As if there were any doubt, it looks like Half-Life 3 is officially some kind of a thing. Enough so, at least, that Valve feels the need to make sure everyone knows who it belongs to. And hey, who wouldn't want Gordon Freeman leading the charge on liberating the living room from the nefarious Combine force that is proprietary, closed console-dom? Also, I did have the fortune of speaking to a Valve source recently who dropped some pretty big hints about Source 2, Left 4 Dead 3, and Half-Life 3. So that's worth noting as well. But even then, nothing's even close to confirmed. Here's what we definitely do know.
Valve has filed a trademark (via NeoGAF) for Half-Life 3 with Europe's Office For Harmonization In The Internal Market. It covers "computer game software, electronic game software, downloadable computer game software via a global computer network and wireless devices, and videogame software." It was filed on September 29th and is currently under examination.
Oddly, Valve has yet to file for the same trademark in the US. I've emailed asking for confirmation and comment. We'll see what comes of it.
It is important to note that trademarks can mean many things. In many cases, it's just developers or publishers protecting their IPs or reserving them for potential future use. This trademark, on its own, doesn't necessarily mean anything - at least, not in the short term.
But we can hope, right? Because frankly, at this point, hope is pretty much all we have. That, and our memories.