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My opponent has three face-down morphs on the battlefield.
I play Sever the Bloodline targeting one of them.
Do we exile all three because they all have the "same name"? Or do we exile just one because "no name" indicates a lack of a characteristic altogether?
They don't have names:
702.35a Morph is a static ability that functions in any zone from which you could play the card it’s on, and the morph effect works any time the card is face down. “Morph [cost]” means “You may cast this card as a 2/2 face-down creature, with no text, no name, no subtypes, no expansion symbol, and no mana cost by paying {3} rather than paying its mana cost.” (See rule 707, “Face-Down Spells and Permanents.”)
If they don't have names, they can't have the same name, so only the morph targeted by Sever the Bloodline will be exiled.
For an official source, see the fourth ruling in Maelstrom Pulse:
A face-down creature has no name, so it doesn't have the same name as anything else.
Bonus: Why "no mana cost" gives us a converted mana cost of 0 instead of no converted mana cost?: That's because there is a specific rule about this:
202.3a The converted mana cost of an object with no mana cost is 0.
