This paper presents an account for vague predicates based on granular partitions, alternative to Bittner and Smith's one (e.g. T. Bittner and B. Smith [1], [2], [3].).
I consider one kind of vague linguistic expressions: adjectives like tall, short, big, small, etc. They are used in problematic and unproblematic cases. Consider two examples. (a) Marc is 160 cm tall and John 190 cm tall. English speakers do not hesitate to assign the adjective short to Marc and tall to John. (b) A hundred men differ for 0.5 cm with respect to their height. That is a borderline case: speakers can be indecisive about what men are tall and what short. This paper presents a model that accounts for the computational operations that underlie speakers' application of vague adjectives both to problematic and unproblematic cases. The model can then be used in a theory of formal ontology based on the notion of granularity as the one developed by Bittner and Smith.
The model is built on two basic ingredients: (i) comparison classes and (ii) granular partitions. (i) Comparison classes are introduced to account for the context-sensitivity of vague adjectives. The extension of the predicate being tall in the comparison class of men is different from its extension in the comparison class of children. (ii) We can look at the elements of a context under different standards of precision, each of them corresponding to a granular level of observation. Finer the level is, more differences between the individuals are detected. Granular partitions as equivalence classes are used to represent indistinguishability relations between objects with respect to the properties expressed by vague adjectives. The elements of each comparison class turn out to be weakly ordered with respect to each vague predicate. Such an algebraic treatment makes the model computational.