Universally free logic and standard quantification theory

Interest has steadily increased among logicians and philosophers in versions of quantification theory which meet the following criteria: (1) no existence assumptions are made with respect to individual constants, and (2) theorems are valid in every domain including the empty domain. Logics meeting the former of these criteria are calledfree logics by Lambert and have been investigated in a series of papers by him and by van Fraassen, and by Leblanc and Thomason.' Although it is natural to impose (2) in the presence of '1), the criteria are independent.2 Hence we baptize logics which meet both criteria universallyfree. The purpose of the present essay will be to supply a simple axiomatization of a pure universally free quantification theory and to show its equivalence to a certain fragment of standard first order quantification theory without identity, in the sense that for every wff of universally free quantification theory we can effectively define a wff A* of standard quantification theory such that A is a theorem of universally free quantification theory iff A* is a theorem of the standard calculus. We shall derive from this a proof that our axiomatization is consistent and complete.3