The container is currently occupied by a material which amounts in volume to that of 50% of the container's total capacity. The terms "full" and "empty" imply that the material within the container is either present in an amount greater than 50% or in an amount less than 50%. Given that, one might even argue that the container is neither full nor empty, given the equality between the two states of presence.

My head hurts now, and the more I read this the more I'm convinced I need help.

Am I drinking or pouring?

I hate this parable. It's the most mathematically illiterate thing that people keep spouting.

The glass is 1. If it's half full, then it's 1/2. If it's half empty, that means it's 1/2/0. You can't have half of nothing. There's no such thing. It's a misnomer.

This and the year 2000 being the last year of the 20th century are two things that probably have mathematicians smashing their heads against brick walls over and over again at the blatant stupidity of people.