- In the Lowell Lectures  Peirce defined the Sheet of Assertion as the representation of a universe that was constructed during a discourse between Graphist and Grapheus.
But that is just one of many ways of using logic. In 1911 he wrote about “whatever universe” and “the whole sheet”:
Every word makes an assertion. Thus ──man means “There is a man” in whatever universe the whole sheet refers to.
This is less restrictive than the definition in the Lowell lectures. For example, it would allow a logician to use a sheet of paper to write a proof by contradiction. In that case, there would be no universe about which the statements on the paper could be true.
In that case we may say that a sign’s set of denoted objects is empty. I think this tactic probably goes back to my earliest algebra courses, where our teachers cautioned us to remember that the “solution set” of a formula could be the empty set. By apt analogy, then, we may well call “a sign’s set of denoted objects” its “denotation set”. Of course an empty set is a subset of every set, but nothing about this requires the universe of discourse to be empty, much less not to exist.