It is common in formal settings to speak of interpretation as if it created a direct connection from the signs of a formal language to the objects of the intended domain, in effect, as if it determined the denotative component of a sign relation.  But closer attention to what goes on reveals that the process of interpretation is more indirect, that what it does is provide each sign of a prospectively meaningful source language with a translation into an already established target language, where already established means its relationship to pragmatic objects is taken for granted at the moment in question.

With that in mind, it is clear interpretation is an affair of signs which at best respects the objects of all the signs entering into it, and so it is the connotative aspect of semiotics we find to embody the process.  There is nothing wrong with our saying we interpret expressions of a formal language as signs referring to functions or propositions or other objects so long as we understand the reference is generally achieved by way of more familiar and perhaps less formal signs we already take to denote those objects.

Resources

• Cactus Language • Semantics
• Survey of Animated Logical Graphs
• Survey of Theme One Program

cc: Academia.edu • BlueSky • Laws of Form • Ma^thstodon • Research Gate
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science