A characterization of logical consequence in quantification theory

In this paper, we characterize the logical consequence relation as the minimal relation in a class of 'adequate consequence relations for quantification theory. These adequate relations are easily seen to include the usual syntactic relations of deductive consequence and thus our results can, perhaps, be considered an abstract version of the completeness theorem. Our work is closely related to the methods of Hintikka [1] and Smullyan [2], [3]. We adopt [3] as our standard reference and we assume the reader is acquainted with Smullyan's marvellous a, β, γ, 6 notation for sentences of quantification theory as well as his Unifying Principle [3], p. 67. In what follows, Kt-X will stand for an abstract binary relation between finite sets of sentences K and single sentences I . We allow K = 0, the empty set, in which case we write, h i . Also if Y is a sentence, we will write K, Y \-X for K U {FJhl , etc.