Mill’s System of Logic

John Stuart Mill A System Of Logic, Ratiocinative And Inductive: Being A Connected View Of The Principles Of Evidence And The Methods Of Scientific Investigation. Eighth Edition (Significant changes since the first) 1843. 

Mill’s system formed the basis of his later works, such as utilitarianism, for which he is better known. His notion of mathematics is considerably broader than ours, overlapping with Physics. Thus he regards the existence of points and lines in the physical world as being asserted by Geometry, rather than Geometry offering a model which it is the responsibility of Physicists to endorse, empirically. While this is confusing, Mill does distinguish between parts of mathematics that are more or less certain, leading on to a more general discussion of uncertainty that underpinned his later work and is of relevance today, even if we do need to adapt it to contemporary concepts.


“There is a real distinction, then, between definitions of names, and what are erroneously called definitions of things; but it is, that the latter, along with the meaning of a name, covertly asserts a matter of fact.”


Mill notes that the ‘map’ of Science typically ‘about’ subjects and their attributes, irrespective of the actual nature of the territory. Mill holds that axioms are empirical, and special only in so far as they have greater grounds and hence are regarded as being less uncertain than other propositions. Nothing about the external world is known a priori. (We may so that the axioms of geometry are not empirical, but that their relevance is.)

Coupling this with Mill’s remarks on definition, above, we need to be careful that just because definitions are couched in terms of objects, attributes and relationships does not mean that such conceptualization is necessarily adequate.

Of The Ground Of Induction

INDUCTION … may … be summarily defined as Generalization from Experience. … [T]here is … an assumption … that there are such things in nature as parallel cases; that what happens once, will, under a sufficient degree of similarity of circumstances, happen again, and not only again, but as often as the same circumstances recur. … . And, if we consult the actual course of nature, we find that the assumption is warranted. …

… To Europeans, not many years ago, the proposition, All swans are white, appeared an equally unequivocal instance of uniformity in the course of nature. Further experience has proved to both that they were mistaken; but they had to wait fifty centuries for this experience. During that long time, mankind believed in a uniformity of the course of nature where no such uniformity really existed.

Before we can be at liberty to conclude that something is universally true because we have never known an instance to the contrary, we must have reason to believe that if there were in nature any instances to the contrary, we should have known of them.”

Mill notes that, in practice, much induction is still of the simplistic kind that Bacon warned against. Has this changed? Mill also discusses causation, including causation of a propensity. Even if it is not known, he assumes that it exists.


“Let any one watch the manner in which he himself unravels a complicated mass of evidence; … he extemporizes … a first rude theory of the mode in which the facts took place, and then looks at the other statements one by one, to try whether they can be reconciled with that provisional theory, or what alterations or additions it requires to make it square with them. In this way … we arrive, by means of hypotheses, at conclusions not hypothetical.”

We might say ‘not so very hypothetical’, since the uncertainty is reduced but not eliminated. We might also want to consider alternative ‘rude theories’ (viewpoints) in case they lead to significantly different possible conclusions.

“[A]n hypothesis of the sort in question is entitled to a more favorable reception, if, besides accounting for all the facts previously known, it has led to the anticipation and prediction of others which experience afterward verified … .”

Again, this is clearly along the right lines, but the term ‘verified’ should not be taken as implying that the hypothesis is actually true, only that it has passed a test.


“A derivative law which results wholly from the operation of some one cause… will always be true except where some one of those effects of the cause, on which the derivative law depends, is defeated by a counteracting cause. But when the derivative law results not from different effects of one cause, but from effects of several causes, we can not be certain that it will be true under any variation in the mode of co-existence of those causes, or of the primitive natural agents on which the causes ultimately depend.”

Of the Calculation of Chances

Mill quotes Laplace:

The theory of chances consists in reducing all events of the same kind to a certain number of cases equally possible, that is, such that we are equally undecided as to their existence ; and in determining the number of these cases which are favorable to the event of which the probability is sought.”


