Model theory: What it is and what it isn’t

Model theory is the branch of logic concerned with relations between formal languages and extralinguistic structures. The most basic technical notion is that of a formula being true in a given structure, and the subject originated with Tarski and his famous definition of truth for classical first-order languages. On a common usage followed here, a model is just a structure of the kind sentences are true or false in. Model theory contrasts with proof theory, which is concerned only with formal deducibility relations among sentences, regardless of any connection with extralinguistic structures. Important post-Tarski contributors to the model theory of classical first-order languages have included Abraham Robinson, who gave the subject an orientation towards applications to abstract algebra, where the pertinent structures are groups and rings and fields, and Saharon Shelah, who introduced methods of great technical sophistication. Many mathematical logicians today use “model theory�? to refer only to technically sophisticated, algebra-oriented, first-order model theory, a subject with little bearing on philosophy. Model theory has been developed also, however, for nonclassical formal languages (tense, modal, and other), especially by Saul Kripke and his successors. This branch of the subject claims applications to philosophy of language and theoretical computer science.