What's worse than "taking off" points?

I used to penalize students who didn't complete their assignments by taking off a point or two. After several years, I finally realized it was not changing the behavior of my students. The easiest way out for students is to take a hit by missing a few points every once in a while. This year, I'm trying out an experiment in lieu of a different philosophy towards points, grading and late work. In the students' eyes, it might even be worse than taking off points.

I require students to finish work they do not complete the first time.
Here is my thought process:

If the assignment was important enough to complete the day I assigned it, why wouldn't it also be important enough to finish it, even if it's a day late?

I can foresee several rebuttals to this philosophy - many of which end in something like...

Aren't you teaching your students that it is okay to be irresponsible?

In my mind everything I do should revolve around helping students "learn." I will let the rest of the school "teach" responsibility by taking off points for incomplete assignments. In the meantime, I will require students to be responsible by completing the important math assignments I have carefully selected for them to conquer. By requiring them to do their assignments, I believe that I am helping them learn.