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There are some spells and abilities that subtract from a creature's power/toughness: for example, part of Ulcerate says "Target creature gets -3/-3 until end of turn."

If the target creature is only a 2/2, is that target destroyed (goes to graveyard) from the -3 toughness given to it? If not what happens, does it stick around with negative toughness?

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A creature with 0 or less toughness will be sent to the graveyard immediately, so yes, getting -3/-3 can kill a creature.

Note that this is not technically being "destroyed", it is a different way of sending the creature to the graveyard. This matters for things like regeneration.

From the basic MTG rules:

Your creatures go to your graveyard if the damage they’re dealt in a single turn is equal to or greater than their toughness, or if their toughness is reduced to 0 or less.

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    Just in case anyone is looking for the comprehensive rules, rule 704.5f describes the action that moves a creature to the graveyard, and rule 121.1a explains how counters affect toughness. I didn't post this as an answer because the basic rules are definitely more suitable here. – Rainbolt May 20 '14 at 14:55

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