Arguments Applying Statistics

§4.2

Consider the following argument, and let’s assume that the premises are certain:

Map 40a:

If P1 said that all Frenchmen were Catholic the argument would be a valid deduction, but, as it stands, the argument is invalid. Does it, though, give us any reason to believe its conclusion? Does it make the conclusion even epistemically possible? For all that P1 says, there may be only two Catholics among the tens of millions of Frenchmen. So, if these premises were the only support we had for the conclusion, the conclusion would be unfounded. But suppose we changed the argument as follows:

Map 40b:

This is clearly an improvement. If most Frenchmen are Catholic, then isn’t the conclusion that Pierre is probable? After all, given that he’s French, isn’t he more likely to be Catholic than not? And if we change P1 again to be more specific as to how many Frenchmen are Catholic, can’t we then say more precisely how probable it is that Pierre is Catholic?

Map 40c:

We have here an extremely simple example of statistical reasoning. Statistics is the science that deals with the interpretation and analysis of quantitative data about the prevalence of traits within groups, and it is sometimes used to draw conclusions about how certain we can be that an individual has a certain trait.

Notice that we haven’t yet assessed either of the last two arguments. Since what we’re interested in here is the inference, let’s assume for the sake of argument that we know the premises to be true. If you know that Pierre is French and that most Frenchmen are Catholic, does this mean that the proposition “Pierre is Catholic” is has an epistemic status of probable for you? If you know that 60% of Frenchmen are Catholic, does it have a probability of 60%? Not necessarily.

It depends on what else you know about Pierre. Suppose that Pierre is a Muslim and that you know this about him in addition to knowing the two premises above. Then, far from being probable, the proposition that Pierre is Catholic would be certainly false. Even if you didn’t already know whether Pierre was Catholic, there are lots of other things you might know about him that would make it unreasonable for you to infer that he was probably Catholic, from the knowledge that most Frenchmen are. Suppose you knew, for example, that Pierre was a Communist and that most communists are atheists, or that he comes from a particular part of France that is predominantly protestant or that he’s extremely intelligent and that intelligent people often hold views outside of the mainstream of their society.

From the premise that 60% of Frenchmen are Catholic, you can infer with 60% probability about a Frenchman selected absolutely at random, that he is Catholic. But, as any decent statistician knows, if Pierre is not a random Frenchman, you cannot infer it about him without first consulting the rest of your knowledge about him to see whether any of it is relevant to determining his religion. It is important to keep this in mind, since we are so often bombarded with statistics. For example, if you are a woman, and you read online that unmarried women are fifty percent more likely to die in any given year than married women, then even if this statistic is true, it would be unreasonable to infer that you will probably increase your longevity by accepting a marriage proposal.

There is an important point to be make here that applies far beyond statistical arguments. It is a special feature of deductive inferences that one can analyze them in isolation from the rest of one’s knowledge. This is because deductive arguments are just those in which, if the premises are true, then, no matter what else may be the case, the conclusions must also be true. But this is not the case with any other kind of argument. Because of this, when assessing any non-deductive inference, you need to make a point to consider whether you have any knowledge other than the premises that impacts the degree to which the premises support the conclusion. This makes assessing non-deductive inferences much more difficult than assessing deductive ones.