Examples can be useful forms of evidence. In fact, in conversation, we often ask people to "give me an example," meaning: support the point you just made with an example. For instance, your child comes home and says, "My teacher is not fair and does not like me." You would most likely ask your child to give you some examples to support this claim. What could your child say back to you that would support the claim about the teacher?
Here are some possibilities: "She never calls on me," "He laughs when I give the wrong answer," and so on. These examples clearly support your child's claim, and so they would be considered useful. What if, instead, your child said, "She wears the same outfit every day" or "He drinks too much coffee"? Unless the teacher wears a shirt that says "I hate little Bobby" on it every day, you would say, "What does that have to do with how much he/she likes you?" These examples do not support the claim. This example is an exaggeration, of course, but oftentimes people will use examples as evidence even though they really do not support their claims. You have to be on the alert for this as a writer and reader.
Examples should be used sparingly. They work best as additional supporting evidence, but should not be relied on exclusively. The reason why is that the example may not be representative of the general situation. This is referred to as an anecdotal fallacy. For example, "Smoking cannot be bad for you—my uncle smoked two packs a day and lived to be 102." This one example is an oddity; it does not represent the normal experience of a life-long heavy smoker, plus it was chosen specifically to the exclusion of all other evidence. It ignores all the counterexamples out there.
The last tool used in developing arguments comes from within. Authors use logic to build one known or proven fact upon another, leading the reader to agree that a certain point is true. For example, consider the case of Frank Frank has six cats. To prevent having even more cats, all Frank's cats are female. Now consider his cat Zoe. What is Zoe's sex? We know that ALL Frank's cats are female, and we know Zoe belongs to Frank, so we can conclude that