbles, and they are all black. On this basis, I conclude that the 100th marble will also be black. My argument looks like this:

P1. Marble 1 is black.
P2. Marble 2 is black.
P3. Marble 3 is black.
                  ⋮
P99. Marble 99 is black.
                                                    
[Probably] C. Marble 100 is black.

I can’t assign a precise degree of probability to that conclusion. There are infinite possibilities for what (shade of) colour the remaining marble could be. But it certainly seems more reasonable for me to suppose that it is black, given the contents of the bag so far, than to suppose it is some other colour.

In ordinary life, we reason like this all the time. When I suppose that the sun will come up tomorrow morning, I’m extrapolating from many cases of the sun coming up. This has happened every day of my life so far, and I expect it to continue. I suppose that if I get hit by a bus, I will get hurt, and I suppose this on the basis of what usually happens when people get hit by buses, and on the basis of what has happened to me in the past when I have been hit by large heavy objects. You might think that all I need to work out what would happen if I was hit by a bus is to apply the laws of physics, but even the belief that the laws of physics will continue to apply is justified through an enumerative inference. Such arguments are very important, and very useful.

Not all enumerative inferences are strong. They are often difficult to assess. Consider once again the marbles case above. When I know that there are 100 marbles in the bag, and the first 99 have been black, it seems reasonable to conclude that the 100th marble will be black. But what if I didn’t know how many marbles were in the bag? What if I have only drawn 10 marbles? Can I still (justifiably) conclude that the next marble will be black?

There are a number of things to take in consideration when assessing an enumerative inference.

• How big is the sample?

The more data that is collected, the stronger the enumerative inference. That is why an inference about the colour of the next marble is stronger when you have tested 99 marbles than when you have tested only 9. The sample of times the sun has come up in the morning is huge, and we feel very confident that there will be a tomorrow.

• How big is the sample compared to the total population?

If I know that there are a million marbles in the bag, and I’ve tested 99, I’ll feel less confident about the next marble I pull out than if there are 100 marbles in the bag and I’ve tested 99.

The size of the total population can also vary depending on what the conclusion is claiming. Sometimes a conclusion is about the next case alone. So, consider:

Argument 1

P1. The sun has come up every day of my life.
                                                    
[Probably] C. The sun will come up tomorrow.

I feel very confident about this conclusion. The sample size is all the days of my life, and the total population with which the argument is concerned is all-the-days-of-my-life +1. The sample is a very big proportion of the total population.

Compare:

Argument 2

P1. The sun has come up every day of my life.
                                                    
[Probably] C. The sun will come up every day for the rest of my life.

and

Argument 3

P1. The sun has come up every day of my life.
                                                    
[Probably] C. The sun will come up every day forever.

In argument 2, the total population is unknown, but I’m going to be optimistic and suppose I’m about halfway through my life. That means I’m extrapolating from known cases to about the same number of cases again.

In argument 3, the conclusion is so broad that it’s unlikely to be true. We know that the world will end some day, so there will, one day, be a day that’s the last day. A conclusion which goes too far beyond its sample ends up having a sample which is a very small proportion of the total population.

• How was the sample collected?

Suppose I am given 10 bags of marbles, with 10 marbles in them each. If I take 10 marbles from the first bag, and they are all black, I have some reason for thinking all the marbles in all the bags are black, but it’s not a particularly good reason. If I take 1 marble from each bag, and they are all black, then I have a much better reason for thinking all the marbles are black.

Generally speaking, the more data you have collected, the stronger the enumerative inference. The more random the data collection, the stronger the enumerative inference. If you find yourself rejecting almost all enumerative inferences, your standard for reasonableness is probably too high. We do use these inferences all the time. And, in fact, it would be impossible to function without them. You’d have no reason not to step in front of a bus.

Here are some for you to try.

It’s important to note that the possibility of being wrong is not sufficient grounds for rejecting an enumerative inference. This possibility of inferring a false conclusion from true premises is a feature of all non-deductive arguments. Consider again the turkey from the beginning of this section. The turkey is justified in his conclusion even though one day he will be wrong. One day the sun won’t come up. That doesn’t mean you are unjustified in believing it will come up tomorrow.