The claim for anonymous blogging would not be held because of the limitation of TCP/IP, where the origin of packets, without special treatment, would be exposed to your server. Nobody can be certain that your server is not compromised at any time (which we cannot even assume that you are not spying on the logs, for example), a claim in such degree may actually got believed by less technical users of your app.

Please consider this as anonymity may be live or death situation to some people in the world.