Fuzzy Sets and Spatial Choice∗

Abstract

Vagueness and ambiguities in individual perception pose conceptual and technical problems in the study of spatial choice. These problems may be attributed partially to deficiencies in the set-theoretic structures underlying our models of spatial cognition and evaluation. L. Zadeh's theory of fuzzy sets provides an explicit account of such ambiguities. The potential of this account includes applications in the areas of algebraic and probabilistic choice theory, distance perception, and in temporal and population aggregation.NTENSIVE study of spatial patterns and be-I havior in terms of individual decision-making has been under way for more than a decade. Recent preoccupations and results are epitomized in the volume by Golledge and Rushton.' This, and most other recent work in behavioral geography, leans heavily on several crucial assumptions regarding 1 ) the existence or at least the constructability of action-spaces, life-spaces, mental maps, cognitive opportunity sets and the like; and 2) a value-based epistemology integrating these constructs with postulates on preference, utility, and choice. The power of these assumptions is evident: for example they underpin the scaling methodologies which have dominated recent work. The Golledge and Rushton volume, however, juxtaposes such technical analyses with other contributions, such as those of Burnett and Gould, which reveal continuing insecurity in the ontology and epistemology on which these assumptions are based, especially as they pertain to our most conspicuous single theme-spatial preference and choice.2 Geography and the Philosophy of Mind," in Golledge and Rushton, op. cit., footnote 1, pp. 23-48; and P. Gould, "Cultivating the Garden: A Commentary and Critique on SomeIn view of the continuing scrutiny of the foundations of behavioral geography, it is surprising that little attention has been paid to the implications and limitations imposed by the set-theoretic bases of the assumptions mentioned above. The categories of our descriptive terminology are Boolean: the principle of the excluded middle holds. This principle governs our descriptive partitions of geographic space (into opportunity sets, for example), and the partitions we impute to cognitive space (into choice sets, for example). It also governs the descriptive predicates we assign to sites (for example, the attribute dimensions of shopping alternatives) . Our value-based explanatory frameworks, such as choice functions and binary preference algebras, obey a truth-functional language based on the same principle. The salient assumption of traditional set theory is that set assignments are unambiguous, that is, set characteristic functions assume the values 0 or 1. Using sets and their complements this assumption implies that all universes of entities can be partitioned into exhaustive and mutually exclusive sets.Set-assignment problems in the analysis of spatial choice and repetitive travel are wellknown. Examples are the difficulty of establishing classes of destinations or of their attributes, and...

Sorites paradox and vague geographies

“…This can be shown to enable a logic, which is the extension of traditional binary set logic, allowing negation, union, intersection, etc. The full set of fuzzy set operations is large, and is covered in a number of recent textbooks [43,46], as well as numerous introductory articles [1,12,35,50,66,71]. Review of these operations is not the purpose of the current article.…”

“…A suite of early papers in the geographical literature advocated fuzzy sets especially as a basis for behavioural geography and for geographical decision making [35,50,66,67]. More recent treatments of similar topics show a continuing and developing interest in this same basic area [52,63].…”

Fuzzy Sets Approach to Spatial Analysis and Planning, a Nontechnical Evaluation

“…Pipkin (1978), Leung (1979aLeung ( , 1980b, Nijkamp (1979), and Ponsard (1979, 1980a, b, 1982) all argue that with prevailing fuzziness in our information system, preference and utility are generally vague. Individuals may not be able to discriminate perfectly between different goods or places.…”

“…Though a number of analysts (Gale, 1972;Pipkin, 1978;Ponsard, 1980a;Leung, 1982a) have argued that fuzzy sets theory can serve as an appropriate foundation for spatial analysis, the theory has developed in a slow and fragmentary way. This is true, of course, of the application of many other mathematical methods.…”

An agent system for managing uncertainty in the integration of spatio-environmental data

“…Clearly, in order to address the integration of the various spatial data sources for large-scale environmental systems, approaches such as these must be considered. Some early work by geographical scientists in the 1970s utilized fuzzy sets in topics such as behavioral geography and geographical decision-making [20,28,35]. However, the first consistent approach to the use of fuzzy set theory as it could be applied in GIS was developed by Robinson [38][39][40].…”