“Except, however, in such cases as games of chance …I can conceive no case in which we ought to be satisfied with such an estimate of chances … .

In the case of a witness, persons of common sense would draw their conclusions from the degree of consistency of his statements, his conduct under cross-examination, and the relation of the case itself to his interests, his partialities, and his mental capacity, instead of applying so rude a standard … as the ratio between the number of true and the number of erroneous statements which he may be supposed to make in the course of his life.”


“When a fact has been observed a certain number of times to be true, and is not in any instance known to be false, if we at once affirm that fact as a universal truth or law of nature, without either testing it … or deducing it from other known laws, we shall in general err grossly; but we are perfectly justified in affirming it as an empirical law, true within certain limits of time, place, and circumstance, provided the number of coincidences be greater than can with any probability be ascribed to chance. The reason for not extending it beyond those limits is, that the fact of its holding true within them may be a consequence of collocations, which can not be concluded to exist in one place because they exist in another; or may be dependent on the accidental absence of counteracting agencies, which any variation of time, or the smallest change of circumstances, may possibly bring into play.”

Thus an excess of apparent coincidence begs an explanation, but even the most obvious an appealing one might not be true. For example, sudden infant deaths (SIDs) were explained by the empirical law that one death might have been an accident, two were murder. But the deaths of interest were ‘unexplained’, and there is a trend for new explanations to arise as a ‘counteracting agency’. Mill draws our attention to the possibility for more discoveries.

Remaining Laws of Nature

“The mathematical solutions of physical questions become progressively more difficult and imperfect, in proportion as the questions divest themselves of their abstract and hypothetical character, and approach nearer to the degree of complication actually existing in nature ; insomuch that beyond the limits of astronomical phenomena, and of those most nearly analogous to them, mathematical accuracy is generally obtained ” at the expense of the reality of the inquiry … .

The value of mathematical instruction as a preparation for those more difficult investigations, consists in the applicability not of its doctrines, but of its method.”

On the grounds of Disbelief

“The ground for abstaining from belief is simply the absence or insufficiency of proof; … . By disbelief is here meant .. that … we are fully persuaded that some opinion is not true; insomuch that if evidence, even of great apparent strength …  were produced in favor of the opinion, we should believe that the witnesses spoke falsely, or that they … were mistaken.

We are seldom, therefore, without the means (when the circumstances of the case are at all known to us) of judging how far it is likely that such [an unseen] cause should have existed at that time and place without manifesting its presence by some other marks, and (in the case of an unknown cause) without having hitherto manifested its existence in any other instance.

The improbability … , or, in other words, the unusualness, of any fact, is no reason for disbelieving it, if the nature of the case renders it certain that either that or something equally improbable, that is, equally unusual, did happen. …  We are told that A. B. died yesterday; the moment before we were so told, the chances against his having died on that day may have been ten thousand to one; but since he was certain to die at some time or other, and when he died must necessarily die on some particular day, while the preponderance of chances is very great against every day in particular, experience affords no ground for discrediting any testimony which may be produced to the event’s having taken place on a given day.”

We see here an important distinction between improbability and surprise. Thus, for example, we ought to have been open to the possibility of two unexplained infant deaths in a row due to as yet unknown causes.

Operations Subsidiary to Induction

“Whenever the nature of the subject permits our reasoning processes to be, without danger, carried on mechanically, the language should be constructed on as mechanical principles as possible; while, in the contrary case, [language] should be so constructed that there shall be the greatest possible obstacles to a merely mechanical use of it.”

For example, loose talk about ‘the probability of a hypothesis’ does nothing to encourage people to take account of the wider issues, as in this work.

“In our direct inductions we can not for a moment dispense with a distinct mental image of the phenomena, since the whole operation turns on a perception of the particulars in which those phenomena agree and differ.

