This is primarily expanding on my comment on the question. This answer gets you about as close as Scala lets you get to what you want, but it also shows why it's probably not a good idea to do what you're doing.
You can express (something close to) the type you want, but I'm not sure what you intend to gain. First, if you want to take different arglist types, then Column needs to be generic.
trait Column[-A] {
def rule(arg: A): String
}
Then we can implement your case objects as subclasses of an appropriate parameterization of this.
case object FirstColumn extends Column[String] {
def rule(arg: String): String =
"stub implementation"
}
case object SecondColumn extends Column[(String, String)] {
def rule(arg: (String, String)): String =
"stub implementation"
}
Note that FirstColumn and SecondColumn do not inherit from the same Column[A] as they don't implement the same method. We can get them to have a common type, but... not in a very useful way.
One option is to find a common supertype of Column[String] and Column[(String, String)], which (since the argument is contravariant) is akin to finding a common subtype of String and (String, String). The closest common subtype is... Null. That's not helpful unless you're only ever planning to pass null to your rule.
Instead, we can use existentials.
val foo: Column[_] = FirstColumn
val bar: Column[_] = SecondColumn
Now we've lost all type information. You can access the foo.rule slot and you can print it, but you can't call it because we don't know what we need to pass it. You'll have to do a cast to get it back to a usable format.
The point that I'm making here is that, yes, it's doable, but once you've lost as much type information as you're giving up, there's not much point. The type system is correctly telling us that foo and bar have virtually nothing in common except the existence of a method named rule which takes... some kind of argument. From a type theory perspective, it's hard to get more uninteresting than that.
ruleto take variable arguments in the trait, then you're promising that the implementation works for any number of arguments, not just some specific implementation-dependent number of arguments. So aFirstColumnwhich provides a 1-ary version of the function does not correctly implement the trait.Column, how would I callruleon it? If I pass one arg and it turns out to be aSecondColumn, that's an error. And vice versa if I pass two and it's aFirstColumn. So if you have aColumnthen you can't do anything type-safe with it, which defeats the point of it even being a trait.