Theories with the Independence Property

Abstract

A first-order theory T has the Independence Property provided T (Q)(Φ ⇒ Φ1 ∨ · · · ∨ Φn) implies T (Q)(Φ ⇒ Φi) for some i whenever Φ, Φ1, . . . , Φn are formulae of a suitable type and (Q) is any quantifier sequence. Variants of this property have been noticed for some time in logic programming and in linear programming.We show that a first order theory has the independence property for the class of basic formulae provided it can be axiomatised with Horn sentences. This condition, called crispness, is to some extent also necessary, but the properties are not equivalent.The existence of so-called free models is a useful intermediate result. The independence property is also a tool to decide that a sentence cannot be deduced. We illustrate this with the case of the classical Carathéodory theorem for Pasch-Peano geometries.