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The distinction between sets and proper classes has interesting ontological consequences.

One way to define a proper class, in the context of set theory, is to state that it is not an element, in a set. As a consequence, if c is a proper class, $\neg\exists x(x=\{c\})$. If we take an object to exist just if it is an element, it follows that proper classes do not exist.

Nevertheless, we may quantify over classes. If e.g. R is Russell's class $\{x|x\notin x\}$, we have that $\exists x(x=R)$; but $\neg\exists x(x=\{R\})$. One may say that Russell's class subsists, but does not exist.

Do "empty names" behave similarly, so that e.g. Sherlock Holmes subsists, but does not exist?

Does the element-strategy have noteworthy advantages? Has it been explored in the literature?

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    $\begingroup$ What you describe sounds formally just like a two-sorted set/class theory (e.g. Gödel–Bernays and similar) but with different terminology, saying “$a$ exists” for what would conventionally be read as “$a$ is a set”. Is this what you have in mind, or if not, what difference besides terminological are you thinking of? Systems with terms denoting not-necessarily-existing things have certainly been defined (e.g. Fourman/Scott’s partial logic), but in all such systems I know, “$a$ exists” is equivalent to “$\exists x\ (x=a)$”, i.e. what you’re calling “$x$ subsists”. $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented Aug 30, 2021 at 18:46
  • $\begingroup$ @PeterLeFanuLumsdaine It indeed has such a provenance. The terminology I opted for has bearings upon how we think about some important problems. Yes, there are many free logics, with non-denoting terms. The approach I suggest has advantages, it seems. To wit: (1) The mere subsistence, and thus also non-existence, of proper classes, combines well with the idea that classes merely abbreviate formulas. (2) It provides a nice semantics for fictional characters, by including them in the domain of discourse, while making statements as SH is F and SH is not-F false (if we have choice-negation). $\endgroup$
    – Frode Alfson Bjørdal
    Commented Aug 31, 2021 at 0:38

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