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**1 - 2**of**2**### A Theory of Classes: Proofs and Models

, 1996

"... We investigate the proof structure and models of theories of classes, where classes are `collections ' of entities. The theories are weaker than set theories and arise from a study of type classes in programming languages, as well as comprehension schemata in categories. We introduce two langua ..."

Abstract
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We investigate the proof structure and models of theories of classes, where classes are `collections ' of entities. The theories are weaker than set theories and arise from a study of type classes in programming languages, as well as comprehension schemata in categories. We introduce two languages of proofs, one a simple type theory and the other involving proof environments for storing and retrieving proofs. The relationship between these languages is defined in terms of a normalisation result for proofs. We use this result to define a categorical semantics for classes and establish its coherence. Finally, we show how the formal systems relate to type classes in programming languages. 1 Introduction The classes of the title are collections of entities. We choose this term as more neutral than sets, as we have a precise idea of the rather primitive theory we wish to explore which is quite distant from a set theory. The theory focuses on the extent (or extension or abstraction) or, as...