The fallacy of the undistributed middle is a logical fallacy that is committed when the middle term in a categorical syllogism isn't distributed .

For example:

All students carry backpacks.

My grandfather carries a backpack.

Therefore, my grandfather is a student.

The middle term is the one that appears in both premises - in this case, it is the class of backpack carriers. It is undistributed because neither of its uses applies to all backpack carriers. Therefore it can't be used to connect students and my grandfather - both of them could be separate and unconnected divisions of the class of backpack carriers.

However, if the latter two statements were switched, the syllogism would be true:

All students carry backpacks.

My grandfather is a student.

Therefore, my grandfather carries a backpack.

In this case, the middle term is the class of students, and the first use clearly refers to 'all students'. It is therefore distributed across the whole of its class, and so can be used to connect the other two terms (backpack carriers, and my grandfather).

It is a syllogistic fallacy.