It is only after … the question has been (to speak technically) reduced to an equation, that the unmeaning signs become available, and that the nature of the facts themselves to which the in restitution relates can be dismissed from the mind. Up to the establishment of the equation, the language in which mathematicians carry on their reasoning does not differ in character from that employed by close reasoners on any other kind of subject.

[G]eometry and algebra are the only sciences of which the propositions are categorically true ; the general propositions of all other sciences are true only hypothetically, supposing that no counteracting cause happens to interfere. A conclusion, therefore, however correctly deduced, in point of form, from admitted laws of nature, will have no other than an hypothetical certainty. At every step we must assure ourselves that no other law of nature has superseded, or intermingled its operation with, those which are the premises of the reasoning … . [W]e must be constantly studying them; making ourselves acquainted with the peculiarities of every case to which we attempt to apply our general principles.

This approach, similar to that of  Shackle, might have been helpful prior to the financial crash of 2007/8. I have also notice in multi-disciplinary activities two quite different approaches. One follows Mill’s advice. The other is to try to work at the level of the results of each discipline. Perhaps one domain takes the lead and others try to fit in, or perhaps there is an attempt to rationalize the results of the various domains. But, as Mill says, unless the situation is quite simple, it is better if the various contributors have enough insight into their own domains so that they can – between them – determine how far each domain’s findings and recommendations can be taken, and how to modify or reform them in the light of the particulars. The same seems to be true for crisis management: people need to be educated in their domains (e.g. via experience), not just trained in its results.  

“[Apart from mathematics] we ought to wish for contrivances to make it impossible that we should ever lose sight of that meaning even for an instant.

… If any one, having possessed himself of the laws of phenomena as recorded in words …, is content from thenceforth to live among these formula, to think exclusively of them, and of applying them to cases as they arise, without keeping up his acquaintance with the realities from which these laws were collected, [he will] continually fail in his practical efforts … .

[H]e will apply his formulas without duly considering whether … other laws of nature do not modify or supersede them; but the formulas themselves will progressively lose their meaning to him, and he will cease at last even to be capable of recognizing with certainty whether a case falls within the contemplation of his formula or not. It is … necessary… that the things on which we reason should be conceived by us in the concrete, and ” clothed in circumstances”  … .”

Science of history

“[E]very human action … is the concurrent result of two sets of causes. On the one part, the general circumstances of the country and its inhabitants … . On the other part, the great variety of influences special to the individual … . If we now take the whole of the instances which occur within a sufficiently large field to exhaust all the combinations of these special influences, or, in other words, to eliminate chance; and if all these instances have occurred within such narrow limits of time, that no material change can have taken place in the general influences constituting the state of civilization of the country; we may be certain, that if human actions are governed by invariable laws, the aggregate result will be something like a constant quantity.”

It is often efficient to take these preconditions for granted, but following Mill’s advice could help avoid, manage or cope with a crisis. (There is a kind of uncertainty principle here: ignoring the preconditions leads to better short-run performance at the expense of crises. Looking out for threats and opportunities in the context helps avoid problems (and take advantage of opportunities) but diverts intellectual resources away from the short-run.)

Logic of the moral sciences

“[T]he intellectual element in mankind, including in that expression the nature of their beliefs, the amount of their knowledge, and the development of their intelligence, is the predominant circumstance in determining their progress.

… [T]he volitions of exceptional persons, or the opinions and purposes of the individuals who at some particular time compose a government, may be indispensable links in the chain of causation by which even the general causes produce their effects … .

Certain truths can not be discovered, nor inventions made, unless certain others have been made first; certain social improvements, from the nature of the case, can only follow, and not precede, others. The order of human progress, therefore, may to a certain extent have definite laws assigned-to it: while as to its [speed], or even as-to its taking place at all, no generalization, extending to the human species generally, can possibly be made … . “

Most of Mill’s book has been about induction and laws, which tend to support incremental progress. But, as Shackle covers more fully, there is another side: exploiting opportunities to bring about more radical change. One needs to spot trends, then support or inhibit them to bring about the conditions under which change is possible. But it often still needs a brave overt act (e.g., by a ‘Great Man’) to trigger change.

