MR: Please clearly state at least one “distinctive quality of sign relations”.
Sign relations are triadic relations.
Can any triadic relation be a sign relation?
I don’t know. I have pursued the question myself whether any triadic relation could be used somehow or other in a context of communication, information, inquiry, learning, reasoning, and so on where concepts of signs and their meanings are commonly invoked — there’s the rub — it’s not about what a relation is, “in itself”, intrinsically or ontologically, but a question of “suitability for a particular purpose”, as they say in all the standard disclaimers.
What Peirce has done is to propose a definition intended to capture an intuitive, pre-theoretical, traditional concept of signs and their uses. To put it on familiar ground, it’s like Turing’s proposal of his namesake machine to capture the intuitive concept of computation. That is not a matter to be resolved by à priori dictates but by trying out candidate models in the intended applications.
I gave you what I consider Peirce’s best definition of a sign in relational terms and I pointed out where it needs filling out to qualify as a proper mathematical definition, most pointedly in the further definitions of correspondence and determination.
That is the current state of the inquiry as it stands on this site …