In the novel "1984," how is logic manipulated and corrupted, particulary in, part two ?

In our understanding of the word, there is no logic in 1984.  The party states that the external world does not exist; that "reality" exists only in the mind.  Our concepts of logic are generally based on the belief in the existence of an outside world that is consistent and allows us to be "logical."  If you can manipulate "reality," the manipulation of "logic" is nothing.

So at the end, when Winston is made to believe that 2 + 2 really does equal 5, and this is not something that he just says, he actually believes it, he is consenting to the notion that the logic of math does not exist and that reality can be manipulated to suit specific needs, in this case the needs of the party.

We see the same thing in the Party's manipulation of history.  Something that we tend to think is real, even though most of us know practically nothing about it and can be made to believe anything we are told, is subject to manipulation.  The Party's "truth" is substituted for he "logic" of history, and after a while, who knows.  It is reminiscent of Our Ford's "History is bunk" in Brave New World.  It's bunk because we never know if it's true, and perhaps never can know.