Quaternio terminorum is the fallacy of four terms. The standard three-line argument requires that one term be repeated in the first two lines, and eliminated from the conclusion. This is because it works by relating two things to each other by first relating each of them to a third thing. This ‘syllogistic’ reasoning depends on one term, the ‘middle term’, being repeated in the premises but disappearing from the conclusion. Where there are instead four separate terms, we cannot validly draw the conclusion, and the quaternio terminorum is committed.
John is to the right of Peter, and Peter is to the right of Paul, so John is to the right of Paul.
(This looks reasonable, but one line has ‘to the right of Peter’ where the other one simply has ‘Peter’. These are two separate terms, and the four-terms fallacy is involved. The conclusion is not validly established. After all, they could be sitting round a table.)
We might just as easily have said:
John is in awe of Peter, and Peter is in awe of Paul, so John is in awe of Paul.
(The error is more obvious. John might respect Peter for his intellect, and Peter could respect Paul for his Mercedes. Since John has a Bentley, he might not transfer his awe from Peter to the other cheap upstart.)
The fallacy arises because, strictly speaking, the terms in this type of argument are separated by the verb ‘to be’. Whatever comes after it is the term. It can be ‘the father of, or ‘in debt to’, or many other things. Unless the whole term appears in the next line, there is a quaternio terminorum. Of course, with four terms we cannot deduce new relationships between terms by using a middle term common to both – there isn’t one.
John is the father of Peter, and Peter is the father of Paul, so John is the father of Paul.
(Even your grandfather can see this is wrong.)
Now look at the example where there is a middle term repeated:
John is the father of Peter, and the father of Peter is the father of Paul, so John is the father of Paul.
(There are three terms, and this is valid.)
Quaternio terminorum can result is endless confusion in daily relationships. If John is in debt to Peter to the tune of 45 dollars, and Peter is in debt to Paul (who saved him from drowning), John might be very surprised to find Paul on his doorstep demanding money with menaces. On the other hand, if John is in love with Mary, and Mary is in love with Paul, no one except a theatre dramatist would attempt to complete the fallacious deduction.
The four-terms fallacy is more likely to appear as a source of genuine error than of deliberate deception. People may fool themselves with arguments constructed around it, but they are unlikely to fool others. There is something about the odd look of it which alerts the unwary; it is like a cheque without the amount filled in. No date perhaps; maybe even no signature; but everyone looks at the amount.
China is peaceful towards France, and France is peaceful towards the USA, so China must be peaceful towards the USA.
(You do not even need to know anything about China to know this is wrong. Just remember not to trust any relationship with France in it.)
One way to use the fallacy with a fair chance of success is to smuggle it in amongst a group of comparatives. Comparatives, such as ‘bigger than’, ‘better than’, ‘stronger than’, or ‘fatter than’, do work because they are transitive, despite the four terms. After a few of these, slip in the non-transitive relationship and it might get by.
Darling, I’m bigger than you are, stronger, and richer; yet I respect you. You stand in the same relationship to your mother, so I, in turn must respect your mother.
(I just don’t want her in the house.)