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Returning to the Barcan formulas, where what is at stake is a comparison between $\bigcirc \exists x P(x)$ and $\exists \bigcirc P(x)$. Given our set up, strictly these are ill-formed. The monad $\bigcirc_W$ sends $W$-dependent types to the same. A quantifier such as $\exists_{x:A}$ change context, sending $A$-dependent types to plain types.
Two thoughts then.
(1) Say I have $w: W, x: A(w) \vdash \phi(w, x): Prop$. Then I can form $\bigcirc_W \sum_{x: A(w)} \phi(w, x)$, the world-dependent (constant) type containing all possible $A$s which are $\phi$. Evaluated at a specific world, this amounts to $\sum_{w:W}\sum_{x: A(w)} \phi(w, x)$. Then I can also form $\sum_{(w, x):\sum_{w:W}A(w)}\phi(w, x)$.
A version of the Barcan formulas then amounts to the equivalence of taking dependent sum in one or two stages.
$\sum_{w:W}\sum_{x: A(w)} \phi(w, x) \simeq \sum_{(w, x):\sum_{w:W}A(w)}\phi(w, x)$It’s just the rebracketing of the three-part terms in pairs: $(w, (a, p)) \leftrightarrow ((w, a), p)$. We might say:
Possibly there’s an $A$ which is $\phi$ and there is a possible $A$ which is $\phi$.
(2) Alternatively, we are imagining some trans-world identity of $A$s, perhaps by the pulling back to the constant world-dependent type $W^{\ast} A$. Then what should I say strictly?
If I have $w: W, x: A$ as my context, I guess strictly I shouldn’t rely on this being the ’same’ as $x: A, w: W$, and so interchanging $\bigcirc_W$ and $\exists_{x:A}$ is not allowed.
Maybe it just amounts to the previous solution but with $W^{\ast} A(w)$ instead of $A(w)$.
If you find the metaphysics of possible worlds unintuitive, turning to temporal logic and the Future operator may help. Some people would want to deny the assertion “in the future there will be a ruler of the world” entails “there is something (now) which in the future will be ruler of the world”. My solution (1) is about reworking the latter claim into “there is a future person who will be ruler of the world”, and is presumably unexceptionable as the rebracketing account would suggest.
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