(1) Paternal government and despotic government. Paternal government is good, so too will despotic government be good. This is a false analogy, says Mill, because it implies that despotic government shares with paternal government that feature which makes it good, namely, that there is a responsible authority (who cannot be the children in the familial case or the general populace in the case of the state); but, in fact, the goodness of paternal government is due to two features it does not share with despotic government, namely the superior experience of the parents and the affection of the parents toward their children.
(2) Individual stages of life and political stages of life. As human beings and animals go from youth to maturity to old age to death, so do nations. But in human beings and animals this progress is due to "the natural progress of those very changes of structure which, in their earlier stages, constitutes its growth to maturity" but political bodies die not because of the progress of their structures but because of the stagnation and regression of their structures.
(3) Immutability and unpunishability. An argument made by Hooker: As natural bodies require an unmovable mover in order to have motion, so political bodies require an unpunishable member in order to have punishment. Mill doesn't give a specific reason for the falseness of the analogy; he gets distracted by the fact that he also thinks its premise (about immovable movers) is false.
(4) Monetary utility and utility of resources. An example taken from Whately: “It would be admitted that a great and permanent diminution in the quantity of some useful commodity, such as corn, or coal, or iron, throughout the world, would be a serious and lasting loss; and again, that if the fields and coal-mines yielded regularly double quantities, with the same labor, we should be so much the richer; hence it might be inferred, that if the quantity of gold and silver in the world were diminished one-half, or were doubled, like results would follow; the utility of these metals, for the purposes of coin, being very great. Now there are many points of resemblance and many of difference, between the precious metals on the one hand, and corn, coal, etc., on the other; but the important circumstance to the supposed argument is, that the utility of gold and silver (as coin, which is far the chief) depends on their value, which is regulated by their scarcity; or rather, to speak strictly, by the difficulty of obtaining them; whereas, if corn and coal were ten times as abundant (i.e., more easily obtained), a bushel of either would still be as useful as now. But if it were twice as easy to procure gold as it is, a sovereign would be twice as large; if only half as easy, it would be of the size of a half-sovereign, and this (besides the trifling circumstance of the cheapness or dearness of gold ornaments) would be all the difference. The analogy, therefore, fails in the point essential to the argument.”
(5) Metropolis and heart. Also an example taken from Whately: As the metropolis is like a heart, and as significant increase of the heart is a disease, so too significant increase in the size of the metropolis is a disease of the political body.
(6) Pythagorean analogies. There are proportions in the monochord that give music; there are similar proportions in astronomy, and thus a music of the spheres, "as if the music of a harp had depended solely on the numerical proportions, and not on the material, nor even on the existence of any material, any strings at all." Likewise there are various numerological and cosmological speculations through history that later are held to be fanciful. These often identified real resemblances (proportions and the like) but drew the wrong conclusions from them.
(7) Technical defect and moral defect. From arguments of the Stoics, an example taken from Cicero. Suppose a number of lyres that are out of tune; we can say that they are equally out of tune; therefore all departures from the rule (of right living) are equally departures from it, and so all are equal. Likewise a navigator has equally failed at his task whether he loses a ship carrying straw or he loses a ship carrying gold; likewise, it doesn't matter whether a man beats his father or his slave if he does it without due cause.
Some of these are interesting; with regard to (7), for instance, Cicero himself identifies the problem as equivocation, which makes for dissimilar similars. Mill rejects the navigator/abuser analogy because he holds that a failure of skill and a moral failure differ precisely in that interest is irrelevant to the former but not the latter; Cicero rejects it because the actual cargo doesn't affect our assessment of skill but who gets beaten does affect our assessment of justice. (But Cicero also denies the premise of the comparison, saying that when we consider negligence, it does matter whether the cargo was straw or gold.) But what's notable is that every single one of the responses that clarifies why the analogy is false argues that it is a false analogy because the causes involved in each are not really and truly the same kinds of causes. As he explains:
In these and all other arguments drawn from remote analogies, and from metaphors, which are cases of analogy, it is apparent (especially when we consider the extreme facility of raising up contrary analogies and conflicting metaphors) that, so far from the metaphor or analogy proving any thing, the applicability of the metaphor is the very thing to be made out. It has to be shown that in the two cases asserted to be analogous, the same law is really operating; that between the known resemblance and the inferred one there is some connection by means of causation.
That is, each analysis purporting to show that the Fallacy of False Analogy has been committed is simply an argument that the similarities have been used improperly as a basis for generalization.
This Fallacy [of False Analogy] stands distinguished from those already treated of by the peculiarity that it does not even simulate a complete and conclusive induction, but consists in the misapplication of an argument which is at best only admissible as an inconclusive presumption, where real proof is unattainable.
It looks very much like analogical inferences are not allowed defeasibility on Mill's account; the very failure of the inference to get the right conclusion is itself sufficient evidence that it commits the Fallacy of False Analogy. This is an unusual use of the word 'fallacy'; fallacious reasoning isn't usually taken to be proven by the mere fact that you got a wrong answer. But this broad usage is not unique to Mill; Bentham regularly uses the word in something like this way, too. And it ties in with the fact that Mill thinks of 'fallacy' as naturally connected to evidence: a fallacy is a case where apparent evidence is not real evidence or apparently conclusive evidence is not actually conclusive, so that a false conclusion is drawn. Thus the very test of a fallacy in Mill's sense is that the evidence results in a false conclusion, or involves a false assumption. So, for instance, Mill says (3) is doubly fallacious: it involves false analogy and it inolves the scholastic error of thinking that motion requires an immutable mover. A great deal of philosophical content is being packed into this understanding of fallaciousness. When you take it in this sense, one sees that all the false analogies are analogical inferences that Mill thinks result in incorrect conclusions. This is all well and good for people who agree with Mill about (for instance) the nature of the body politic, but it means that Mill's conclusions about the Fallacy of False Analogy -- including the claim that there is such a fallacy -- are not necessarily consistent with non-Millian views.