Valences

Every BQN function can be called with one or two arguments, possibly doing completely different things in each case. The Valences (⊘) 2-modifier grafts together a one-argument function 𝔽 and a two-argument function 𝔾, with the resulting function calling one or the other as appropriate. It's the tacit equivalent of a block function with two bodies. So the function {÷𝕩 ; 𝕩-𝕨} can also be written ÷⊘(-˜). A full definition of Valences as a block is {𝔽𝕩;𝕨𝔾𝕩}.

      -⊘+ 6  # - side
¯6

    3 -⊘+ 2  # + side
5

Valences provides one way to check whether 𝕨 is present in a block function. The expression 𝕨0⊘1𝕩 always ignores the values of the arguments, resulting in 0 if 𝕨 isn't given and 1 if it is (if you want 1 or 2, then ≠𝕨⋈𝕩 is shorter, but I'm not sure if I like it).

        {𝕨0⊘1𝕩} 'x'
0

    'w' {𝕨0⊘1𝕩} 'x'
1