Will inflation rise or fall when the Fed increase their interest rate? When the Fed lower the interest rate, there will be a rise in GDP and thus a decline in unemployment, which in turn cause the inflation to go up according to the Phillips curve. But Fisher’s equation tells me that real interest rate = interest rate - inflation. So if real interest rate does not change, an increase in interest rate should also cause a increase in inflation, a contradiction.

Any help or comment is appreciated.
Hint:

If you're keeping real interest rates constant in the second scenario, you should keep real interest rates constant in the first scenario.

If you do this, lowering the nominal interest rate will decrease inflation off the bat (by the Fischer's equation) by the same amount as the change in nominal interest rates.