The satisfaction relation $\vDash$ plays a major role in set theory and model theory. The usual exposition is to define the notion of a variable assignment, then define $M\vDash\phi[s]$ for a structure $M$, formula $\phi$, and variable assignment $s$, then finally to define $M\vDash\phi$ as "there exists a variable assignment $s$ such that $M\vDash\phi[s]$". This is done in various sources (e.g. in Drake, Enderton, and on OpenLogicProject's build Sets, Logic, Computation: An Open Introduction to Metalogic) with varying verbosity. For example, OpenLogicProject explicitly defines (p.114) the function obtained by replacing the value of a variable assignment $s$ on a given input $x$ and calls the resulting function an "$x$-variant", Enderton describes this but without naming it, and Drake does not explicitly write the piecewise definition at all.
But the notion of variable assignment seems to be thought of as convoluted by people who are learning formal logic. In "An Autobiography of Polyadic Algebras" Halmos recounts this:
When I asked what 'interpretation' meant, I was answered in bewildering detail (set, correspondence, substitution, satisfied formulas). In comparison with the truth that I learned later (homomorphism), the answer seemed to me unhelpful - forced, ad hoc.
There are some other possible methods of presenting semantic truth, for example an Ehrenfeucht–Fraïssé-style game between two players. If the idea of variable assignment appears overcomplicated from a first look, is it a good idea to explain semantic truth a different way? If so, is there a presentation that might teach better than the usual Tarskian $\vDash$?
