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The SEP article on indexicals mentions a lot of the seemingly logical complications that arise in connection with them. Indexicals are also comparable to variables and hence objects of schematism, so again there seems to be a "logical" quality to them. The way Korsgaard and Darwall (for the latter, see his The Second-Person Standpoint) talk about the "personal perspectives" sounds like a discussion about some obscure alternative in a theory of logic.

More specifically, suppose that semantic reference is related to extensions of intensions and then quantification, all bundled up as a theory of existence-talk. Self-reference exists, and this seems like self-quantification, then, or so a circularly founded elementhood-like relation. Etc. and so forth and so on; so would it be possible to conceive of a "first-person logic" with a fundamental sentence of self-reference I ∈ I, i.e., "I am an element of myself"? Or things along that line, anyway. Is the difference between first=, second-, and third-person perspectives a logical difference, a cognitive/epistemic difference, a mathematical difference, something else, all of the above...?

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    Radulescu, The logic of indexicals:"It has been thought that the only way to construct a logic for indexicals is to restrict it to arguments which take place in a single context... I propose a logic which does away with these restrictions, and thus places arguments where they belong, in real world conversations... This enrichment of the notion of logical form leads to some seemingly counter-intuitive results: a sequence of sentences may make up a valid argument in one sequence of contexts, and an invalid one in another such sequence."
    – Conifold
    Jan 20 at 23:43
  • @Conifold that's super-interesting, I forget to ask what the effect would be on theories of proofs/validity. Is this thing about real-world conversations related to dialogical logic? That SEP article doesn't itself cite Radulescu, but the theme sounds similar to the above quote.
    – Kristian Berry
    Jan 21 at 0:13
  • He supplements logical form with relations between contexts and defines validity accordingly. Dialogical logic mostly focuses on game theoretic aspects rather than complications with indexicals, so there is no direct relation, but they do share attention to contexts and both introduce devices to account for them.
    – Conifold
    Jan 21 at 9:32

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