Exploring essential occurrence

An expression occurs essentially in a formula (or sentence) when it occurs in every formula equivalent to the given formula, taking equivalence as logical equivalence relative to the logic in play in the discussion. Setting aside various niceties, this amounts to provable equivalence if that logic is presented via some proof system, and to valid equivalence if the salient characterization is couched in semantic terms. This notion of essential occurrence, or an informal analog thereof, has found its way into numerous philosophical discussions over the past seventy or more years, and here we tease out some issues of specifically logical interest it presents, stretching that description somewhat so as to subsume under it the frequently mooted connection between the essential occurrence of a singular term in a sentence and that sentence's being genuinely about what the term denotes. This connection, stressed originally by Nelson Goodman, is touched on in several sections in the main body of the paper, but especially in 4, where it is contrasted with an alternative suggestion due to R. Demolombe and L. Fariñas del Cerro. Some issues raised by this and other parts of the discussion are also treated in several longer notes (referred to by means of letters A, B, ... , K) which are postponed to an Appendix (5) of roughly the same length as the main body of the paper. This enables readers with a special interest in one or more topics to consult them selectively, while allowing those with no such interest to avoid involvement with the further details supplied in the associated longer note(s).