I doubt he will lose his mind. Cortana did not keep him sane. He would have been fine without here but he is just too used to her being with him most of the time. If you read the books than you would know what training he had gone through in his training, Spartans are too strong to fall apart. He could be damaged physically, but nothing will weaken his mental state. He worked fine with the other Spartan-IIs before he got Cortana, he can also work fine by himself. He was just shocked about losing Cortana. He was used to losing soldiers on the battlefield. When it was a Spartan though, he would do anything to save them, even if that meant risking his and others lives. It was different with Cortana though, he couldn't save her this time, he felt helpless. He should be fine on his own, his performance may not be as strong without an A.I though. Until then, we will have to see who his ally will be, if he has one.