When I teach logic, in order to offend everyone equally, I sometimes say that both sides of the abortion debate, at least in their slogans, commit the strawman fallacy. One commits this fallacy when, instead of addressing the actual argument being presented, one erects a strawman: something that is superficially similar to the argument but is actually as dissimilar to it as a scarecrow is dissimilar to an actual human being. Moreover, by being made of straw, the strawman is much easier to knock down than an actual person.
So how does the pro-life side commit this fallacy? By saying abortion is murder. Murder involves the intentional killing of what someone recognizes as an innocent human being. Many women who have abortions have been told that the fetus is not even alive, much less a living human being, much less an innocent human being. The abortion doctors are in a different position: they know the fetus is alive -- but then again, so is an individual cell in your liver. But I doubt that they believe the fetus is a distinct human being, a person (for simplicity's sake I'll treat "human being" and "person" as interchangeable, although many distinguish them in this debate). However, this is an assumption on my part: I don't know of any polls as to whether abortionists tend to believe that the fetus is a person. But even if there were, I don't think I'd trust the results: if you thought fetuses were human beings, but thought abortion was a necessary evil, would you acknowledge this in a poll? I'm sure there must be some abortionists who do think the fetus is a person. Kermit Gosnell, in the manner of serial killers, kept trophies of all the babies he killed -- I say babies, not fetuses, since he delivered them and then killed them outside the womb. A doctor wouldn't keep trophies of the tumors he'd removed from patients, so obviously Gosnell recognized that he was taking the lives of innocent human beings.
I think the strongest objection one could make is that if the person should have known that she was killing a human being, then their act could still be considered murder. If a philosophy student who had read her Peter Singer and Michael Tooley killed a newborn baby and argued afterwards that she honestly didn't think it was a human being, she would probably still be convicted of murder. That's because nearly everyone recognizes that a newborn baby is a human being and has a right to life, and that same intuition, that same source of knowledge, would have been available to the person being accused, since they lived in the same culture and context. With abortion, however, there is no such consensus. We have to be careful here to not commit another fallacy, argumentum ad populum, appeal to the masses. But I think we can avoid this as my point is a more modest appeal to humility: given that there is widespread disagreement on abortion, we shouldn't assume that a particular person knew the fetus was a human being and killed them anyway.
Anyway, the closest parallel I can think of to this -- the intentional killing of something that one recognizes is alive but does not recognize to be an innocent human being -- would be a hunting accident. A hunter sees movement in the brush ahead, thinks it's a deer, and intentionally aims and shoots with the goal of killing it. But to her horror, she discovers it wasn't a deer but another hunter. She recognized the thing ahead of her was alive, and deliberately killed it. But she didn't recognize that it was a human being she was killing. Perhaps a court of law might determine that a hunter should have known that it was a person she was shooting at, but since she wasn't deliberately trying to kill another human being, it is unlikely the hunter would be convicted of murder. Similarly, even if we grant the pro-life position (as I do) that fetuses are distinct human beings, insofar as the abortionist and the woman do not recognize this fact, they are not guilty of murder, whatever else one might say of them. So to call abortion murder is to erect a strawman to the effect that the abortionist and the woman are intentionally killing what they recognize to be an innocent human being.
How does the pro-choice side commit the strawman fallacy? By saying the woman has a right to do what she wants with her own body. Well, yeah, of course she does, as long as she doesn't harm someone else. The old saying is, your right to swing your fists ends where another person's nose begins. That is, one person's right to do what she wants with her body only extends to the point that she harms someone else or restricts the other person's right to do what he wants with his body. And the claim of the pro-life side is that the fetus is another person. A woman does not have the right to do what she wants with someone else's body, and the fetus is someone else's body (namely, the fetus's), not her body. That's the claim. Perhaps that claim is false, perhaps it's even absurd, but that's the claim being made. Slogans like "Keep your laws of my body" or "Keep your rosaries off my ovaries" may be clever, but their goal is to defend a right that no one is challenging. Thus such claims are complete strawmen.
Of course, the relationship between the pregnant woman and the fetus is a unique one. The only real world scenario I can think of that's even remotely analogous is conjoined twins. I'm unaware of a situation where one conjoined twin deliberately killed the other, but it seems to me that it would be considered murder (assuming all of the conditions discussed above). Judith Jarvis Thomson presented an interesting thought experiment: a woman wakes up in the hospital and finds herself connected to an unconscious violinist. The violinist was suffering from kidney failure, and the only way to save his life was to hook his circulatory system up to the woman's so her kidneys can do the work that his kidneys couldn't. It's only temporary, just nine months, and then she can be unplugged from the violinist and go on her merry way. Thomson argues that, even granting that the violinist is a human being, a person, the woman has the right to unplug herself from him, even knowing it would cause his death. The violinist's right to life does not include the right to use someone else's body.
When I first heard this argument, I thought it left out an important element: except in the case of rape, the pregnant woman engaged in an activity which has been known from time immemorial to lead to pregnancy. You'd have to add to Thomson's scenario that the woman went of her own volition to the hospital for some ostensibly pleasurable reason (maybe they were throwing a Christmas party and serving bacon wrapped shrimp) and signed a paper acknowledging that, by entering the hospital, she is accepting there is a nontrivial chance that she would be hooked up to a violinist for nine months. This changes the scenario dramatically. In fact, I first thought that Thomson was presenting this as an argument against abortion. So Thomson herself has commited the strawman fallacy: rather than include an element that would make her thought experiment more accurately track the abortion issue, she has excluded it in order to make the intuition she's appealing to more commanding.
Having said that, Thomson's point is still very astute and important: in the case of rape, which more closely parallels her thought experiment, does the fetus's right to life not include the right to use the woman's body as an incubator for nine months? Some people are opposed to abortion even in this case, because the evil of intentionally killing an innocent human being is greater than the evil of significantly, but temporarily, disrupting the woman's life. A lot of issues come into play here: what takes priority, a right to life or right to a lifestyle? What about the psychological effects on the woman? These could very easily ruin her life, they can't just be dismissed. What if we shortened the period of time the woman's life was disrupted? What if it was only three months? Or three weeks? How about three minutes? At some point, even though we might agree with the principle that one person's right to life doesn't include the right to use another person's body, most of us would think the inconvenience becomes trivial and the life of the fetus so much more important that we would no longer think the principle takes precedence. For that matter, couldn't we reframe the principle the other way around? Does the woman have the right to do what she wants with the fetus's body in order to continue her lifestyle? I mean, by killing the fetus, she's using its body for her own ends. On the other hand, how probable is it that the fetus is really a person, a human being? If you think it's just one chance in two, you might think the potential evil of killing the fetus is greater than the evil of disrupting the woman's life for nine months. But what if it's one chance in ten? Or a hundred? Or a million? At some point, even though we might agree that killing an innocent person is a greater evil than disrupting a woman's life for nine months, most of us would think that the probability that we really are killing an innocent human being becomes so low as to become trivial. I'm not even going to try to answer these questions, but I think they show that there's a reason why the abortion issue is controversial, and we should treat those who disagree with us respectfully and assume they are acting in good faith.