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It's fairly easy to conjure a set theory with a modal logic in it. I'm wondering whether the construction is incoherent or useless. I'm also curious what ways of extending set theory with a modal logic already exist.

Let $\varepsilon$ denote the empty set.

There are many questions on this site that ask about what ordered pairs are and the answer usually boils down to this is what a Kuratowski pair is, although there are alternatives.

I think that the standard way to express the above statement is to use model theory. Suppose we are working in ZFC. My understanding of model theory is a bit weak, but I think this amounts to taking the signature of ZFC and its theory, extending the signature with an ordered pair function symbol, and extending the theory with the characteristic property of ordered pairs. We then show that original ZFC with an ordered pair defined as a Kuratowski pair is a model of our extended signature + theory. I'm not exactly sure how to deal with a model whose domain is a proper class rather than a set, but I think the above approach works.

I'm wondering whether it makes sense to instead consider an extended set theory that uses modal logic and is thus capable of hosting non-primitive symbols itself.

For the sake of this example, I'm considering a modal set theory that is like an ordinary set theory such as ZFC, but all of the usual set axioms and axiom schemas of set theory are necessary truths.

The meta-statement a Kuratowski pair is a possible implementation of an ordered pair in some set theory has an obvious translation into modal logic, shown below. The particular modal logic I am using here is S5, but I'm not sure what the consequences are of using different modal logics.

Assuming we do some work off to the side to define the notation $(\cdot,\cdot)$ and assert that it is a legal term if its two arguments are terms, we can express the characteristic property of ordered pairs as follows (10).

$$ \Box \bigg( (a, b) = (b, a) \iff a = b \bigg) \tag{10} $$

We can assert that we are choosing the Kuratowski pair in this world by not using a modal operator at all (11).

$$ ( a, b ) = \{\{a\}, \{a, b\}\} \tag{11} $$

We can assert that a Wiener pair also works

$$ \Diamond \bigg( (a, b) = \{ \{\{a\}, \varepsilon\} , \{\{b\}\}\} \bigg) \tag{12} $$

I think the main thing we've bought by adding modal operators to our logic is the ability to make things that would have been metatheorems into theorems.

For instance if we wanted to express the fact that an ordered pair can't be associative, we can express it as follows

$$ \lnot \Diamond \bigg( (a, (b, c)) = ((a, b), c) \bigg) $$

Greg Nisbet
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  • If axioms were necessarily true, it would mean that no other axiom would be possible. But, there is not only one set theory. - Saying that a definition is necessarily true may be seen as redundant, for a definition is ( usually) a logical equivalence ( $\equiv$). –  Jun 03 '20 at 15:03
  • The modal operators are just logical symbols within the theory, like $\lnot$ or $\forall$. I'm not using them to make meta-statements about whether set theory is necessarily true or which things are necessarily true about all set theories. A set theory equipped with a modal logic is not meant to talk about different possible set theories in the example given but rather a single, fixed theory (such as ZFC) with non-primitive notions like ordered pairs implemented in terms of sets. I was intending both (10) and (11) to be axioms in the new, expanded system. – Greg Nisbet Jun 03 '20 at 15:31
  • As far as I got it, you would like to use a first oder extension of modal logic (which one?) to talk about different possible 'realizations' of the axioms of ZFC. You'll end up with a Kripke frame where for each possible world the non-modal part of the semantics is given by classical first order logic, e.g., the ordered pair might be interpreted as in (11) or as in (12). But for what purpose? Do you expect any deeper insight from analyzing the Kripke structure than from analyzing these possible realization individually? t.b.c – Willem Hagemann Jun 25 '20 at 12:03
  • I mean, do you have any meaningful example in mind, like 'if (11) holds, then necessarily ... holds'? – Willem Hagemann Jun 25 '20 at 12:04
  • Please replace 'realizations of axioms' by 'realizations of semantics' in the comment above. – Willem Hagemann Jun 25 '20 at 12:12
  • @WillemHagemann, I picked S5 as the modal logic to use, but didn't commit to it. I'm asking about something kind of like fuzzy set theory, but with modal logic in place of fuzzy logic. My initial motivation is to be able to express a) a meta-theorem like {{x}, {x, y}} is a possible implementation of a pair as a conditional theorem, where the characteristic property of ordered pairs is a premise, and b) to come up with a formal setting where we can distinguish arbitrarily chosen definitions from other facts. – Greg Nisbet Jul 07 '20 at 16:15
  • More generally, I'm interested in how you can use set theories to "host" arbitrary logics as a way of studying the logics. – Greg Nisbet Jul 07 '20 at 16:16

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