Institutions and Peirce's Triadic Semiotics
I tend to think of the interpretant as the relation between the token and its object, implicitly including all the contextual information needed to make that particular connection. However, I depart from Peirce in thinking that this relation should include not just one token and one object, but all the potential tokens and meanings for that context, including those where the relation does not hold. This extension is consistent with a fundamental insight of the linguist Saussure (during the same time frame as the logicians), that signs come in systems, not as individuals; this is particularly evident in language, but it also seems to hold in general (e.g., consider alphabets, traffic signs, ...), and it is consistent with much of Peirce's own writing, e.g., his (largely undeveloped) "Speculative Grammar," but not with his writings on the triadic nature of the sign. Institutions formalize this broad triadic point of view, with the added refinement that meaning should change consistently with changes in context. Thus, the "signature" argument of the institutional satisfaction relation should be thought of as a context; this is particularly well illustrated by a database example in an appendix to Data, Schema, Ontology, and Logic Integration, in which queries are sentences, answers are models, and databases are contexts. An additional possible refinement of the institution notion supports vagueness, by allowing the relation of satisfaction to hold "to some extent," e.g., by taking values in a partial order.