The sentence "When you give them an inch, they will take a mile" illustrates a form of flawed reasoning often related to the slippery slope fallacy, but among the provided options, it does not directly correspond to any of them. However, it suggests an emotional appeal in terms of fear or concern about allowing someone to take advantage of a small concession, aligning somewhat with appeal to emotion.
So, the best fit from the given options would be appeal to emotion.