What is the logic of the interest rates and bonds being inversely proprtional (if interest rate goes up, bond price goes down)?

I know that PV = FV/(1+interest)^year ; since interest rate is in the demoninator, as the interest rate goes up, the bond price goes down, which makes sense mathematially.

But logically, if the interest rate goes up (you get a higher return every 6 months), shouldn't the bond price go up, because there is more demand for this asset that is returning more money every 6 months?