The standard three-line argument called a syllogism has two premises and a conclusion; the premises are the evidence and the conclusion is deduced from them. If both of the premises are negative, no conclusion can be validly drawn from them and the fallacy is called the fallacy of exclusive premises.
No handymen are bakers, and no bakers are fishermen, so no handymen are fishermen.
(It seems innocent enough, but the logic is fishier than the handy-men. If we had used ‘tax-dodgers’ instead of fishermen, we would have ended up saying ‘no handymen are tax-dodgers’, which everyone knows is untrue. The fault lies with the two negative premises.)
The source of the fallacy is fairly clear. The three-liner relates two things to each other by means of the relationship which each has with a third. When both premises are negative, all we are told is that two things lie wholly or partly outside the class of a third thing. They could do this however they were related to each other, and so no conclusion about that relationship can be drawn:
Some brewers are not idiots, and some idiots are not rich, so some brewers are not rich.
(Did you ever hear of a poor one? With two negative statements, the idiots who are not rich do not need to be the same ones who don’t include the brewers among their number. If this sounds confusing, remember two things: two negative premises do not prove anything, and all brewers are rich.)
The fallacy tends to occur because some people genuinely believe that if a group is excluded from something, and that group in turn is excluded from something else, then the first group is also excluded from it. If John cannot get into the Masons, and the Masons cannot get into the country club, it seems plausible to assume that John doesn’t stand a chance for the country club. Of course, since the Masons cannot get in, John might stand a better chance because he isn’t one of them.
No pudding-eaters are thin, and some smokers are not pudding-eat so some smokers are thin.
(Many of us are negative about puddings, but two negative statements about them will not tell us anything about smokers. If smokers are thin, it might well be from worrying about the health warnings, and a lack of money to eat puddings after paying for the cigarettes.)
When you want to use the fallacy of exclusive premises, you should try to make your negative statements ones which the audience will accept the truth of. When you slip in a conclusion which seems plausible, they will assume that you have proved it. You will not get very far if you start out with statements such as ‘No council workers are lazy’, but should try instead to keep within the bounds of your audience’s experience. Use obvious truisms such as ‘No removal men are careful.’
The above article is from the book How To Win Every Argument by Madsen Pirie. The article is only for educational and informative purposes to explain and understand formal logic and logical fallacies. It is a great book, definitely worth a read!