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Republican Sens. Ted Cruz of Texas and Rand Paul of Kentucky — both potential presidential contenders in 2016 — voted for Mrs. Gillibrand’s proposal.

Mr. Cruz said he made his decision to support the Gillibrand plan as a father thinking about how he would want the military justice system to work if his two young daughters decided to join the military some day.

“I tried to approach this issue from the perspective of a dad,” he told reporters. “Heidi and I have two little girls, and I spent a lot of time in those hearings thinking about, they’re right now 3 and 5, but thinking about if 20 years from now, Caroline or Catherine chooses to serve. What are the rules that I’d want in place for my little girls?”