So...he IS happy when you hold him?
Does he sleep...go to sleep on his own?
If he's happy when you're holding but screaming when down, then he needs to learn to self soothe. If he's screaming all the time, no matter what, I would guess his formula doesn't agree with him, or he has reflux. etc. My own kids couldn't take high-iron formula, for instance. When I'd reintroduce it, they'd all 4 act like someone flipped a switch!
So, once you've ruled out a physical issue, I'd do this:
I hold and talk to them for a bit each day, and when I'm feeding or changing them. Other than that, they are on the floor to play, or occasionally in a bouncer/carseat.
If baby has been fed, changed and given attention, then crying = you must be tired. The end. "You are really crying. You must be tired. Lets go night...night". Sweet voice, smile...snuggle...bed...pacifier...close door.
Sleep, no sleep, it doesn't matter. He could be overstimulated, he could be tired, but either way, a nice cozy bed is the answer.
If he cries, then go in and very quietly reassure him every once in a while. If he's comming up on another "meal", then get him up, and try the above routine yet again.
