you see it in the mathematics solving for the volume of a black hole.
you can win the arguements its not that hard. look at it the way I do.
1. let her talk, listen closely to everything she says. you will need the information later.
2. repeat to her what she said, but do it in a way that also makes her logic contradict itself, do not say where the mistake is before recapping. it must be done as you are repeating what she says.
3. she may then say something else, repeat steps 1 and 2 to expose her flawed logic
4. when you echo her logic enough in a way that destroys the logic even by her standards, she will soon give up on it. if not, again, just keep repeating the process.
5. comfort her afterwards because now she will feel stupid about herself (make her smile, laugh etc.)
6. when she is made to smile due to your comforting that means she is seeing you as a good guy instead of somebody who has to be "right all the time".
7. now she has newfound respect towards you
8. you better reflect the same.
9. profit