This is a fairly complicated subject. Titanium alloys in the wild are generally not harder than a knife steel, not even close - that's not what they're made for. But you're right: some titanium alloys can be hardened into the mid-50s on the
Hardness, Rockwell C scale, while retaining high toughness and a very fine grain, and as a blade, function better than you would expect at that hardness unless you start cutting highly abrasive things containing even harder particulates.
Some can be hardened into the mid 60s HRc but at that point you tend to see a lot of bad effects, such as embrittlement, and weird grain structures that can't get sharp despite being "hard." This is better for say, a sacrificial ballistic armor strike plate. The Air Force tested bi-alloy titanium armor plating for aircraft in the '70s that had a ti alloy at HRc 65 mated to a much tougher and softer 6al4v plate. The armor was stellar but cost too much to make and the project was canned.
It's a very shock-resilient material which is one of the reasons it's so good as a long thin blade, in addition to being a nice weight for the volume of a very large blade, among other things. There's a lot to it. I also think it would be just fine as a kitchen knife, too.
Otherwise, everything
Larrin
said above is spot-on. The only thing I would add is that I think those cera-titan type blades are a sintered material, that don't really equate to a real titanium alloy. I had one and thought it was surprisingly brittle but worked well, but then again I'm a terrible cook and a brute with knives in the kitchen.
Silver is a "beta phase-stabilizing" element in titanium, meaning (among other things) it helps harden titanium in the solution-treated and aged condition. It's effects are probably not much different than any other such element, such as vanadium and molybdenum, other than silver isn't toxic to the human body and is thus good for dental work. Human joint-replacement implants made from titanium alloys have been substituting vanadium with niobium for this very reason.