This is a bad question because "Logic" can mean many things. Primarily, it could be a branch of Philosophy or a branch of Mathematics. However, the answer is still "no". A smart kid can learn programming without any prerequisites whatsoever.
I started to program when I was 7. My dad taught me a few basic things about how his programmable calculator worked, and I played around with it until I figured it out. If instead of teaching me what "if" and "goto" do he had started by introducing Logic and whatever other things people think should be prerequisites for programming, I would have been overwhelmed and I wouldn't have learned anything.
yeah. I mostly started out fiddling with QBasic and later Turbo C and DJGPP and similar back when I was in elementary school (mostly on my own).
at the time, I didn't even really know what the order of operations were or similar, but it didn't take long to learn them from observation.
I did ok in high-school level algebra, but basically sucked at everything else.
got to college, pretty much sucked at everything.
and, as well, suck at pretty much anything life-in-general related.
not particularly good with dealing with abstract thinking either.
...
still reasonably good at programming though at least.
like, in life, there are relatively few things that actually make sense. programming seems to be one of them.
I suspect in many ways, it may be a lot more fundamental than all of the "meta" and academic disciplines and similar that people seem to try to glue onto it.