While that has traditionally been true, it is less true today, and I'd wager that it's out and out false now.
For reference on the following discussion, refer to
Steel Guide
Consider that the carbon content of a high carbon stainless steel knife is right around 1%, on average. 440A is a common stainless steel for kitchen cutlery. Really cheap stuff is probably 420.
The good "carbon" steels also hang out around 1%, but very rarely higher. 01, 1095 being very popular in hand crafted knives and tough stuff. It's not what you see in the cheap carbon kitchen knives. Just as with stainless, lower carbon knives are the dominant steel. Both for cost and maintenance reasons.
Carbon is what turns iron into steel. The more carbon,, the better the steel can be. Not automatically is, however.
The critical piece in the comparison you're making is in the heat treatment of the steel. Stainless was harder to heat treat. And fabrication methods didn't create as good of a crystalline matrix in stainless steels. Those flaws are exposed in the heat treat.
But with the economics of stainless compared to carbon, there has been great progress in the heat treatment of stainless steel. Additionally, steel can now be fabricated from fine powders and vaccuum "forged" creating very consistent crystalline structures.
Only at the very lowest end of the scale with knives no one on this board would consider for kitchen use would I consider your statement to be true in todays' kitchen knives.
And a big factor in the test is the user's ability to sharpen.
On that list, I own knives in various steels. From the staining steels, i have D2 and M2. From the stainless, I have 440C, ATS-34, 154CM, AUS 6, AUS 8, 12C27 and S30V.
The S30V is the best of the bunch and the most expensive. I'm not impressed with the D2. Very hard to sharpen and not particularly good at holding the edge. The M2 is also hard to sharpen but will hold the edge longer than ATS34 and lesser steels. I use it for very hard use only when I absolutely need the edge and can't stop to touch up the edge. S30V in stainless can keep up with it with much easier resharpening and costs only somewhat more than the M2.
The other hassle with carbon steel in the kitchen is the staining of the food. Forget about the knife. Cut artichokes iwth a carbon blade and watch them discolor.
Chop a bunch of potatoes and look for the black streaks from your blade.
Carbon isn't worth it in the kitchen.
Phil