The fallacy of the undistributed middle is a form of syllogistic fallacy. Syllogisms consist of three categorical propositions, two assumptions and a conclusion, which are statements of the form (all, some,or none) (items in category A) (are or are not) (items in category B). For example, “all dogs are animals,” would be an example of a categorical proposition. If we combine this with the categorical proposition, “all poodles are dogs,” we can conclude from these two assumptions that, “all poodles are animals.” The logic we have used to make this conclusion is known as syllogistic logic. Here is another example:
No vegans eat meat;
Some raw foodists are vegan;
Therefore, some raw foodists do not eat meat.
If we accept that the first two assumptions are true, then it necessarily follows from those assumptions that the conclusion, some raw foodists do not eat meat, is true as well. There are a limited number of forms syllogistic statements can take. All vegans are not people who eat meat, is the same as saying no vegans eat meat, and vice versa, so there are in actuality 4 different forms a categorical proposition can take and a syllogism consists of 3 individual categorical propositions, so there are 4^3 or 256 possible forms a syllogism can take. Not all of these forms result in conclusions that are always true. For example:
Some raw foodists are vegan;
Some meat eaters are raw foodists;
Therefore, some meat eaters are vegan.
While both of the assumptions above are true, the conclusion is clearly false. Up to this point we have kept the three different categories being used in the syllogism in the same order, but the order can move around so long as three different categories are introduced in the two assumptions, the category that repeats does not repeat in the same assumption, and the conclusion incorporates the two categories that had not yet repeated. For example:
All dogs have tails;
I do not have a tail;
Therefore, I am not a dog.
In this case, the categories, dogs, (things that) have tails, and me, appear in a different order, but in this case we still have a syllogism which (if we accept the premises) yields a conclusion which is necessarily true. Changing the order in which the categories appear does change which syllogisms work and which do not.
Syllogistic logic is something you likely do every day without being consciously aware of it. Notice that statements like, “If someone is vegan then they do not eat meat,” means the same thing as, “all vegans are people who do not eat meat.” The fallacy of the undistributed middle is an example of a syllogism that does not work. It takes the form:
All A are B;
All C are B;
Therefore, all C are A.
All A are B;
Some C are B;
Therefore, Some C are A.
All fruitarians are vegan;
I am a vegan;
Therefore, I am a fruitarian.
In this case the two assumptions may both be true, but the conclusion does not necessarily follow from the assumptions (I assure you that I am not a fruitarian). In this instance it is easy to see that the logic is flawed, but when we incorporate this logic into everyday speech it becomes more difficult to see:
If someone is vegetarian then they never eat meat,
Bob never eats meat,
Therefore, Bob is vegetarian.
In this case the conclusion is necessarily true, but the reason why is unclear. When we said, “if someone is a vegetarian then they never eat meat,” we also meant to imply, “if someone never eats meat, then they are a vegetarian.” This extra assumption can then be used to form a valid syllogism. Sometimes when we use the “if” statement it doesn't work this way:
If someone wears fur, then they do not care about animal rights;
Bob does not care about animal rights;
Therefore, Bob wears fur.
The conclusion that Bob wears fur does not necessarily follow from the assumptions because in this case we did not imply that everyone who doesn't care about animal rights also wears fur.
Here is an example were this may come up in reality.
If someone eats too much meat then they are going to be unhealthy. Bob is unhealthy. We cannot immediately assume that Bob is eating too much meat and that is the source of his poor health.
Check out next week's fallacy, illicit minor.