Mr. Lukeman defends his position by citing Horrid Examples. Not a single one of them is convincing, let alone persuasive. He posits this sequence: "He ran with his shirt over his head. He had forgotten his umbrella once again." The passage would be vastly improved, he suggests, by replacing the period with a semicolon. The emendation "lends an appropriate feeling of connection."
With appropriate deference, pooh! With or without the surreptitious semi, you wind up all choppy. Any good editor could suggest half a dozen things to do with those two sentences, such as shooting them. One could try, "Having stupidly left his umbrella in the trollop's boudoir, he pulled his shirt over his head and fled." The sentence still needs some polishing, but c'mon! At least you don't have to back up and read it twice.