In a categorical syllogism, the middle term occurs in both premises and is absent from the conclusion. It connects the major predicate with the minor subject.
For example: “This metal is being heated”; “Metal heated under these conditions expands”; therefore “This metal expands.” Being heated supplies the shared link in this controlled illustration.
Simply repeating a word does not establish an inference. The form must be valid and the relevant premises justified. We must then ask whether the middle term is also a cause in the world.