Edit: I do realize this is a MacBook Air subforum, but wanted everyone to know it's not limited to the Air or laptops.
Same thing here. Mac Pro Early 2008 3,1 running 10.9.3, 22GB Ram, Samsung 840 EVO SSD, Bootcamp partition, more stats available upon request. I get random shutdowns, with nothing meaningful or repeatable to be found in the console/error logs. It's been happening on and off since the install "upgrade" of Mavericks, and it happens on all version of Mavericks so far with no updates solving the problem yet. It seems to happen more frequently when video-intensive programs are running, but it's certainly not limited to that, and seems to occur completely at random most times, regardless of what programs (or none) are running, or how much or little system resources are being used. Sometimes it happens multiple times per day, other times I'll get away without a shutdown for a week if I'm lucky. When it shuts down no video is output to the monitor, but it doesn't shut all the way down. Instead, the fans go full speed, but otherwise it's non-responsive and I have to do a hard reboot. Sometimes it shuts down again immediately after reboot, or rather never gets to the "chime" startup sound and goes straight back into full speed fans.
At first I thought it might be the video card, the ram or external drives, but everything tests well, and I've tried removing all drives except my boot drive and all peripherals and externals. None of that helped. I've also attempted cloning my drive with SuperDuper, which I use for backups to other drives, and the same issue persists. I've tried verifying and repairing permissions, checking error logs for problems prior to shutdown like kernel panics or other red flags, running Apple Hardware test, running 3rd party RAM test and drive tests, SMC resets, paramater ram resets, left everything unplugged and disconnected for 24 hours when I was away over the weekend, changed out the logic board battery (CR2032 coin battery), tweaked the power management settings, ran all scripts with Onyx, made sure all drivers are up to date, cleaned all the leads to the ram, video cards, HD's and anything/everything else that plugs into the computer, ensured all software up to date, and about a hundred other changes in hopes that something, anything, might make a difference. Nothing has helped, and the problem persists.
I never had this issue prior to Mavericks. All of the previous OS' worked without any major issues that were unable to be fixed with a little research or basic troubleshooting. I think the best performance and stability I had was with Snow Leopard on my machine, but I have upgraded OSX pretty regularly after waiting at least few months for the major bugs to (hopefully) be worked out. Some programs and features needed Mavericks to run, so I "upgraded" when I needed those features. It seemed unlikely that this type of problem would stem from a software issue, so I assumed it was hardware, but my experience lines up with many others here and on other forums, and I have no reason to suspect any of the other components as nothing else changed between the switch to from Mountain Lion to Mavericks.
I hope Apple acknowledges this issue, or someone comes up with a fix. Someone in an earlier post recommended smcFanControl, so I might try that.
Sorry for the long post, but I've had trouble finding what others have tried and what does and does not seem to help. Hopefully this helps someone else as a place to work from with new troubleshooting approaches I haven't thought of.