It may not be possible to draw a sharp line between things that exist from the things that do not exist. Surely there are problematic referents ("the smallest triple of numbers in lexicographic order such that a^3+b^3=c^3", "the historical jesus", "the smallest pair of numbers in lexicographic order such that a^3+24=c^2", "shakespeare's firstborn child") that need considerable working with before ascertaining that they exist or do not exist. Given that difficulty, it seems like we work with existence explicitly, as a theory; it's not "baked in" to human reasoning.

Guy Steele wrote a talk called "Growing a Language", where one of his points is that building hooks (such as functions) into the language definition to allow the programmer to grow the language is more important than building something that is often useful, say, complex numbers or a rich collection of string manipulation primitives. Maybe talking about the structure of "theories of X" would be valuable. Perhaps all theories have examples (including counterexamples as a specific kind of example) and rules (including definitions as a specific kind of example) - thats the kind of thing that I'm suggesting might be more like a hook.