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So, Syracuse Law School's LOR instructions are to, "Include all available letters/evaluations." I have 3 LORs, but one of them pulled out, so I will never get the third one. If I assign the two I have, will Syracuse never see the third unfinished one? Or can they see that I am missing one and see the application as incomplete?

They only see what you assign them

Having a third unrecieved letter does not make your LSAC file - and therefore your application - incomplete unless that third letter is assigned to the school. I had the same situation come up - just didn't assign the missing letter to any school - and had no issues having my applications go complete.