Of course the President can lie, just like almost any other human. However, if a President is proved to have lied to the public on a matter of public interest, that constitutes grounds for impeachment.
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the cake is a lie... the cake is a lie... the cake is a lie... the cake is a lie... the cake is a lie... the cake is a lie...
Abraham Lincoln
After agreeing to take the vial and insert it into the purifier, you can tell President Eden that you have his self destruct code. He will take this as a lie, but you can then tell him the code, and he self destructs.
There is a famous apocryphal story that George Washington cut down his father's prize cherry tree with his new axe and said when questioned ,confessed with the words, "I can not tell a lie.". Lincoln was known as "Honest Abe".
A lie, which is a lie