Theories with the Independence Property, Studia Logica 2010 95:379-405

A first-order theory T has the Independence Property provided deduction of a statement of type (quantifiers) (P -\u3e (P1 or P2 or .. or Pn)) in T implies that (quantifiers) (P -\u3e Pi) can be deduced in T for some i, 1 \u3c= i \u3c= n). 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 formulas provided it can be axiomatized with Horn sentences. The existence of 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 Caratheodory theorem for Pasch-Peano geometries