Can my PID controller have negative p value and positive i value for a PID logic
Except when you're driving a very strange plant, you want the sign of all of your gains to be the same -- all positive, or all negative.

If you have a fairly normal plant (i.e. it has a minimum phase response in dynamic systems terms, meaning that all of its poles and zeros are in the left-half plane) then you need to use negative gains for both proportional and integral gains, or you need to take @gregb212's suggestion and do your calculation, then multiply it by -1 at the end.