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A graph theoretic approach to the problem of the "just noticeable difference" or threshold in visual perception and in making choices. Because we cannot distinguish between points that are too close, in order to study perception and preferences experimentally, we must replace classical geometric primitives--such as straightness, parallelism, perpendicularity, and betweenness--with more general notions obtained from the classical by substituting closeness for identity: a tolerance geometry. This paper presents and proves axioms for tolerance geometry on the line where the classical ternary relation of betweenness is replaced by the tolerance relation of epsilon-betweenness. The axioms give the necessary and sufficient conditions for the existence of a real-valued function satisfying the desired inequality. 22 pp. Ref. (See also P-4423.) (MW)

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