Situation Theory

ion is a natural operation in semantics. By abstracting Paul from the information σ that Raymond cooked the omelette for Paul, we obtain the property of being someone for whom Raymond has cooked the omelette. By abstracting both Paul and the omelette from σ, we obtain the relation which holds between x and y just in case Raymond cooked x for y. This relation and the previous property differ from the relations and properties we have considered so far in that they are clearly not atomic. In this section we see how the infon structures can be used to model these complex relations. There are two parts to abstraction over a structural object: the object after abstraction containing holes where the abstracted object used to be, and something indicating where the holes are so that they may subsequently be filled again—we call these the ‘abstract’ and the ‘pointer’, respectively. The relationship between abstract and pointer may be captured by a binary structural relation Abs with Abs(a, x) meaning that x is a pointer to the abstract a. Definition 3.24 An abstraction structure A is an extensional structure with structural relation Abs such that if Abs(a, x) and Abs(a′, x) then a = a′. The objects of sort Abs∗ are called pointers. If Abs(a, x) then x is said to point to the abstract a. The objects of interest in abstraction structures are the abstracts, but they do not form a structural sort; rather, it is the pointers that are structurally determined. The intention is that each pointer x of an abstract a is a hereditary component of a. We obtain the results of ‘applying’ an abstract a to some argument b by substituting b for a pointer x in a. To see how this achieves the desired effect we must consider how abstracts arise by abstraction from other objects in the domain. Suppose, for example, that we want to abstract b from the infon σ = 〈〈 r; i : b, j : c 〉〉. We would expect to obtain an abstract a with exactly one pointer x such that the result of substituting b for x in a in just σ. Furthermore, the result of substituting b′ for x in a should be the infon 〈〈 r; i : b′, j : c 〉〉, if this infon exists. This is captured in the following definition. Definition 3.25 Suppose A is an abstraction structure. Given elements a and b in |A|, an abstract λb.a is the abstraction of b from a if there is an x such that 1. [b 7→ x].a = λb.a, 2. Abs(λb.a, x), and 3. the only sort of x is Abs∗. 14Analogues of pointer in the literature are ‘indeterminates’ and ‘parameters’.