Inflation is fought by lowering the money supply. Raising interest rates has a weaker effect on squashing inflation that lowering the money supply.
If the Fed truly wanted to reduce inflation, they would reduce the money supply.
All this “we are raising interest rates to reduce inflation expectations” is just talk: they are using a weak method to reduce inflation, because they hope they can control it by changing the *expectation* of inflation. That is my theory at least. I think they hope to pull the wool over peoples’ eyes in this way.
So if they are serious, they should also require higher reserve lending.