Disagreement to terms

I think that part of the problem with many arguments is that people try to take your position and apply the rules of their belief system in an attempt to prove that you’re wrong, rather than arguing against your specific points.
It’s controversial, but take abortion as an example. I recently read a letter to the editor where the writer took exception to the notion that a woman’s right to choose is more important than the unborn child’s right to life. They went on to explain that the opposition only had the “right to choose” argument to fall back on when presented with the “facts” about the rights of an unborn child.
The problem is that the pro-choice side of this argument didn’t stipulate to any of the “facts” that the opposition presented, and actually based their arguments on a completely separate set of “facts”.
If you argued that the speed limit on major highways should be 100 miles per hour because driving at high speeds is safer and all roads are straight, then I could make a valid argument that high speeds are not safe, regardless of the nature of the road. But what if I attempted to discredit your argument about the safety of high speed driving by providing evidence that the roads are more curved than you believe them to be? Could I reasonably claim that your position has been invalidated because you didn’t take my evidence about the road into account? Would my argument be more definitive because I filtered it through my own set of facts?