(N.B. This is a slightly modified form of an argument I first wrote in one of Mark Shea's comment boxes. The original is available here.)
There are a number of people who have pointed out that the acceptance of contraception will lead to the acceptance of abortion and euthanasia. Think, for example, of the prophetic words of Pope Paul VI in Humanae vitae. Yet there are many who argue that such a stance is an example of the fallacious slippery slope form of argument.
The argument that the acceptance of contraception logically leads to the acceptance of abortion and euthanasia is not a slippery slope. It entails necessary conclusions drawn from what the acts of contraception, abortion and euthanasia are, and from the principles that must be accepted to view these acts as morally acceptable.
Sex causes babies. This is what sex does. To say that there is a right to contraception entails holding that one has a right to the cause of babies - sex - without the result that naturally follows - the babies themselves.
Principle 1: a right to sex without babies.
There is, however, a problem: no form of contraception is 100% reliable. Barrier methods fail, pills fail &c. But we already necessarily hold that there is a right to sex without babies. Thus there must be an all purpose backup that will eliminate babies in the cases were contraception fails. This is abortion. This is how abortion was historically argued for by a number of people, i.e. as a necessary backup to contraception.
Moreover, it is still how abortion is argued for today. Please see this HuffPo article that Mark Shea links to here: "Others find that their dignity depends on being able to end the pregnancy." Human dignity depends on Principle 1, on the right to sex without babies. Thus human dignity depends on abortion.
There is, however, a problem. From the moment of conception there is a genetically unique, self-contained, self-directed, genetically human life. This is a fact that science - today's great god-king of all that is knowable - confirms. There is no logical way to say that this life is anything other than a human being. But Principle 1 necessarily demands that this innocent human being can be killed if it is not desired. Thus it must follow that human life only has worth dependent on circumstances. It has no inherent worth in itself.
Principle 2: Human life has no inherent worth.
Now, the chronically sick and the elderly can be difficult to care for. Caring for them can be just as difficult, if not more difficult, than raising children. But we have already posited a right to sex without children. How can we have a right to be free of the latter "burden," but not the former? We cannot, and Principle 2 gives us the way out.
The chronically sick and the elderly live lives that are of a lesser quality than others do. They live with pain. They suffer a loss of qualities such as speed, strength, agility, beauty &c. They are a "burden" on those who have to take care of them and are usually without a means giving much of anything back as compensation for being such a "burden."
We would certainly appraise the value of anything else that had so many detriments with so few benefits as being of little worth. And since Principle 2 holds that human life has no inherent worth, we can appraise the value of a human life the same way we would appraise the value of anything else. Thus we can appraise the life of the chronically ill and the elderly as being worthless and eliminate them is we so choose.
Contraception necessarily requires abortion because the principle behind the acceptance of contraception is that we have a right to sex without babies, and the only way to fully guaranty sex without babies is to have abortion as a backup for the failure of ontraception. The acceptance of abortion necessarily requires that some innocent human life is worth less than others and can thus be taken if it is undesirable. It thus becomes a necessary principle that human life has no inherent worth. And if human life has no inherent worth, then there is no reason why the life of the chronically ill and elderly cannot be appraised as having little worth and eliminated so as to relieve the burden that would otherwise be imposed on those who would have to care for them.
In proof form:
If you accept a right to contraception, then you accept Principle 1.
If you accept Principle 1, then you accept that abortion is a right.
Therefore, if you accept a right to contraception, then you accept that abortion is a right.
That is a valid hypothetical syllogism.
If you accept that abortion is a right, then you accept Principle 2.
If you accept Principle 2, then you accept euthanasia.
Therefore, if you accept that abortion is a right, then you accept euthanasia.
That is another valid hypothetical syllogism.
If you accept a right to contraception, then you accept that abortion is a right.
If you accept that abortion is a right, then you accept euthanasia.
Therefore, if you accept a right to contraception, then you accept euthanasia.
This is yet another valid hypothetical syllogism.
The conclusion is thus proved: If you accept contraception, logical adherence to principles demands that you accept abortion and euthanasia.