Rick Santorum has tried very, very hard to make arguments regarding why gay marriage would be such a terrible, terrible thing. One of his go to arguments is discussing how, if gay marriage is legal, things like marrying animals, bigamy and incest must also be legal. Outside of the fact that the slippery slope is called a logical fallacy for a reason, these stances ignore the very basics of both contract and Constitutional law.
The bestiality one is the quickest, so I'll get that out of the way first. At its heart, marriage is simply a contract. In exchange for you agreeing to join your life to mine, including a whole host of property considerations that entails, I also agree to join my life to yours. As such, all divorce actually provides is a process to more efficiently breach the marriage contract. Since it is a contract, marriage requires something very important: consent from both parties. A dog cannot consent to a contract under any circumstances, so a person cannot marry a dog. Or a cat, or a horse, or whatever other man/animal marriages Rick Santorum is envisioning.
The bigamy and incest issues, on the other hand, require a quick summary of some basic Constitutional Law. I'm just going to be detailing the most exacting and least exacting tests for challenging a law on the basis of the Fourteenth Amendment, which applies the Bill of Rights to the states. Prior to the passing of the Fourteenth Amendment, a state could have, for example, had an official religion if its state constitution did not prohibit it.
The most exacting test is strict scrutiny. Strict scrutiny applies if a law infringes a fundamental constitutional right or if a suspect classification is involved. Suspect classifications include race, sex and national origin, but to this point have not included sexual orientation. I strongly believe that sexual orientation should be a suspect classification, but to this point the higher courts have not agreed. For a law to pass strict scrutiny, it must do three things: (1) be justified by a compelling government interest; (2) be narrowly tailored to achieve that goal or interest; and (3) be the least restrictive means for achieving that interest.
Rational basis scrutiny, on the other hand, is the least exacting test and applies to challenges to laws under either the Due Process Clause or Equal Protection Clause of the 14th Amendment when a more exacting test does not apply. At this point, the body enacting the law must only have a rational basis for enacting the challenged provision. That rational basis only needs to not be arbitrary.
Currently, it is most likely that laws prohibiting gay marriage will be reviewed under rational basis scrutiny. If that is the case, a court would need to determine that preventing gay marriage is arbitrary, or in other words that there is no rational basis it provides and is only prohibited because gay people give some straight people the heebie jeebies. But clearly laws prohibiting incest and bigamy do have a rational basis. Incest has a unique problem because can lead to children with all sorts of nasty birth defects, but bigamy and incest share a big problem as well: oftentimes engaging in those act lead to child victims. Protecting children from the terrible situations that bigamy and incest often lead to is clearly a rational basis that is not arbitrary. At this point, believing children are harmed by gay marriage is the among the worst kinds of prejudice, and has been shown to be flat out false.
Even if the courts determined that laws preventing gay marriage should be subject to strict scrutiny, that still would not prevent the enactment and enforcement of laws that prevent bigamy and incest. On the one hand, if courts determined that sexual orientation was a suspect classification, its determination in that case would only apply to gay marriages. On the other hand, even if the courts determine that marriage is a fundamental right, protecting children from the harm that often results from bigamous and incestuous relations is a compelling government interest, and the only way to enforce that is to make those sorts of relationships illegal.
Santorum's analogies simply don't hold up to any sort of legal scrutiny. If he wants to make an argument against gay marriage, let him go ahead and be on the wrong side of history. But his slippery slope argument is just intellectually lazy.