I have never really understood it either. I just go with common sense.
But after thinking of a good rule of thumb this very moment, the following has occurred to me:
P2 has to invoke the subject of P1 and not its predicate, given that P1 is universally quantified and P2 is an example of an individual (which it attempts to induce into the group represented by the predicate of P1).
I still have a problem with what "distributed" means. Wiki says:
In a categorical proposition, we assert that all (or some) members of the subject category are in (or not in) the predicate category.
For example: All S are P. Here S is the subject term and P is the predicate term.
There are four classes of categorical propositions:
All S are P
All S are not P
Some S are P
Some S are not P
Each category can be distributed or undistributed. If the proposition asserts something about
every member in the category, then we say it is distributed, and otherwise it is undistributed.
So for the four classes:
All S are P: S is distributed. P is undistributed (as we are not claiming something about
all members of P).
All S are not P: S is distributed. P is distributed (as we are saying that All P are not some S).
Some S are P: S in undistributed. P is undistributed.
Some S are not P. S is undistributed. P is distributed (as we are saying that that all P are not all S).
A syllogism connects two premises with three categories, generally of the form A -> B and B -> C. But if the B term is undistributed in both premises, then we can't guarantee that the subset of B in the first premise overlaps the subset of B in the second premise. Therefore, the conclusion may be false, and we have the fallacy of the undistributed middle.
So for example:
Underseer said:
All teabaggers are morons.
I am a moron.
Therefore, I am a teabagger.
If S = class of teabaggers, P = class of morons, Q = class of Underseer. The syllogism is
All S are P
Some Q are P (or All Q are P, it doesn't matter)
Therefore, some/all Q are S.
Here, the goal is to link Q -> S by saying Q -> P and P -> S, so P is the middle. However, P is undistributed in both premises so we don't know that the morons that are teabaggers are guaranteed to be the same morons that are Underseers (sorry Underseer
). That is the fallacy of the undistributed middle.