Have you ever begged a question? You probably have and perhaps didn’t even know it at the time. It is a technical term in logic, also known as petitio principii, meaning “a fallacy in which a premise is assumed to be true without warrant.” You could also call it circular reasoning, as in the following unsuccessful attempt to prove a statement:
Dick (accusingly): Jane committed perjury!
Perry Mason (with aplomb): How do you know Jane committed perjury?
Dick (sputteringly): Because...she wasn’t telling the truth!
If that’s all the testimony Dick can muster, Perry will win yet another case, and Jane will get off scot-free.
In recent years, however, some people—even some who possess college degrees and unabridged dictionaries—have begun to use “beg the question” to mean “avoid asking the proper a question, or fail to provide an answer.” After providing the technically correct definition (i.e. the logical fallacy), The Oxford Companion to the English Language adds this second definition--with a caveat: "Avoiding giving an answer or facing an issue. Henry Fowler in Modern English Usage calls the second sense 'a misapprehension of which many writers need to disabuse themselves.'”
A Dictionary of Modern American Usage is absolutely dogmatic: "'Begging the question' does not mean 'evading the issue' or 'inviting the obvious questions,' as some mistakenly believe. The proper meaning is 'basing a conclusion on an assumption that is as much in need of proof as the conclusion itself.'"
All this suggests rather conclusively that using "beg the question" to mean "avoid answering a question" is sub-standard.
But wait! Help for the sub-standard among us is on the way!
In Webster's Collegiate Dictionary" (2007) this is the entire definition for "beg the question": "1. to pass over or ignore a question by assuming it to be established or settled. 2. to elicit a question logically as a reaction or response."
The meaning of the term has evolved, and linguistic purists, kicking, screaming and pulling their thinning hair, must forgo attempts to put this wayward genie back into the bottle.
The Bard of Buffalo Bayou, whose unerring logic beggars all description, begged that this meretricious bit of doggerel be appended to this otherwise scholarly monograph:
I tried to beg the question,
But you said “Not a chance.”
And then I begged to differ,
As you just looked askance.
I asked to beg a favor,
Then I was double-crossed.
And when I begged your pardon,
You told me to get lost.