For almost its entire existence, a hamburger has meant ground beef served on a bun. To admit ground beef on toast as a hamburger is to make the idea of a "hamburger" so loose, so abstract, so semiotically promiscuous as to have no meaning.
Do you know why we insist on a narrow definition of what a slider is here at AHT? Because pretty soon anything stuck between two slabs of bread will be considered a slider. Just look at what was called a slider last night on the Food Network's reality cooking show Chopped. That does not even qualify as a hamburger, let alone a slider.
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