Other counterexamples are not decisive. Indeed, some purported coun- terexamples miss their targets entirely. If a person claims that all snakes except rattlesnakes lay eggs, someone might respond with another counterexample: male snakes. This counterexample does not really refute the intended claim, since that claim was meant to be about the methods by which female snakes of various species give birth when they do give birth.
When a counterexample can be answered with a simple clarification or modification that does not affect the basic force of the original claim, it is a shallow counterexample. A deep counterexample is one that requires the orig- inal claim to be modified in more important or interesting ways. Shallow counterexamples can sometimes be fun as jokes, but they are usually not much help in refuting arguments, since basically the same argument can be resurrected in a slightly different form. Indeed, people who give too many shallow counterexamples can be annoying. If you really want to understand a subject matter, you should look for counterexamples that are deep.
Deep and decisive counterexamples are not always easy to think up, but Socrates was a genius in this respect. In one dialogue by Plato, Theaetetus and Socrates are trying to define “knowledge.” They notice an important difference between knowledge and mere belief: It is possible for someone to believe something that is false, but it is not possible for someone to know something that is false. This leads Theaetetus to suggest a simple definition of knowledge: Knowledge equals true belief. This proposed definition is re- futed in the following exchange:
Socrates: [There is] a whole profession to prove that true belief is not knowledge.
Theaetetus: How so? What profession?
Socrates: The profession of those paragons of intellect known as orators and lawyers. There you have men who use their skill to produce con- viction, not by instruction, but by making people believe whatever they want them to believe. You can hardly imagine teachers so clever as to be able, in the short time allowed by the clock, to instruct their hearers thoroughly in the true facts of a case of robbery or other vio- lence which those hearers had not witnessed.