The basic idea behind an argument from silence (argumentum ex silentio), is to argue for a conclusion based on lack of evidence. Of course, given that this is the skeptic's favorite form of argument, we shouldn't be surprised to see it cropping up often. In fact virtually any argument can be made based on silence or lack of evidence (or supposed lack of evidence) - which is exactly why skeptics love it so much. Now, an argument from silence can be a good argument under the right circumstances. For example, if you had an exhaustive list of U.S. Presidents, then you could prove that Benjamin Franklin was never the President of the U.S. by pointing to the fact that his name is not on that list. That would be a good argument from silence.
One skeptic I saw tried to "prove" that there was no Nazareth at the time of Jesus (a rather popular skeptical argument, especially among the looney-tune "Jesus was a myth" crowd) by means of various arguments from silence. The arguments for this are that 1) Nazareth is not mentioned in the OT, 2) historical sources from the time of Jesus don't say anything about Nazareth, 3) Josephus doesn't mention Nazareth, but does mention 40-50 cities and towns of Galilee. Notice that all three of these are forms of argument from silence - some particular source doesn't say anything about Nazareth, therefore we conclude that Nazareth didn't exist at the time that source was from.
For the moment we'll leave aside the archaeological evidence that Nazareth was a city which was populated during Jesus' time. I'll get to that at the end. Let's just consider these arguments on their face.
1) The fact that Nazareth is not mentioned in the OT means absolutely nothing. The last of the OT was written 400 years or so before Jesus was born, so even if we could read into the silence of the OT on Nazareth, it wouldn't mean that there was no city of Nazareth in Jesus' day. This is an awful argument from silence, completely brainless. Strike one.
2) In fact we do have historical sources from that time period that talk about the city of Nazareth - namely Luke and Matthew. While the skeptic might want to say that these don't count, in fact they count just as much as any other historical source from the time period. Elsewhere I've presented some of the copious amounts of evidence for Luke's accuracy as a historian. Even in many cases where scholars thought Luke made this or that error, it later turned out that Luke was right and the critics were wrong.
Now, the skeptic might be tempted to argue that this is circular reasoning - to use the Gospels to prove that the Gospels are accurate. But this is a separate question than simply asking whether there was a city of Nazareth in the time of Jesus. It would be completely bogus, for example, to take a historical source that talks about some city, and say that this particular source doesn't count in determining whether that city existed. And in this case you have four sources saying that Jesus was from Nazareth, two of them saying that Nazareth was a city (polis). In any other context, this would be about as conclusive as you can get that Nazareth existed in Jesus' time (keeping in mind that the argument is not whether there was a city of Nazareth, but whether it was populated in the time of Jesus or not).
So this second argument fares little better than the first one, particularly given the archaeological corroboration. Strike two.
3) This third argument sounds more plausible on the surface. Josephus never mentions Nazareth, and he was the governor of Galilee for awhile. He mentions by name 45 cities and villages from Galilee, and yet Nazareth is not among them. Could this be a good argument from silence?
To answer that, the first question we have to ask is whether or not he mentioned all or even most of the cities and villages of Galilee. It was actually in trying to find the answer to that question that I came upon a discovery. Josephus himself, governor of Galilee, tells us how many cities and villages there were there. In his autobiography (The Life of Flavius Josephus), 45th chapter, he writes that there were 240 cities and villages in Galilee. He wrote this in a letter to one of his enemies, saying he would be willing to meet him in any of those places except for two of them which were in league with him.
This simple fact changes what looks like a plausible argument from silence into an utterly disastrous one. While it might sound like a big deal that Josephus mentions 45 cities and villages in his writings, that's not quite so impressive when you realize that that is less than 19% of the total. In other words, more than 80% of the cities and villages of Galilee have not been mentioned by him, or 4 out of every 5. It isn't even necessary to argue that Nazareth wasn't a very big city in his time to explain why it might not have been mentioned by Josephus - the majority of them weren't mentioned. This likewise renders moot the related argument sometimes used by skeptics that Nazareth is not included among the 63 Galilean cities and villages mentioned in the Talmud, either. Even 63 is only about 25% of the total. Even if you assumed that the ones mentioned by Josephus don't overlap at all with the 63 mentioned in the Talmud (which seems pretty unlikely though I haven't seen an actual comprehensive list of either), that would still be less than half! It would still only be 45%. So neither of these arguments are actually any good, in spite of their apparent plausibility (and the fact that they get widely repeated on the internet by skeptics who simply copy and paste bad arguments from each other). Strike three. That's called a strikeout.
Finally, a word about archaeology. The following three links deal with recent archaological finds in the area around Nazareth. Some scholars have suggested that it was inhabited at the time of Jesus, but was a small village. In that case, Jesus still could have been from Nazareth, but calling it a city would have been a mistake. However, as the last two links show, evidence of an early Roman bath house in Nazareth suggests that it may have been larger than believed. As the article in the Guardian (that well-known fundamentalist rag [irony alert]) notes, there actually hasn't been much archaeological work done in Nazareth for some reason, which is all the more reason why arguments from silence based on lack of archaeological evidence from Nazareth is a bad idea.
The skeptic will likely argue that the archaeological evidence doesn't count for this or that reason, such as the fact that it involves religious artifacts and could be a boon to tourism. Well, certainly finding a bath house that Jesus might have used (though that would be difficult to prove) would be a boon to tourism, but that has nothing to do with the authenticity of the find. Virtually any important archaeological find is going to be a boon to tourism for somebody. And while I don't suggest that these finds are conclusive proof (history rarely works that way), they certainly give enough reason to say that the conclusion of the skeptic that Nazareth didn't exist at the time of Jesus is unwarranted to put it mildly. So remember this the next time some skeptic repeats one or more of these bad arguments.