On A Logical Argument In Favor of Abortion

The Original Argument:

P1: If abortion is not legal, there will be women who would be desperate enough to find a specialist to abort her fetus illegally.
P2: She would be putting herself at risk of an abortion operation from a quack.
P3: She could die along with the fetus.
C: For the life of the woman, abortion should be legal.

Firstly, I will comment on the logic of the argument formally, that is, with regard to form of the argument, apart from truth and falsity:

P1: The first "premise" proposes a hypothetical, highly probable situation, that, in the past, has been a historical fact. This statement does not have any logical flaws. It merely presents the conditions of the syllogism that follows it. To simplify it for the sake of analysis: If abortion is illegal, it will still be performed. (Let it be taken as given throughout the course of this commentary.)
P2: The second premise, while not presented formally, can be restated formally as such:
P2: Illegal abortion is a health risk to women. There is nothing illogical about this statement. It can be used as a premise in a syllogism. The only question is whether the other premises used in that syllogism will lead to a logically valid conclusion.
P3: Placing this statement in its logical form, that is, two terms connected by a copula (is), it looks like this: Illegal abortion is a potential cause of death to abortion patients. From this we can see that it is being used in conjunction with P2 to give support to the conclusion that is drawn from some unmentioned premises.
C: The conclusion does not explicitly follow from the preceding "argument." The argument stated its conclusion in this way: For the life of the woman, abortion should be legal. This is not an appeal to logic, since this conclusion does not follow from the premises. A logical argument must connect two different terms, a major and a minor, through a common term, the middle term. This argument's conclusion is not derived directly from what is stated in the premises, nor could it have been since there was no middle term in the argument, and new terms are being introduced in the conclusion. There are a few steps in the argument that are made implicitly. If we examine the argument more carefully we will be able to see what is going on more clearly:

Given: If abortion is illegal, it will still be performed. 
P1:  Illegal Abortion is a health risk to women.
P2: Illegal Abortion is a potential cause of death to abortion patients.
C1: If abortion is illegal, there will be health risks to women and potential death for women who have abortions.

C1 is the logical conclusion based on the given conditions and the two premises (P1 and P2).

So how are we supposed to arrive at the conclusion that "Abortion ought to be legal."?

The term "ought" indicates that according to the conclusion of the argument, legalized abortion should be an obligation of the state. On what grounds? The grounds of the argument are P1, P2, and the given conditions concerning illegal abortion. How did we move logically from C1:  That there will be health risks to women on account of illegal abortion, all the way to the moral obligation to have legal abortion without any connecting premises or terms whatsoever? The argument arrives at its conclusion through the following implicit or hidden premises:

Hidden Premise 1: Abortion is necessary for the life of women.
Hidden Premise 2: Legal Abortion is not a risk to the life and health of women.
Hidden Premise 3: It is necessary to protect women from risks to their health and life.

It was explicitly assumed that abortions will continue to be done if it is illegal. What is implicitly assumed in order to arrive at the conclusion is that abortion needs to be done even if it is illegal (HP1). The reason that this is implicitly assumed is that the conclusion mentioned above (C1) does not dictate any action or obligation on the part of the state, rather it merely concludes that illegal abortions are serious health risks to women. If illegal abortions are serious health risks to women than all that can be concluded from that with respect to the state is that the state has done well to make such life endangering procedures illegal. However, a different conclusion is drawn: Abortion ought to be legal. By brushing away P1 and P2, which serve only to support HP1, we can look at the argument as it actually is formulated, albeit implicitly:

HP1:   Abortion is necessary for the life of women.
C1:     Illegal Abortion is a risk to the life and health of women.
HP2:  Legal Abortion is not a risk to the life and health of women.
HP3:  For the life of women it is necessary to protect women from risks to their health and life.
Conclusion: Therefore, it is necessary for the life of women that there be legal abortions.

Formally speaking, this argument is logical. If abortion is necessary for the life of women, and the only kind of abortion that does not harm the life and health of women is legal abortion, then surely legal abortion is necessary. The scary thing is that the logical argument is not presented explicitly, rather it is hidden underneath a vacuous argument that appeals to emotions in order to lead people to conclude that the best thing for the health of women is abortion.

Secondly, I will comment on the logic of this argument materially, that is, with respect to its truth and falsity:

HP1:   Abortion is necessary for the life of women.
Why? This would only be the case if the conditions for abortions were necessary. What are the conditions for an abortion? Unwanted pregnancy. Why are there unwanted pregnancies? People have sex and do not want to have children. Is it necessary to have sex and not want to have children too? Certainly not. Then neither is abortion a necessity for the life of women. This is an outright falsehood which assumes that women and men are unable to have sex in a responsible, rational, and consequently, human way.  False.

C1: Illegal Abortion is a risk to the life and health of women.
True.

HP2:  Legal Abortion is not a risk to the life and health of women.
False. Many women have died because of complications from abortion, e.g., bleeding, infections from baby's body parts which remain in utero after the procedure is completed, anesthesia complications, etc. Also, women who have had abortions are 60% more likely to have breast cancer. Women who have had abortions regularly experience Post-Abortive Trauma, psychological damage that takes years (if ever) to recover from.

HP3:  For the life of women it is necessary to protect women from risks to their health and life.
True.

Conclusion: Therefore, it is necessary for the life of women that there be legal abortions.
False. Even if only one of the premises from which the conclusion was drawn was false the entire argument would fail to produce a true conclusion. But this argument has two false premises that both need to be true if the conclusion is going to be true! The argument itself does not conclude logically and the latent argument does not conclude in reality because it is blatantly false to say that abortions are necessary for the life and health of women.

In fine, this argument ought to be rejected as a manipulative attempt to steer its audience towards the evils of abortion. For it attempts to reach its conclusion by evoking emotions and relying on a degrading concept of women as incapable of reasonable and responsible actions in the sphere of human sexuality in order to sway the audience from a rational analysis of the evils of abortion.  AMDG.