I've written about how the fears that recognizing same-sex marriage will lead to recognition of incestuous marriage and polygamous marriage are, as a practical matter, ill-founded. Nor do I think that there's a logical equivalence between the three that means that, to be consistent, those who accept the first must accept the others. Recognizing same-sex marriage, I think, is likely to help society; recognizing the other kinds, I think, is more likely to hurt society.
Nonetheless, the slippery-slope concerns are made more plausible (though in my view still unpersuasive) by the way some advocates of same-sex marriage argue. For instance, consider this item from law professor Dan Pinello, a forceful supporter of same-sex marriage:
Republican State Senators Serphin Maltese and Frank Padavan are longtime foes of gay rights....
I call for a Maltese-Padavan Patrol whereby gay men, lesbians, and other supporters of marriage equality do the following:
1. At regular intervals of every business day -– even better, every hour of every business day -– telephone both the Albany and Queens district offices of Senators Maltese and Padavan to ask them questions such as “Why don’t you support the right of all New Yorkers to marry the person of their choice?” and “Why do you think that lesbians and gay men should be second-class citizens?”
2. At regular intervals, e-mail the two senators with the same questions.
3. At regular intervals, visit both their Albany and Queens district offices to ask the same questions.
4. Whenever either Senator Maltese or Senator Padavan makes a public appearance, be in the audience to ask the same questions.
Now I'm pretty certain that most supporters of same-sex marriage (quite likely including Prof. Pinello himself) do not support the right of all New Yorkers to marry the person of their choice. They would presumably deny this right to New Yorkers who want to marry their siblings. They would also presumably deny it to New Yorkers who want to marry someone who is already married, even if the New Yorker gets permission from the other person's other spouse. (I don't think this can be dismissed by arguing that this doesn't involve the right to marry the person of their choice, which is to say only one. First, the new spouse would be exercising her right to marry the one person of her choice. Second, though the old spouse would be marrying more than one person, why shouldn't people who have the right to marry one have the right to marry two, if that's their choice? Saying that the right to marry doesn't work that way, because it inherently, by its nature, must involve only one marriage at a time per person, invites the response that the right to marry by its nature involves only one man marrying one woman.)
On the other hand, if indeed same-sex marriage wins using the "right of all New Yorkers to marry the person of their choice" argument -- if this right is therefore accepted, either by a court or by a legislature, or perhaps by courts and legislatures throughout the country -- then that would, it seems to me, strengthen the argument in favor of recognizing same-sex marriage or incestuous marriage.
True, defenders of same-sex marriage who oppose the other kinds of marriage could then say "no, we didn't really mean such a right that categorically, we meant the 'right of all New Yorkers to marry the person of their choice unless it's a sibling or unless their chosen person is already married.'" But that wouldn't be a trivial argument to make, and it's possible that at least some people who had become persuaded that there is a "right of all New Yorkers to marry the person of their choice" would feel that they ought to apply that right as it was initially enunciated, rather than to throw in qualifiers. So if you make the argument that Prof. Pinello is making, it's hard to ridicule the slippery-slope concerns that the other side raises. (As I said in my article, it's still possible to argue against those concerns, because of the practical unlikelihood that the pro-incestuous-marriage or pro-polygamy movement will have the force that the pro-same-sex-marriage movement now has; but the argument is not open-and-shut.)
And even beyond this, isn't the questioner who raises the "right of all New Yorkers" argument inviting a quick and effective slap-down, precisely because his argument on its face entails a consequence that most listeners wouldn't approve of, and that even the questioner probably doesn't mean?