*ahem*

This is quite easy to solve using discrete math. Let 'x' equal the phrase "opposite day" spoken by person 'A'. Now, let 'y' equal all inverse logic that happens on opposite day.

We can set up the equation. A * x = y. (Meaning, a person saying it's opposite day, on opposite day, thus the inverse logic of 'y' for the phrase comes into play.)

Now, what we want to do, is to PROVE whether it truly is opposite day or not, using this equation. To do that, we need to get 'x' by itself.

x = y/A

Now this proves that the phrase "opposite day" is TRUE if person A is using inverse of his/her words. (basically lying)

This still doesn't answer our question to prove if it's truly opposite day or not though. So let's take it further. Now let's see what happens if we can make our lying person, tell the truth. To do that, we need to get 'A' by itself, and be sure it's not 1/A. Like so;

y/x = 1/A <-- not quite right so we take the reciprocal.

x/y = A <--- that's better.

So now person 'A' is telling the truth when he uses phrase "opposite day" inversely (which, it can only be used inversely on opposite day.)

And when someone uses the inverse of "opposite day" they are saying its NOT opposite day. Therefore, we can conclude opposite day (the day for inversed logic) does not exist.

And this children, is how you use math. DISCRETE MATH. Properly. Thank you. :)