Logic Part 6:  Mill’s Methods: Logical arguments taking us from repeated observations to causal explanations

In the 19th century, the English philosopher JS Mill provided some algorithms or rules of thumb that show us how to take a set of repeated observations and infer causes.  If we go to chapter 8 of the free 8th edition of  ‘A System of Logic’ Mill suggests forms of argument (or methods of analysis) that might be called inductive, in the sense that they are derived from collections of observations. These ideas have become popularly known as Mill’s Methods. These are:

1) The Method of agreement which operates  “by comparing together different instances in which the phenomenon occurs”. Mill says “this method proceeds by comparing different instances to ascertain in what they agree”

Mill’s methods are well suited to the practice of medical research, for example. To apply the Method of Agreement in this field we might look for a factor that might be a cause of a disease. In other words, if epidemiologist wishes to discover what risk factors lead to a particular disease they might look for a factor that is common to people suffering from that condition. Respiratory exposure to asbestos fibres in workers dealing with this substance has been found to be a common factor in the rare form of lung cancer known as mesothelioma. Indeed it is thought that most mesotheliomas of the lung are almost always caused by asbestos fibres being taken into that organ. Here we have an almost perfect example of the Method of Agreement.

2) The Method of Difference arrived at  “by comparing instances in which the phenomenon does occur, with instances in other respects similar in which it does not”. “This method compares an instance of its occurrence with an instance of its non-occurrence, to discover in what they differ”

The ‘Method of Difference’ does the exact opposite of The Methods of Agreement and seeks to determine a cause by looking for something that has distinguished particular instances (or people). So if a medical condition (or a healthy trait) was found in certain members of the population but not others we might seek not to find out what is common to the affected group but what is different in those unaffected in the wider population. Of course, in practice,  we might conjoin the methods of difference and agreement, particularly when looking for multiple risk factors in conditions such as heart disease or bowel cancer.

Mill says of Methods 1) and 2) “Both are methods of elimination”. “The Method of Agreement stands on the ground that whatever can be eliminated, is not connected with the phenomenon by any law. The Method of Difference has for its foundation, that whatever can not be eliminated, is connected with the phenomenon by a law. He also compares this with the deductive process of elimination in dealing with mathematical equations.

The three remaining methods are:

3) The Joint method of agreement and difference

4) The Method of Residue  “Subducting from any given phenomenon all the portions which, by virtue of preceding inductions, can be assigned to known causes, the remainder will be the effect of the antecedents which had been overlooked, or of which the effect was as yet an unknown quantity”. In other words when we exclude known or at least accepted causes for particular effects what are we left with by way of explanation?

5) The Method of Concomitant Variations looks at the magnitude of proposed effects and causes.  This would apply “in the cases in which the Method of Difference, strictly so called, is impossible”. In other words, the difference is not complete but instead varies in quantity.  “The quantity (my italics) or the different relations of the effect follow those of the cause”.

In the Method of Concomitant Variation, we are looking for what we now think of as a statistically deduced correlation between the magnitude of the effect and the proposed cause. The dose-response relationship in pharmacology is perhaps an obvious example. Give patients a greater dose of a drug such as paracetamol (Tylenol) and they will experience more pain relief or a longer pain-free period. Mill uses the example of friction and says that we can only experimentally reduce frictional effects in slowing down moving bodies we cannot remove them entirely. He also warns that we should not extrapolate outwith the range of the observed limits, which is in effect bringing in another logical constraint.

Of course, Mill was well aware that the potential causes we ascribe might not be the only ones we could find or the only which might apply exclusively. In other words, there is an additional process of logic here that tells us that we cannot reasonably expect that our explanation may have only identified a partial cause

Notice also we have a starting premise, or a “canon” as Mill calls it.  In the Method of Agreement, the premise is “If two or more instances of the phenomenon under investigation have only one circumstance in common, the circumstance in which alone all the instances agree, is the cause (or effect) of the given phenomenon”. We now see that this kind of inductive inference also needs a starting premise, which in itself we might label as having a ‘logical’  quality. Deduction and Induction then begin to appear as if they have underlying characteristics in common; argumentative premises. Mill himself says “the Method of Residues, as we have seen, is not independent of deduction”.

Nowhere in these ideas above need we be over-concerned with absolute truth or certainty of belief in our conclusions since we are not applying some form of highly restricted classical logic. Nevertheless, we are adopting what might be referred to as a rational approach.

Steve Campbell
Glasgow, Scotland
2017, 2019

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