… Suppose that we have … discovered that a particular cause will produce the desired effect, but have not ascertained … all the circumstances which, if present, would prevent its production. If … we attempt to frame a rule of art, we perform that operation prematurely. Whenever any counteracting cause, overlooked by the theorem, takes place, the rule will be at fault ; we shall employ the means and the end will not follow. No arguing from or about the rule itself will then help us through the difficulty; there is nothing for it but to turn back and finish the scientific process which should have preceded the formation of the rule. …

[I]n the complicated affairs of life, and still more in those of states and societies, rules can not be relied on, without constantly referring back to the scientific laws on which they are founded. To know what are the practical contingencies which require a modification of the rule, or which are altogether exceptions to it, is to know what combinations of circumstances would interfere with, or entirely counteract, the consequences of those laws; and this can only be learned by a reference to the theoretic grounds of the rule.

By a wise practitioner, therefore, rules of conduct will only be considered as provisional. Being made for the most numerous cases …  they point out the manner in which it will be least perilous to act, where time or means do not exist for analyzing the actual circumstances of the case, or where we can not trust our judgment in estimating them. But they do not at all supersede the propriety of going through, when circumstances permit, the scientific process requisite for framing a rule from the data of the particular case before us.”

Logic of practice, or art

“The grounds, then, of every rule of art, are to be found in the theorems of science. An art, or a body of art, consists of the rules, together with as much of the speculative propositions as comprises the justification of those rules. The complete art of any matter includes a selection of such a portion from the science as is necessary to show on what conditions the effects, which the art aims at producing, depend. And Art in general, consists of the truths of Science, arranged in the most convenient order for practice … . Science groups and arranges its truths, so as to enable us to take in at one view as much as possible of the general order of the universe. Art, though it must assume the same general laws, follows them only into such of their detailed consequences as have led to the formation of rules of conduct; and brings together from parts of the field of science most remote from one another, the truths relating to the production of the different and heterogeneous conditions necessary to each effect which the exigencies of practical life require to be produced.

A scientific observer or reasoner, merely as such, is not an adviser for practice. His part is only to show that certain consequences follow from certain causes, and that to obtain certain ends, certain means are the most effectual. Whether the ends themselves are such as ought to be pursued, and if so, in what cases and to how great a length, it is no part of his business as a cultivator of science to decide, and science alone will never qualify him for the decision.


It seems to me that, apart from its quaint taxonomy, this provides some good insights:

  1. It has a good conceptual framework for probability and statistics, the lack of which – or perhaps a misunderstanding of which – has been implicated in many serious avoidable failures and crises. Something like this should be obligatory reading for anyone involved in all but the most routine decisions.
  2. It provides a context for Mill’s Utilitarianism which makes it seem much more reasonable.
  3. It gives some insight into a problem that I have often witnessed: of sometimes heated disagreement between scientific advisers who wish to impose their view of ‘the facts’ , and those whom they should be advising. Mill’s advise seems sound. It is not because they are scientists  that their counsel is unsound: it is because they are only used to dealing with the routine and lack experience at ‘close reasoning’ in the face of uncertainty. Mill suggests that mathematicians (as close reasoners rather than computers or solvers of puzzles) might be useful here. (I may have a bias, here 😉 .)
  4. Finally, it (indirectly) draws our attention to the need to distinguish not just between the short and the long-run, but between the short, medium and long-run. The short-run is where conventional rationality, essentially extrapolation works. The long-run is beyond our horizon. But in-between Mill describes a zone where short-sighted rationality fails but we can, usefully, do something, through ‘careful reasoning’. Maybe we should try to understand what this would mean.

See Also

Rationality bibliography

Dave Marsay

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