Systems between First-Order and Second-Order Logics

Abstract

The most common logical system taught, used, and studied today is Elementary predicate logic, otherwise known as first-order logic (see Hodges’ chapter in this Volume). First-order logic has a well-studied proof theory and model theory, and it enjoys a number of interesting properties. There is a recursively-enumerable deductive system D2 such that any first-order sentence Φ is a consequence of a set Γ of first-order sentences if and only if Φ is deducible from Γ in D1. Thus, first-order logic is (strongly) complete. It follows that first-order logic is compact in the sense that if every finite subset of a set Γ of first-order sentences is satisfiable then Γ itself is satisfiable. The downward Lowenheim-Skolem theorem is that if a set Γ of first-order sentences is satisfiable, then it has a model whose domain is countable (or the cardinality of G, whichever is larger). The upward Lowenheim-Skolem theorem is that if a set Γ of first-order sentences has, for each natural number n, a model whose domain has at least n elements, then for any infinite cardinal κ, Γ has a model whose domain is of size at least κ (see Hodges’ chapter, and virtually any textbook in mathematical logic, such as Boolos and Jeffrey [1989] or Mendelson [1987]).