I thought about editing tag wikis once, when I got a notification that I could do it. I wasn't sure if it was worth the time, so I put it off. Then, strangely, the reputation required to edit them went up, and I couldn't edit them anymore. I thought that was strange at the time.
More recently, I thought about editing them again, when I noticed it says I need 5,000 to "vote" on tag wiki edits. I assumed that "vote" meant "approve", and since I don't have 5,000 yet I thought I would wait until I do so I could edit them and approve my edits. Now, based on the comments here, it seems that I can't edit the tag wikis outright without 20,000 rep.
20,000 seems like an almost absurd amount of rep to require; someone can edit other people's posts with 2,000, but they have to have double the points of the moderation tools to edit a summary of what a tag means? Frankly, editing tag wikis with the current setup doesn't seem worth the trouble to me. I can spend time on other things that don't need someone standing over my shoulder to "approve".
I would suggest lowering the rep needed to edit tag wikis to 2,000, the same as required to edit other people's non-community-wiki posts. It takes enough time to get 2,000 to limit that to somewhat dedicated users.
I tried editing the one for "logic" now that I have 5,000 rep, and I can confirm that I can't edit it. Strangely, it said that my edit needs "peer review", but I don't think any of the four editors with over 20,000 rep is an expert in logic, so I'm not sure how they are supposed to "peer review" it. Looking for vandalism is not the same as peer review (or else my refereeing would be a lot easier) - that message ought to be changed in the interface.