The Logic of Totality
Some expressions are used to explicitly assert the completeness of a description or report. Such “totality operators” are approximately expressed, in English, by locutions such as “and that’s it”, “and that’s all”, or “and that’s the whole truth”. We need them because saying “p, and that’s it” typically goes beyond merely asserting p. In order to give a systematic and precise account of what extra inferences are licensed, we need a logic of totality operators.
Some of the most interesting features of the logic of totality emerge when we consider the interaction of operators that express different concepts of totality. For instance, if p is complete in the sense of entailing every positive truth, then “p, and p is complete in the sense of entailing every positive truth” is complete in the stronger sense of entailing every truth. A logical system with different symbols for different totality operators allows us to represent such relationships in a perspicuous way.
Outside the context of fundamental metaphysics, totality operators will typically be restricted: not all truths are always relevant, after all. We will thus also study the logic of totality operators that are restricted to a subject matter.