Sometimes the best way of coming to understand something is by seeing an example of what it isn't, or as in this case, someone else's misunderstanding. Peter S. Williams provides us with this opportunity. For someone who claims to be a philosopher, to make such "school boy errors" in logic is extraordinary. That's why I used the word "claims". Perhaps I'm being overly harsh. He writes a lot, and the more you write, the more opportunity for error. However, in this article, Darwin’s Rottweiler and the Public Understanding of Scientism, he claims Richard Dawkins is guilty of making many fallacies in his arguments. I think Williams is wrong in nearly everything he says. (Here's a guy who claims the non-answer: "God did it" is scientific - he even "trumpets" peer reviewed paper on Intelligent Design. The worth of peer review depends on your peers. It was published in Philosophia Christi and is available via Discovery.org. Was it that he was still waiting for some experimental data - so wasn't quite ready for Nature or Science?) As such, I have neither the time, inclination, or respect to deal with all his misidentifications of fallacies. (Besides the above, he wrote a book on angels - gimmie a break, and a fan guide on both Doctor Who and The Matrix... see the first link.) I will probably return to it in the future, as "there's gold in them there hills".
For the moment, I'll deal with this specific example, where Williams thinks he has spotted an example of Begging the Question. He is wrong. The following italicised text is his work:
2. Begging the Question - ‘this fallacy occurs when a disputant uses his conclusion as one of the premises employed to establish [his] conclusion.’ 
Dawkins asserts that: ‘As time goes by and our civilization grows up more, the model of the universe that we share will become progressively less superstitious, less small-minded, less parochial. It will lose its remaining ghosts, hobgoblins and spirits, it will be a realistic model, correctly regulated and updated by incoming information from the real world.’  How can Dawkins know this assertion is true before all the evidence is in? Dawkins assumes that his conclusion is true and then promises that it will be justified on evidential grounds at some unspecified point in the future.
This is not an example of Begging the Question as Williams claims. There is no premise in Dawkins’ statement. There is also no conclusion. Ergo, it’s not Begging the Question. It is simply an assertion. You can choose to agree or disagree with it, but it’s certainly not Begging the Question. Begging the Question would be more like:
‘As time goes by and our civilization grows up more, the model of the universe that we share will become progressively less superstitious, less small-minded, less parochial. It will lose its remaining ghosts, hobgoblins and spirits, it will be a realistic model, correctly regulated and updated by incoming information from the real world.’ [My added bit that would turn this into Begging the Question follows] This will occur because as we build more realistic models that are based on good information, we rely less on finding supernatural, small-minded and parochial answers for hitherto unexplained phenomena.
Note the bolded word in my added section - because. This clearly shows this is the sentence I’m using to try and justify the claim in the initial section. But of course, it’s simply the initial claim reworded to sound like a justification. Now it is a Question Begging argument. The actual statement itself may or may not be true (I happen to think and hope that it is/will be true). But the point is it can only be Begging the Question if a claim is made, and then attempted to be justified using the same point as was made in the original claim.
As I said above, Dawkins could only be accused of making an unsupported assertion. Every sentence simply repeats this assertion in a different manner for more emphasis. Reading it again, with emphasis on the right words makes this quite clear:
‘As time goes by and our civilization grows up more, the model of the universe that we share will become progressively less superstitious, less small-minded, less parochial. It will lose its remaining ghosts, hobgoblins and spirits, it will be a realistic model, correctly regulated and updated by incoming information from the real world.’
By only saying “it will” Dawkins is only making a claim. "It will be this, it will be that..." We need a “because”, or an “as”, or a “since” or whatever synonym Shift F7 delivers, to have some kind of justification. In saying this, for all we know Dawkins has been taken out of context and did forward some reasons for this belief. Further to this, my version would not be Begging the Question if the “because” I added was demonstrably true. I’d suggest this is possible to do. The most obvious would be we can see this occurring as a simple historical fact. Over the centuries, with more and more science, or more and more education, humans become less and less superstitious. The data support this – once someone has a university degree their superstition dies of dramatically. If we accept this premise, and add to that as we move forwards in time we will have the required economic and social capital to continue with scientific research and (universal) education, the conclusion – less superstitious etc., follows.
Of course, off the top of my head I can’t remember where I read this. I think it was some really impressive journal, a paper by some super smart researcher, so you’ll just have to take my word for it… you trust me don’t you?
Update: see Red Flax Faux Pas