The Washington Post has corrected itself:
A Jan. 19, 2008, Metro article incorrectly described the Korean language as using symbols. It has an alphabet.
It's true that Korean uses a writing system that represents individual phonemes in words.
And certainly the editors at the Post know that an alphabet is a system of symbols. So I figure they're correcting their assumption about what those symbols represent.
Most simply: writing systems can represent the sounds in a word, the meaning of a word, or both. Those that primarily represent sound are phonographic and can be either phonemic or syllabic. Those that primarily represent meaning are morphographic.
If the system represents the sounds, it can represent each identifiable sound in an ideal 1:1 relation with the symbol. So the sound we make when referring to that thing I wear on my head, has three sounds that have an effect on our understanding of the utterance. It starts with a [h] then there's an [æ] and finally a [t]. When I write it out you can see that the symbols I use represent the sounds I would make if saying the word instead of writing it. If I want to spell out the word for the wooden contraption that I hang my hat on, I would add three more symbols to represent the three added sounds. in addition to [hæt] I would add [ɹæk] (Notice that the 1:1 relation between English spelling and sound is notoriously imperfect.)
If I'm going to create a syllabary, I would use only one symbol for the first word [hæt] and two symbols for the second word [hæt.ɹæk]. In a well ordered system, I would go ahead and use the symbol I used for the first word because the syllable is the same. The second syllable would have its own symbol, and here's a telling analysis—there would be no clue that that vowel is the same in both syllables. Even if the syllables were almost identical—imagine I keep my hat on a hatmat—the syllabary would merely represent these as Syllablex and Syllabley. The components of the syllable aren't represented.
Regardless of the semantic intention of the sounds or syllables, these symbolic systems represent the sounds that are uttered. The last syllable of diplomat and the first syllable of Matlock would look the same within a syllabic and an alphabetical system. (Exceptions do exist because of factors outside the writing system. We'll not get into that here.)
A logographic system does not represent the sounds, but represents the meanings of words. A purely semantic representation would ignore the sounds and even the syllables of a word, instead representing either the whole concept with a symbol or symbols in combination. (This fact is of course abused by such ongoing silliness as the crisis/opportunity canard perpetuated by nobel prize winners who should know better than to repeat something that has been sufficiently discredited.)
So let's create our own little logographic system here: The thing on my shoulders (my [hɛd]) might be written out as 'ɑ'. Then I need a word for that saintly glow also known as a halo. So I write @. The symbol for 'halo' has within it the symbol for 'head' but the symbol isn't guiding my pronunciation in any compositional way. It's pure convention that allows me to use the logogram to figure out the sounds that represent the same thing.
If I could access the original story by the Post, I would have a better understanding of the original claim that was made. It's likely that the story did misrepresent the Korean writing system by claiming that it's not a phonetic system. In that case a correction is warranted. But a more precise correction wouldn't imply that the alphabets and syllabaries are not also symbolic systems.
I noticed (as q-pheevr also mentions in the comments) that the correction comes along pretty late. But if you note the other corrections on the Post's page, this one hasn't had the longest wait. The comments on the article mentioned this error as soon as it came out -- that very day. Way to get on that people.]