axiomatic method originally, a method for reorganizing the accepted propositions and concepts of an existent science in order to increase certainty in the propositions and clarity in the concepts. Application of this method was thought to require the identification of (1) the ‘universe of discourse’ (domain, genus) of entities constituting the primary subject matter of the science, (2) the ‘primitive concepts’ that can be grasped immediately without the use of definition, (3) the ‘primitive propositions’ (or ‘axioms’), whose truth is knowable immediately, without the use of deduction, (4) an immediately acceptable ‘primitive definition’ in terms of primitive concepts for each non-primitive concept, and (5) a deduction (constructed by chaining immediate, logically cogent inferences ultimately from primitive propositions and definitions) for each nonprimitive accepted proposition. Prominent proponents of more or less modernized versions of the axiomatic method, e.g. Pascal, Nicod (1893–1924), and Tarski, emphasizing the critical and regulatory function of the axiomatic method, explicitly open the possibility that axiomatization of an existent, preaxiomatic science may lead to rejection or modification of propositions, concepts, and argumentations that had previously been accepted. In many cases attempts to realize the ideal of an axiomatic science have resulted in discovery of ‘smuggled premises’ and other previously unnoted presuppositions, leading in turn to recognition of the need for new axioms. Modern axiomatizations of geometry are much richer in detail than those produced in ancient Greece. The earliest extant axiomatic text is based on an axiomatization of geometry due to Euclid (fl. 300 . .), which itself was based on earlier, nolonger-extant texts. Archimedes (287–212 B.C.) was one of the earliest of a succession of post- Euclidean geometers, including Hilbert, Oswald Veblen (1880–1960), and Tarski, to propose modifications of axiomatizations of classical geometry. The traditional axiomatic method, often called the geometric method, made several presuppositions no longer widely accepted. The advent of non-Euclidean geometry was particularly important in this connection. For some workers, the goal of reorganizing an existent science was joined to or replaced by a new goal: characterizing or giving implicit definition to the structure of the subject matter of the science. Moreover, subsequent innovations in logic and foundations of mathematics, especially development of syntactically precise formalized languages and effective systems of formal deductions, have substantially increased the degree of rigor attainable. In particular, critical axiomatic exposition of a body of scientific knowledge is now not thought to be fully adequate, however successful it may be in realizing the goals of the original axiomatic method, so long as it does not present the underlying logic (including language, semantics, and deduction system). For these and other reasons the expression ‘axiomatic method’ has undergone many ‘redefinitions,’ some of which have only the most tenuous connection with the original meaning. See also CATEGORICITY , DEDUCTION , FOR- MALIZATIO. J.Cor.