If I connected two outputs with a wire, one with the information A+B and the other with C+D what would these two combined signals make?
For typical gates, if one wants to put out a high voltage and the other wants to put out a low voltage, they fight each other.  Power supply current is high, the chips get hot, and the voltage on that node is at some indeterminate value between logic low and logic high.

That's what will happen in your scenario any time A+B does not agree with C+D.