Modern Mathematical Logic

Suppose that we have a (first-order) language ℒ. As emphasized in , the elements of Formℒ are just syntactic sequences of symbols, and we only attach meaning to these formulas once we provide an ℒ-structure together with a variable assignment. The fundamental separation between syntactic formulas and semantic structures is incredibly important, because it opens up an interesting way to find both commonalities and differences across structures. That is, given two structures with variable assignments (ℳ1,s1) and (ℳ2,s2), we can compare the two sets {φ∈Formℒ: (ℳ1,s1)⊨φ} and {φ∈FormL:(M2,s2)⊨φ}. Although the two structures and variable assignments likely live in different worlds, these two sets both live inside the same set Formℒ. In other words, the syntactic nature of the formulas provides a shared substrate where we can perform comparisons.