My understanding of logic is limited to first order logic without functions with finite set of domain constants, and with herbrand semantics. Now in this setting, I would like to understand the independence property of logical theories --- My goal is to understand VC dimension in this setting.
So let us say I have a FOL language with only the predicate $R$, and domain constants $n$, represented by $[n] = \{1,...,n\}$. Does this language have the independence property?
In my understanding of independence property thus far, this language will have the independence property, if we can write a formula for picking out any subset of the directed graphs on $[n]$. Is this correct?
Furthermore, if I do not put any restrictions, then I can always axiomatize any graph on $n$ in this language, by simply writing out the graph edges in this language.
However, if I restrict that I only allow formulas with no constants, then I do not have independence property, as I can not distinguish between isomorphic graphs. Is this correct?
Finally, How can I compute the VC dimension of a theory/formula?
I am sorry if my question is all over the place, but I am finding it hard to get a rigorous understanding of independence property in herbrand logic.