1. Definition

Case-by-case argumentation is a technique of inquiry developing in several stages, from questions like “What happened, what can happen?”:

— First, make an inventory of possible cases.
— Second, consider each of these cases.
— Third, sum up the cases considered and see if examination leads to the elimination of all possible cases but one.
— Fourth, conclude that the last remaining case should be chosen or is real and true.

S1       — All this money, either comes from a legacy or is your labor, or has been stolen. If it comes from your labor income or from a legacy, it’ll be easy for you to prove it by showing us the relevant documents. No documents of that kind available? So you stole it.

This argument illustrates the classical law of negation of a disjunction, S. Connectives:

“P or Q or R” is true; but P is false and Q is false; so necessarily, R is true.

Definitions can be given on a case-by-case basis. A crime, such as impiety, might be defined as a lack of respect for either the gods, their priests or their shrines. To accuse someone of impiety (or to exonerate oneself from that crime) one must show that one of the three defining conditions has been disrespected (or not) (after Aristotle, Rhet., II, 23, 1399a5; RR p. 367).

2. Argument by division

Argument by division is illustrated by the following example:

The tyre exploded because it was worn out, because there were nails on the road, or because of a manufacturing defect. Now, the tyre had just been bought and no nails were found in it. So there was bad workmanship. (Perelman, 1977, p. 65)

This shows that the label “argumentation by division” is homonymic: it can refer either to the argumentation by composition or division, or to the case-by-case argument.

3. Refutation of the case-by-case argument

A case-by-case argument is perfectly conclusive if all cases have been considered; it can be rejected on the same case-by-case basis by showing that the enumeration of cases is incomplete:

S2 (as a reply to S1, supra): — No Sir, I just won the lottery, here is the winning ticket!

S3 (as a reply to Perelman, supra) — Well, Sir, here are some other possibilities. The tire might have exploded because it was badly inflated, because there was a pothole on the road, because it hit the curb, because it has been overheated (if the driver happens to have just used a torch to unscrew a wheel bolt), because the brake was glued, because it had been brought into contact with an electrical source, because the car was too loaded or was running too fast… My conclusion is that the investigation must go on.