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But you can also say: "Do you know what 11:59PM means? 12:00PM is one minute after that"

Midday and midnight are the points at which AM/PM change and you can't logically differentiate them by appending AM or PM. You just have to know the arbitrary cultural convention that midday is PM.

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The underlying observation is that the entire hour from X:00 - X:59 is either AM or PM, and not a mixture.

12:00 and 12:01 are both in the 12:XX hour, but 11:59 and 12:00 are not. Of course the whole thing is arbitrary, but the convention is that the 12:00 - 12:59 hour is PM, rather than just 12:00.

I think, though, that your example is less convincing, since the hour changes.

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