Shortcut Booleans

I wonder why, traditionally, shortcut evaluation of boolean subexpressions
has been applied to “and” and “or” connectives, but not any others.

For example, what about “implies”?

    a implies b

Truth table:

    a  b  result
    ------------
    0  0    1
    0  1    1
    1  0    0
    1  1    1   

But C does not have an “implies” operator, I hear you say? Of course this
can be written

    not a or b

and shortcut evaluation would apply. But it can also be written

    a <= b

and shortcut evaluation would *not* apply. Why not, if a and b are
boolean-valued subexpressions?