Something like a Socratic discussion of sex, gender and what they mean in contemporary society broke out at the state Supreme Court Wednesday as the justices heard arguments in a sex discrimination case that some advocates say could change the direction — or at least confuse — Connecticut’s path toward greater equality.

“There was a couple of times where you said there is a clear difference between the sexes,” Chief Justice Richard A. Robinson asked, halfway through nearly two hours of argument. “What does that mean? I am going to ask you. It is a simple question, but probably a very complex answer. What is sex? What is gender?”

The case before the court involves a suit by the state Commission on Human Rights and Opportunities that accuses the owners of two health clubs of breaking anti-discrimination law by providing women with private exercise areas in gyms open, otherwise, to both men and women. The ostensible purpose of the private areas is to protect women from sexual harassment by men.

The suit raises questions about whether the thrust of state anti-discrimination law is to treat everyone equally or whether it was meant to redress historic persecution of some groups, such as women, by other groups, specifically men. Complicating the questions are rapidly evolving definitions of sex and gender and whether, in society’s view, they are preferences rather than biological distinctions.

“If you had a transgender women who had not had gender reassignment surgery, is she a women for the purposes of entry into this restricted area of the gym?” Justice Andrew J. McDonald asked one of the attorney’s for the health clubs.

“Yes,” attorney James F. Shea replied.

“Who’s policing this?” McDonald asked.

“No one,” Shea said. “I mean, no one is checking to confirm that the person is biologically as the person presents.”

The case reached the Supreme Court after the health clubs won the early rounds of litigation, convincing a commission hearing officer and a Superior Court that there is a right to gender privacy in Connecticut law that exempts women-only exercise areas from the anti-discrimination prohibition. Many advocates — the case has attracted wide interest -- predict that if the Supreme Court follows that reasoning and recognizes a gender-based privacy right, other groups will follow, threatening a chain reaction that could erode laws intended to ban discrimination against anyone in any public place or organization.

McDonald suggested that men might need private space in gyms to protect them from “ogling” by gay men.

The parties to the case — the commission and two health club owners -- tried to argue narrowly that the dispute rests on interpretation of the state’s public accommodation law and what the Legislature intended the law to do. The law prohibits sex-based discrimination against anyone for any reason in any place open to the public, with two narrow exceptions — sleeping accommodations such as hospital rooms rented for the exclusive use of persons of the same sex, and separate bathrooms or locker rooms.

The commission, taking the position that all sexes and genders must be treated equally under state law, argued that the Legislature intentionally limited the exceptions and explicitly did not make gender privacy, a concept not articulated in state law, one of them.

“There is no ambiguity in these exceptions,” said Michael Roberts, who argued for the commission.

The health clubs ridiculed that position as absurd, arguing that such a narrow interpretation would prevent, among other things, battered women’s shelters from banning men.

From their questions, the justices sounded divided, but concerned about the practical implications of whatever decision the majority reaches.

Just seconds into his argument, Justice Steven D. Ecker interrupted Roberts with the first of what became a long list of hypotheticals from the court: If women are harassed by men after being denied private exercise areas, can they bring harassment claims against gyms?

Roberts said the solution to that problem could be a requirement that prospective gym members submit to harassment training as part of the membership application process.

Shea said the case is not about harassment, but about the objectification of women, centuries of which have made women susceptible to depression or other disorders.

“This is not a case of sexual harassment,” Shea said. “Our expert talked about the concept of objectification. And women are objectified without harassment.”

Justice Christine E. Keller wanted to know whether a gender privacy right applies to women at swimming pools.

“I think women probably feel more uncomfortable about being looked, at ogled or objectified in a bathing suit than whatever they wear to the gym — baggy sweats or one of those nice expensive workout outfits you can buy now in expensive stores.,” Keller said. “Should we have separate hours for women at public swimming pools? Separate swimming pools for women, screened off from the swimming pool for men? Because you are far more exposed in a bathing suit than you are in exercise clothes.”

Shea said that gyms, unlike pools, are “a traditionally male dominated environment” and so women are entitled to privacy right in gyms as a “remedial” measure to correct past mistreatment.

Keller wanted to know what the remedy should be for an overweight man who felt inappropriate looks from women at his gym amounted to “fat shaming.”

Shea said that the expert commissioned by the health clubs to examine the subject determined that “men don’t feel objectified like women.”

Ecker expressed concern that if women are denied private exercise space in co-ed gyms, they will be forced to work out at home. Some of the justices pressed the commission lawyer with questions about whether private, single sex gyms are legal in Connecticut, under public accommodation law, but did not get a definitive answer.

Robinson said the cases presented the court with a ”tough” question. Some of the judges suggested the solution may lie with sending the health clubs across Capitol Avenue to lobby the Legislature for a clarification.