About Allen’s father in the whole anime of Attack on Titan, there are too much doubts, which also is the very important thing that may easily be overlooked on the comic book. At that time, the situation is the arguement of the investigation Corps between Allen and his mother, Grisha ask Alan the reason, Allen said he want to see the world outside. Then Grisha went out and left important words, “Carol, human’s curiosity can not be interrupted just by some other’s restrain. Allen, I will go to show you the secret of the basement when I come back”.
Though we can not see clearly his expression, it’s not difficult to understand that the message under the words: His father wants to show him the basement because Alan want to go outside. Then there are two possibilities attack on titan: Grisha supports Alan to go outside and want to give him something that can help to protect himself. Or does not support, and tells Carol “human’s curiosity can not be interrupted just by some other’s restrain” So Grisha want to show something that may discourage Allen to go outside.
So in the end we have to guess what is in the basement, if it is the truth of the words may be too general. After all, a less than 40-year-old doctor is not likely to have big background, compared to the truth of political(such as government policies and walls giant king), maybe the truth of pgysiological(such as the generation of giants) is more close to Grisha occupation of doctor.
On the other hand, what will Grisha show to an only 10 years old boy? It can be unlikely to the whole truth, it can’t make sense as there is nother happened at that time. Moreover, Grisha looked very calm, completely different from the expression when Krishna fall down. Based on the consideration above, everyone may have own reason to believe the truth between the two possibilities.