...if adding a negative number yields a smaller number, then subtracting a negative number would do the opposite by increasing that number.... But . . . that's not an explanation.

It might not be an explanation you accept, but saying that "doing the opposite ought to lead to an opposite result"

*is* an explanation, and a fairly reasonable one.

The explanation provided here earlier was more logical than mathematical: the result must be "this" or "that", and "that" leads to a logical contradiction, so the result then must be "this". (A mathematical proof would involve graduate-level field theory.) Your father has provided you with another logical explanation; namely, if "this" leads to "the other", then "not this" ought sensibly to lead to "not the other". And there are various other explanations and mental pictures, such as those provided

**here**,

**here**,

**here**, and

**here**.

If none of these "works" for you, then you may just need to accept the rule for now, and give yourself some time to become comfortable with its practice.