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What is The Undistributed middle fallacy?

Classic among schoolboy fallacies is the argument that because all horses have four legs and all dogs have four legs, so all horses are dogs. This is the simplest version of the notorious fallacy of the undistributed middle. Both horses and dogs are indeed four-legged, but neither of them occupies the whole of the class of four-legged beings. This leaves convenient room for horses and dogs to be different from each other, and from other beings which might also without any overlap be in the four-legged class. The ‘middle’ which carelessly omitted to get itself distributed is the term which appears in the first two lines of a three-line argument, but which disappears in the conclusion. The classic three-liner requires that this middle term must cover the whole of its class at least once. If not, it is undistributed.

All men are mammals. Some mammals are rabbits, therefore some men are rabbits.
(Even though the first two lines are correct, the middle term 'mam-mals' never once refers to all mammals. The middle term is thus undistributed and the deduction invalid.)

Common sense shows why the undistributed middle is fallacious. The standard three-liner (called a ‘syllogism’) works by relating one thing to another by means of a relationship they both have with a third. Only if at least one of those relationships applies to all the third thing do we know that it is certain to include the other relationship.

We cannot say that bureaucrats are petty tyrants just because bureaucrats are meddlesome and petty tyrants are meddlesome. It is quite possible that gin-sodden drunks are meddlesome too, but that does not mean that bureaucrats are gin-sodden drunks. (Life might be more interesting if they were.) This fallacy com-monly appears in the form of ‘tarring with the same brush.’

The worst oppressors of the working class are landlords. Jones is a landlord, so Jones is one of the worst oppressors of the working classes.
(Exit Jones, hurriedly, before it is pointed out that the worst oppressors of the working classes are human. Since Jones is human…)

The great thing about undistributed middles is that you can undistribute new ones as further ‘evidence’ in support of your previously undistributed ones. (The worst oppressors of the working class wear shoes; Jones wears shoes…)

The expert user will take the trouble to find out which terms are distributed or undistributed. He will learn the simple rule: ‘Universals have distributed subjects, negatives have distributed predicates.’ Universals are statements which tell us about all or none of a class, and negatives tell us what isn’t so. Armed with this technical information, the expert is able to inflict upon his audience such monstrosities as:

All nurses are really great people, but it happens that some really great people are not properly rewarded. So some nurses are not properly rewarded.
(It may be true, but has he given an argument? Since the middle term 'really great people' is neither the subject of a universal, nor the predicate of a negative, it is not distributed. We have here, therefore, a very complex fallacy of the undistributed middle.)

Leaving aside these technical uses, the fallacy in its simple form will give hours of pleasurable success if applied systematically. You should use it to gain approval for what you favour by pointing out how it shares qualities with something universally admired. Similarly, opposing ideas can be discredited by showing what qualities they share with universally detested things.

The union closed shop is the will of the majority; and democracy is the will of the majority. The union closed shop is only democratic.
(Where do I sign? [You did.])
Elitism is something only a few benefit from, and tennis is something only a few benefit from, so tennis is clearly elitist.
(Fault!)