There’s an ambiguity to the adjective “modal,” so that sometimes we would juxtapose “modal and deontic logic” but sometimes we’d say instead “alethic modal and deontic modal logic.” But so in the latter way of speaking, let’s say that an application of deontic logic to an entire system is, or might be, along the lines of what you’re asking about.
If that’s acceptable: we have a two-tiered example in Peterson[14], where an attempt is worked out to (A) ground deontic logic in category theory and (B) ground an understanding of Canadian legal ethics in the conditions of (A).
If that’s not quite acceptable: I do wonder how we might adapt a whole metaphysical system to modal logic as such. Kant arguably does the deepest soul-searching, in the main “canon,” when it comes to the epistemology of …
There’s an ambiguity to the adjective “modal,” so that sometimes we would juxtapose “modal and deontic logic” but sometimes we’d say instead “alethic modal and deontic modal logic.” But so in the latter way of speaking, let’s say that an application of deontic logic to an entire system is, or might be, along the lines of what you’re asking about.
If that’s acceptable: we have a two-tiered example in Peterson[14], where an attempt is worked out to (A) ground deontic logic in category theory and (B) ground an understanding of Canadian legal ethics in the conditions of (A).
If that’s not quite acceptable: I do wonder how we might adapt a whole metaphysical system to modal logic as such. Kant arguably does the deepest soul-searching, in the main “canon,” when it comes to the epistemology of modality (or: he’s the hardest hitter until Quine, maybe…). But so Peterson[23] explores Kantian modality theory in light of the technique of Barcan’s modern modal theorizing.
Also, not quite a whole system, or if it is, it’s yet “embedded” in an even greater system: but Hamkins (who is a major philosopher of mathematics, esp. set theory, today; and an extremely effective mathematician, too) and Loewe do address the matter of formulating one kind of multiversal standpoint in terms of modal logic.
Now, you’ve gotten a lot of good examples of what you asked after, throughout our answers to your question, so I can’t expect you to accept mine as the most helpful or “definitive” such answer. However, if I do find a stronger example than I did so far, I would be motivated to edit my post to include it, and then I might change my mind about what to hope for…
Share Improve this answer Follow answered Jul 24, 2024 at 2:39 user40843 Add a comment
There’s Hellman’s modal structuralism which seeks to nominalize mathematics via irreducible modal operators. Instead of abstract structures or entities, there’s a nominalist structure of possibilities and necessities. The possibilities and necssetities are irreducible. It uses only modal logic, S5 to be specific.
It encompasses all of math, from the most basic arithmetic to the most arcane.
https://plato.stanford.edu/entries/structuralism-mathematics/
Here’s an example from SEP:
Possibly, there exists an M such that M is a model of the Dedekind-Peano axioms.
Necessarily, for all relational systems M, if M is a model of the Dedekind-Peano axioms, then 2M+3M=5M
I’m unsure if you will count this, but I think it’s something you should be aware of in case you do. Hellman works in philosophy of math, and his system displaces other entire systems (fictionalism, realism, abstract stucturalism, etc) by philosophers.
As an aside, I an very intrigued by this ability to nominalize vast systems via modal logic, and wonder if there’s similar work outside of mathematics.
Share Improve this answer Follow edited Jul 24, 2024 at 2:33 answered Jul 24, 2024 at 0:40 J Kusin’s user avatar J Kusin 4,45411 gold badge1010 silver badges21