thing about the relation between
     the object and the sign, i.e., that
     the object determines the sign.

Are you under the impression that objects determine signs?
I will have to think about that.  As you know, the proper
reading of the definition, if ever we arrive at it, will
depend on using the author's meanings for "correspondence"
and for "determination", which CSP gives in full, and at
length, needless to say, in many other prominent places.
But I still read this definition as defining a relati