This is the topic of "clone theory": the study of which (multivariable) operations can be created by composing together a starting collection of operations. Pos

This is the topic of "clone theory": the study of which (multivariable) operations can be created by composing together a starting collection of operations. Post's lattice is a complete description of which gates can be generated from which other gates in two-valued logic, but the subject becomes much more difficult in multivalued logic. If you are just interested in whether a given gate would allow you to generate all possible operations in a multivalued logic, then there is a nice result called the Rosenberg Completeness Theorem that says, roughly speaking, that any operation which doesn't have a nice property (from a certain explicit list of nice properties, such as monotonicity, linearity, etc.) will generate all other operations. To answer your question, NAND is special because it isn't monotone, isn't linear (in the sense of not being a linear function modulo 2), isn't self-dual, doesn't send all-0s to 0, and doesn't send all-1s to 1.