This morning the Supreme Court will hear oral arguments about the constitutionality of the Defense of Marriage Act, a 1996 law that prevents the federal government from recognizing state-level same-sex marriages. In an effort to divine how the case might be decided, activists, journalists and legal experts will study each Justice’s questions and utterances with the kind of exhaustive, even absurd, scrutiny typically reserved for wills and ransom notes. But the words of Justice Anthony Kennedy — the swing vote on a panel whose other eight members, four to a side, are relatively predictable — will likely offer the most valuable clues. If yesterday’s arguments around a different gay-marriage case are any indication, however, they may be tantalizingly inconclusive.

When the court heard arguments on Tuesday about the constitutionality of Proposition 8, a 2008 California ballot measure that banned gay marriage in the state, Kennedy occasionally seemed in sync with conservative Justices like Antonin Scalia and Samuel Alito, who warned, among other things, that an institution as old as heterosexual marriage should not be redefined until its potential societal effects are clearer. Redefining marriage would lead society into to “uncharted waters,” Kennedy said, and (mixing metaphors) potentially over a “cliff.” In an exchange with Charles Cooper, the private lawyer arguing for backers of Proposition 8 (which California’s governor and attorney general declined to defend themselves), Kennedy said that “[the] sociological information is new. We have five years of information to weigh against 2,000 years of history or more.”

(MORE: Pride and Prejudice: An Interactive Timeline of the Fight for Gay Rights)

But Kennedy then pivoted to a new perspective, suggesting that same-sex-marriage bans are causing an identifiable harm right now. “On the other hand,” he continued, “there are some 40,000 children in California … that live with same-sex parents, and they want their parents to have full recognition and full status. The voice of those children is important in this case, don’t you think?” Kennedy said those children may be suffering an “immediate legal injury” as a result. With that, he had flipped the conservative talking point about same-sex unions harming children on its head; maybe the real harm to children is from gay-marriage bans.

If Kennedy is internally conflicted, that’s good reason to believe the court will issue a narrow ruling on Proposition 8 that affects only California, perhaps on the grounds that the measure unlawfully removed a right that the state’s supreme court had previously granted. Several Justices also raised technical questions about the plaintiffs’ standing, and at one point Kennedy wondered aloud whether the court should have taken the case. Even the liberal Justices showed little enthusiasm for transforming marriage nationally, and at times their questions were surprisingly heterodox: Justice Sonia Sotomayor asked Ted Olson, the former Bush Administration solicitor general who argued against Proposition 8, why same-sex marriage wouldn’t open the door to polygamous and incestuous unions. Olson responded that the law can prohibit certain kinds of conduct — e.g., marrying your sister or your son or three people at once — but can’t deprive rights to a class of citizens — e.g., gays and lesbians — based on their identity.

Today’s Defense of Marriage Act (DOMA) case requires a clearer decision. Unless the court rejects it entirely (and there is a potential standing argument here), the Justices face a binary choice: uphold DOMA or invalidate it. Narrowly, the case is about an 83-year-old New York City woman named Edith Windsor, who is fighting a $363,000 estate-tax bill on an inheritance from her deceased longtime partner, whom she married in Canada. Windsor would owe no taxes on the estate of a husband. (This is one of several legal and financial benefits denied to same-sex married couples under DOMA.) But the broader question is whether the federal government can treat same-sex married couples differently from heterosexual ones, under a law passed by a Republican Congress — and signed by Bill Clinton — to keep gay marriages permitted in one state from becoming valid nationwide.

There are reasons why Justice Kennedy — who has supported gay rights in the past — might find striking down DOMA easier than striking down Proposition 8. Doing so would not make any same-sex couples immediately eligible to get married, which might limit his concern about “uncharted waters.” And the “immediate legal injury” suffered by people like Windsor, with her very large and very specific tax bill, is perhaps more tangible than the harm to children whose same-sex parents are denied legal marital status. Striking down DOMA, however, wouldn’t suddenly make gay marriage legal across the U.S. Individual states would still be free to set their own policies, although the absence of DOMA would bring wide legal and political confusion, as states like Alabama consider which state laws — including taxes, child custody, inheritance — should apply to, say, two men who were married in Massachusetts. (That’s a problem, even if the airlines don’t book too many Provincetown-to-Mobile itineraries.)