Levellism and the Method of Abstraction

ion can be seen as doing for discrete systems what differential calculus has traditionally done for analogue systems. Likewise, the use of predicates is essential in subjects like information and computer science, where discrete observables are paramount and hence predicates are required to describe a system behaviour. In particular, state-based methods like Z (Hayes and Flinn [1993], Spivey [1992]) provide notation for structuring complex observables and behaviours in terms of simpler ones. Their primary concern is with the syntax for expressing those predicates, an issue we shall try to avoid in this paper by stating predicates informally. The time has come now to combine approximating, moderated LoAs to form the primary concept of the method of abstraction. 2.6. Gradient of abstraction For a given (empirical or conceptual) system or feature, different LoAs correspond to different representations or views. A Gradient of Abstractions (GoA) is a formalism defined to facilitate discussion of discrete systems over a range of LoAs. Whilst a LoA formalises the scope or granularity of a single model, a GoA provides a way of varying the LoA in order to make observations at differing levels of abstraction. For example, in evaluating wine we might be interested in the GoA consisting of the “tasting” and “purchasing” LoAs, whilst in managing a cellar we might be interested in the GoA consisting of the “cellaring” LoA together with a sequence of annual results of observation using the “tasting” LoA. L. Floridi – J. W. Sanders, Levellism and the Method of Abstraction ——————————————————————————————————————————————-—————— ——————— Information Ethics Group – Research Report 22.11.04 ———— 13 In general, the observations at each LoA must be explicitly related to those at the others; to do so, we use a family of relations between the LoAs. For this, we need to recall some (standard) preliminary notation. Notation. A relation R from a set A to a set C is a subset of the Cartesian product A × C. R is thought of as relating just those pairs (a, c) that belong to the relation. The reverse of R is its mirror image: {(c, a) | (a, c) ∈ R}. A relation R from A to C translates any predicate p on A to the predicate PR(p) on C that holds at just those c:C, which are the image through R of some a:A satisfying p PR(p)(c) = ∃a: A R(a,c) ∧ p(a). We have finally come to the main definition of the paper. Definition. A gradient of abstractions, GoA, is defined to consist of a finite set {Li | 0 ≤ i < n} of moderated LoAs Li and a family of relations Ri,j ⊆ Li × Lj, for 0 ≤ i ≠ j < n, relating the observables of each pair Li and Lj of distinct LoAs in such a way that: 1. the relationships are inverse: for i ≠ j, Ri,j is the reverse of Rj,i 2. the behaviour pj at Lj is at least as strong as the translated behaviour PRi,j(pi)