Teaching logic: Tarski vs Mates vs "logical constants"
I'm teaching an intermediate/advanced logic course this semester. So I had to ask myself how to introduce the semantics of quantifiers, with an eye on proving soundness and completeness. The standard approach, going back to Tarski, defines a satisfaction relation between a formula, a model, and an assignment function, and then defines truth by supervaluating over all assignments. The main alternative, often found in intro logic textbooks, is Mates' approach, where ∀xA(x) is defined as true in a model M iff A(c) is true in every c-variant of M, where c is a constant not occurring in A.