Quite recently the following occurred. A website for which I was responsible was in need of an online store (a store on the Internet where one can purchase goods). Establishing such a store typically involves the installation of certain software. The author of the software installed it on the computer hosting the website. The store was tested, and it was discovered that a certain function (so-called secure online ordering) did not work when one to utilize a certain browser (a browser is software used to view pages on the Internet). Why did this happen? Well, browsers are temperamental and frequently have "bugs". Perhaps that was the problem. However this seemed to me an implausible answer. I had purchased the software using secure online ordering and the very browser in question. Well, then, perhaps the store software had been misinstalled. The author of the software had himself done the installation, and recognized that this was a possibility. So he downloaded the software as it had been installed and put it on his own website. It worked there as it was meant to work. Throughout this time, the technical support personnel responsible for the server on which the website was located insisted that there was no problem with their computer. But I was persuaded that the evidence I had suggested that there was a problem with their computer. I presented this evidence, accompanied by a threat to move the website, and they undertook a new investigation. It was after some effort discovered that the new version of certain very standard security software (which was not yet installed on the other computers) had a "bug". Once they replaced this new version with the previous version, all was well.

In the course of this investigation some of the techniques that have come to be known as Mill's Methods were utilized. We will now turn to a very informal and incomplete sketch of those methods. The core ideas, which we find, will then be more rigorously examined in the subsequent sections of this topic.

The Method of Agreement

In some cases we are faced with a number of occurrences of phenomena that we wish to explain. Our interest might be in hypotheses that identify causally necessary conditions. Events or features G will, as you may recall, be said to be necessary conditions for events or features F if and only if whenever F is present, G is present. Not all necessary conditions are ones that we would take as causally necessary conditions. For example, being around 93 million miles from the sun is, I suppose, a necessary condition of catching a cold from your classmates. But I doubt that we would take it as a causally necessary condition. We shall consider this sort of problem a bit later on, but for now we can proceed keeping in mind that causally necessary conditions are at least necessary conditions. We will take the method of agreement to be primarily a means of excluding hypotheses that claim such-and-so to be a causally necessary condition.

One often-mentioned kind of application of the method of agreement occurs in connection with, for example, outbreaks of food poisoning among persons who have been eating at a local fast food outlet. Typically investigators look to see if there was some particular item which was eaten by all of those who became ill. If there is, that item is viewed, at least initially, as a prime candidate for being the culprit.

We will utilize the following notational conventions. We will use capital letters A, B, C …for the candidate explanatory factors, typically called the antecedent circumstances. We will use lower case letters p, q, r … to indicate the presence of the phenomena in which we are interested. We will utilize ~p, ~q, ~r … to indicate the absence of those phenomena. We can now exhibit the structure of applications of this method as follows:
 

Cases 1 . . . n

Circumstances A C D B C F A C E C F G A C E

Phenomena p p p p p

Here we would take C as our initial suspect. But note that we can do little more than say C is a suspect. As we noted in our example of being 93 million miles from the phenomena always have any number of circumstances in common. Most of those will be irrelevant from the standpoint of providing any part of the account of why the phenomena in question occurred. The method does not give us an account of which circumstances to attend to- that must come from elsewhere. That is, we begin with a list of circumstances that have in our judgment a chance of being relevant. Even if we do this we have no guarantee that C, for example, is in fact a causally relevant factor.

There is another sort of problem as well. Note that either A or F is present in all the cases. To stick with a food example, let us suppose that C is a salad and is in fact irrelevant. Finding out that it is irrelevant will typically involve other kinds of investigation. But both A and F are meat dishes served with a mushroom sauce. As it happens, this sauce is the culprit. Had we started with a different list of circumstances we might have identified it as a suspect. The method does not itself provide us with a list. It is something that we bring to the situation. And of course we might not have enough cases to apply this method at all.

The Method of Difference

When we apply the method of difference we are, as the name suggests, looking for what made the difference. Typically we utilize this method when we are looking for causally sufficient conditions. Events or features F will, as you may recall, be said to be sufficient conditions for events or features G if and only if whenever F is present, G is present. But again not every sufficient condition need be a causal condition the structure of this method is:
 

Cases 1 . . . n

Circumstances A B C A B C A B C B C B C

Phenomena p p p ~p ~p

Notice here that the phenomena in which we are interested is absent when A is absent. Here we might suspect that A is, at least in such circumstances, causally sufficient.

If you look back to my investigation of the website problem, part of that which was described was in fact an application of the method of difference. One circumstance was the presence of the installed software. Another was its installation on my website's server. We can chart the situation as follows:
 

Cases 1         2

Circumstances Installed Software Running on my server Installed Software Running elsewhere

Phenomena Malfunction         No Malfunction

This, of course, is part of what made me suspect the server. But notice one feature. These circumstances are rather vague. At best they served to identify something which should be examined more carefully.

Problems aside uses of the method of difference abound both in everyday life and in science. In particular we can use this method when we have a unique event, for example, the French Revolution. At a certain point that revolution occurs. What is it that is present at this point that was not present at the other times at which it did not occur? We cannot use agreement here because we have only one French Revolution.

The Joint Method

The joint method is not 'new'. Rather it is simply that we in the course of an investigation we use both the method of agreement and the method of difference. If you look back to the website example you should note that it in fact involved a use of the method of agreement which came up with a negative result. That is, the presence of the software as such was not the problem, as it was present where the malfunction was not present. Similarly, it was not the particular setup of the software that was responsible. That is why the author of the software copied that particular setup to his own website. Had that particular setup malfunctioned on his website we would have had:
 

Cases 1     2

Circumstances Particular Setup   Particular Setup

Phenomena Malfunction     Malfunction

But this did not happen. As noted the method of difference was then appealed to

The Method of Concomitant Variation
The preceding methods were limited to phenomena that were eitherpresent or absent. However as often as not we are interested in phenomena which vary in certain ways. The variations may be among individuals in a population. Some persons are taller than others, and some dogs are more responsive to training than others. In other cases we might notice variations among populations. Smokers, as we know, are more liable to lung cancer than are non-smokers. Suicides are more common at some times than others, more frequent in some populations than others. The method of concomitant variation enables us to say something about some of these claims.

Let us briefly look at smoking. Suppose we divide a population of similar persons into subpopulations of smokers of non-smokers. We note that the incidence of lung cancer is high in the subpopulation of smokers, at least by comparison with the subpopulation of smokers. We might represent this as follows.
 

We will not at this point introduce any chart to exhibit the structure of this method. At this point we will simply note that where we find either that an increase in one factor is correlated with an increase in another or that a decrease in one factor is correlated with an increase in another, we suspect the presence of a causal relation. We will in the next section turn to a more rigorous study of correlations and their significance.