Interpretation is the process of evaluating a construct in some context, translating it from one form to another. The symbols that we use for interpretation are,! and. We read K A,!C L as the partial interpretation kind from kind K to kind L by agent A in context C. Likewise, is read full interpretation. When the agent and context are clear, we simply write K L and read it as the full interpretation kind from K to L. An interpretation is a computable functional kind that preserves some type of structure. We define two sort of interpretation kind. Full interpretations are computable functions that preserve all structure from their domain. This means
that the semantics, that is, the validity, of all related constructs is maintained across interpretation. Within kind theory a full interpretation is defined as a functor on a specific category of kind. A partial interpretation is a computable function that preserves some substructure from its domain. Partial interpretations are, categorically, forgetful functors. Interpretation is a transitive operation. The identity interpretation is always defined on all kinds and instances—it is the identity function, denoted with the term id. Any evaluation process is a type of interpretation. Reading this document is one kind of interpretation, evaluating a mathematical expression with Mathematica is another. In each case, data is transformed via an agent (you, the reader, in the former case, and the Mathematica process in the latter) within a specific context.
