Is it possible to ever argue a logical point with the other side, and get them to see your viewpoint? It totally depends on the people involved and the issue being argued. Even intelligent and otherwise rational people (like my family), will often stubbornly refuse to see things from any viewpoint but their own.
For example, I've given up on convincing my family that abortion should remain legal, and I will no longer bother to argue the point with any pro-lifers, since it's an argument that's impossible to win. Why? I'll explain below, in the hopes that it might help you to choose your battles more carefully.
I'm cursed- my entire family is Republican, and, politically, we share almost none of the same viewpoints. Having a political discussions with my family members is like pulling teeth on an elephant without aenesthetic- painful for both parties involved, with a high risk of being stepped on or smashed into a pulp. They don't argue well, and they're stubborn as hell. And I am too, to some extent, but it's not easy when you're outnumbered.
I first realized how much my family was part of the "other side" during the 2000 election, when I made a comment about the "anti-choicers", and the response was icy silence. After a rather liberal college education (despite it being a Lutheran college- California Lutheran University, for the curious), it was somewhat of a shock to discover my entire family was pro-life. Being the naive individual I was at the time, I assumed that by educating them to my viewpoints, they would "see the light" and take up a more reasonable stance. Very presumptious of me, I know.
My carefully wrought argument about the right to choose, and women having control over their own bodies, and the rest of the usual pro-choice stances was derailed when my brother-in-law refuted with, "So, you think it's alright for me to take a shotgun and go down the street and shoot my neighbor in the face?" Faced with that, I replied, "Well, of course not, but..." Except there could be no "but". The logical simplicity of his stance prevented any response. Although we were both discussing abortion, we were not engaging in the same argument. There's a reason one side calls itself "Pro-Choice" and the other "Pro-Life".
You cannot have a rational argument with someone unless you agree on the basic definitions. I define a fetus as a bunch of cells with the potential to form into a person. My brother-in-law defines it as a human life from the moment of conception. It's a religious conviction, and one taken on faith; it's a definition about the basic concept of life that is not shared by both sides. One can be a lot more cavalier about removing a bunch of cells, but to a pro-lifer, it's as if you're talking about shooting a human being in the face. There's really no way to morally argue it's okay to shoot someone in the face without sounding like a complete lunatic. Thus, the argument breaks down. Rather than thinking of them as "Anti-Choice", we'd do well to remember that from their viewpoint, they're "Anti-Murder". I don't think we'll ever be able to win an argument for the "pro-murder" side, so I just rest assured in the fact that I know I'm right, they're deluded and wrong, and although the pro-life side is incredibly noisy, the silent majority agrees with me.
The moral of the story? Well, simply put- be sure you know the definitions both sides are using before engaging in an argument you intend to win. If there's an intrinsic concept in your argument that revolves around a definition that's not common to both sides (such as "The point at which life begins"), then either first come to a consensus on that definition before engaging in any argument, or skip the argument entirely.