Why does one need valuations in order to define the semantics of first-order logic? Why not just define it for sentences and also define formula substitutions (in he expected way). That should be enough:$$M \models \forall x. \phi \iff \text{for all }d\in \mathrm{dom}(M),\ M \models \phi[x\mapsto d]$$instead of$$M,v \models \forall x. \phi \iff \text{for all }d\in \mathrm{dom}(M),\ M, v[x\mapsto d] \models \phi$$
Why valuations when defining FOL?
lo.logic