A Note on W. Donald Oliver's Theory of Order

Abstract

In his book, Theory of Order,' Professor Oliver offers a definition of 'order.' He proposes (a) that this definition corrects certain shortcomings in customary analyses of order in mathematical logic; (b) that it is general enough to cover a large variety, if not all orders; (c) that the definition of 'order' has important deductive consequences for philosophical theories: consequently the axiomatic point of departure in philosophical analysis should include a clarified concept of order. A corollary of (c) is that Professor Oliver regards his own concept of order to be crucial for his entire philosophical outlook. I note in passing that many philosophers will take exception to (c). The problem of justifying induction, the question of the meaning, kinds and limits of justification in general, the analysis of meaningfulness and the related problem of analyticity, the issues in the unity of science program, including the question of what constitutes adequate set of primitive concepts for the and social sciences are, among others, basic philosophical issues, and they have been fruitfully discussed independently of logically prior commitments to a definition of 'order.' This observation, however, is by the way. More to the point are some misgivings as to the formal and empirical adequacy of Professor Oliver's analysis of order. But let us first see how Oliver defines 'order.' On p. 19,2 'order' is defined as an arrangement of a set of entities that is produced by the correlation, according to a rule, of one arrangement of these entities with another arrangement independent of the first. The author chooses the order of the positive integers to illustrate this definition. He distinguishes, correctly, mere succession from mathematical order. The latter is to be accounted for not merely in terms of position but also in terms of the intrinsic nature of the ordered entities. How do ordering relations order? In the case of the positive integers, for example, how do we guarantee the necessity of the natural order of the successive positive integers and how do we, at the same time, guarantee their self-identity and distinguishability when they are encountered outside this natural succession? It is proposed that one principle of arrangement respect to serial position) guarantees the necessity of the succession, while another principle of arrangement (with respect to magnitude), (logically) independent of the first, guarantees self-identity and distinguishability outside of the succession. Order arises out of the one-toone correlation of these two independent arrangements. The essential details are these. Let '0' be defined as the entity which has no predecessor, '1' as the entity which follows 0, '2' as the entity which follows 1, etc. The resulting ar-