I agree about the "logician's dream" bit, but ultimately your reasoning can't get you information from the airline about who is on a flight. You were only able to work out that your friend was on an LX because you had other information that did not come from BA. So even if BA was correct to agree with your interpretation, the only information that they gave you was that your friend was not on their flight. That in itself tells you nothing about which flight he was on or who else was on either airline's flight. You had to know something else to work that out, and that is not BA's responsibility.