When we’re making arguments, a lot of it is putting things in categories and having ideas about them, and things go awry when the ideas are far removed from reality. Think of the Holocaust (I know I keep focusing on extreme examples, but it’s easier to illustrate poor logic this way); the argument was about a group of people and their characteristics. Most people would argue (yes!) that the argument was illogical and it was far removed from evidence (Jewish people being the cause of pre-war Germany’s economic woes).
So there’s a mistake that can be made when making these groups called the non-distributed middle, where we make statements about groups and then an inference, but really we make an unconscious leap away from logic. It becomes obvious when you draw the groups as a venn diagram to see what overlaps. For example
No men are daughters.
So this is the major premise. There are two groups. Men and daughters. Men are a separate group from daughters.
Fig one
You draw a circle with MEN in it, and you cannot write daughters in this circle because then they would be men also, and that isn’t possible as this is a premise, i.e., a statement we are making that we know to be true.
Fig two
If you make another circle with the word DAUGHTERS in it, then you cannot put another circle with the word MEN in it, because some men would be daughters, and the premise (what we know is true) doesn’t allow it (as we’re assuming this is a sound (true) premise now).
Fig three
Also, we can’t overlap these two circles, because then it would contradict the premise and men could be daughters. So, there are two groups that are not connected in any way—there is no overlap.
Fig four
So now we add a minor premise, another thing we know to be true.
No waitress is a man.
Fig five
So we add another circle, ‘waitresses’. We cannot put it inside ‘men’ because then a daughter could be a man, and so it would contradict the first premise.
Fig six
It could be inside ‘daughter’ without contradicting the premises,
Fig seven
So then the last conclusion is:
No waitress is a daughter.
But the conclusion is illogical because the second premise doesn’t disconnect the two groups, waitresses and daughters.
Figure eight
[recap of the statements:
No men are daughters
No waitress is a man – this does not disconnect waitresses and daughters
No waitress is a daughter]