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What’s the relationship between a first-order (Boolean) hyperdoctrine, which
does for (classical) predicate logics precisely what Lindenbaum algebras do for propositional logic,
and a Boolean coherent category, which is the syntactic theory of a classical first-order theory?
Briefly, a hyperdoctrine keeps the type theory separate from the logic, whereas a syntactic category realizes every predicate as a subobject of a type. The subobject fibration of a Heyting category is a hyperdoctrine, but the predicates in a general hyperdoctrine may not be realizable as objects of the base category. So I think it depends on how general a notion of semantics you are interested in.
Thanks for that Mike. I was exploring the thought of making first-order the coalgebraic conception of modal logic as duality between coalgebras on Stone and algebras on Boolean Algebras, so was interested to see these slides on the duals for Boolean hyperdoctrines as indexed Stone Spaces. One might imagine sticking in the modalities on top of this duality.
Then I’d like to see what hyperdoctrine duality had to do with Forssell’s duality between the syntacic categories (Boolean coherent categories) and topological groupoids of models. Then see whether the ionad as modal model fits in.
Got to dash very quickly now, but I wonder if $Set^X$ has something to do with indexed Set (as dual of CABA), rather than indexed stone space (as dual of BA).
So, as far as classical first-order theories go, syntactic categories are Boolean coherent, and these are included in Boolean hyperdoctrines? So, then, Awodey and Forssell’s duality between the category of ($\kappa$-small) Boolean pretoposes and Stone topological groupoids is contained within the duality between Boolean hyperdoctrines and indexed Stone spaces?
So, now I see there’s a strand to the semantics of modal predicate logic which can be viewed as taking a Boolean hyperdoctrine $B^{op} \to BA$, $B$ a category with finite products (the types), then extending to a modal hyperdoctrine $B^{op} \to MA$ (modal algebras are kinds of BAOs).
Just as there are duals for Boolean hyperdoctrines $B \to Stone$ (indexed Stone spaces), now we have duals of modal hyperdoctrines $B \to$ descriptive general frames. Descriptive general frames have been shown to be coalgebras on $Stone$ as modal algebras are algebras on $BA$.
These indexed descriptive general frames, or at least a form of them for specific $B$, have appeared in the literature as metaframes, and it is known that
all modal predicate logics are complete with respect to cartesian metaframes.
Interesting!
So, commonly, they choose the $B^{op}$ to be the natural numbers with mappings. Intuitively, sitting above $n$ there is the Boolean or modal algebra of formulas whose free variables are contained in the set $\{x_1, ..., x_n\}$. Then on the dual side, $0$ is mapped to a Stone space of worlds, $H(0)$, and $1$ to the space of individuals, $H(1)$. The map $1 \to 0$ gives a fibring of individuals onto worlds. $H(n)$ are similarly fibred over worlds. Particular attention has been paid to cases where a fibre of $H(n)$ is the $n$-fold product of the fibre of $H(1)$.
The frame aspect of the construction captures ’counterpart’ relations between individuals. This is a notion due to the philosopher David Lewis, who took possible world talk very seriously. If ’I might have had eggs for breakfast’ is true in this world, it is made true by the existence of a close world in which my counterpart did have eggs for breakfast. The sorts of complexity one would need to capture include an individual in this world having no counterparts or many of them in an accessible world. With some homotopic nontriviality thrown in, one could imagine a loop in the space of possible worlds along which taking the counterpart relation permutes individuals.
In ’cartesian’ cases where $H(1)$ determines $H(n)$, the information is presumably encoded in the ’bundle’ $H(1) \to H(0)$, which maps the counterpart relation for individuals to the accessibility relation for worlds. This would explain interest in Kripke sheaves.
Awodey and Kishida’s work extend Kripke sheaves to sheaves over general topological spaces, rather than those generated using the Alexandroff topology from a preorder.
Using different categories for $B$ would allow for typed modal logics.
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