Unified Correspondence

Abstract

The present chapter is aimed at giving a conceptual exposition of the mathematical principles underlying Sahlqvist correspondence theory. These principles are argued to be inherently algebraic and order-theoretic. They translate naturally on relational structures thanks to Stone-type duality theory. The availability of this analysis in the setting of the algebras dual to relational models leads naturally to the definition of an expanded (object) language in which the well-known ‘minimal valuation’ meta-arguments can be encoded, and of a calculus for correspondence of a proof-theoretic style in the expanded language, mechanically computing the first-order correspondent of given propositional formulas. The main advantage brought about by this formal machinery is that correspondence theory can be ported in a uniform way to families of nonclassical logics, ranging from substructural logics to mu-calculi, and also to different semantics for the same logic, paving the way to a uniform correspondence theory.

Correspondence Theory may be applied to any kind of semantic entity.

(J. van Benthem, Correspondence theory, Handbook of Philosophical Logic, p. 381).

36.11 Appendix

1.1 36.11.1 Distributive Complex Algebras and Frames

An element $c\ne \bot$ of a complete lattice $\mathbb {C}$ is completely join-irreducible iff $c = \bigvee S$ implies $c \in S$ for every $S\subseteq \mathbb {C}$; moreover, $c$ is completely join-prime if $c\ne \bot$ and, for every subset $S$ of the lattice, $c\le \bigvee S$ iff $c\le s$ for some $s\in S$. An element $c\ne \bot$ of a complete lattice is an atom if there is no element $y$ in the lattice such that $\bot < y<c$. An element $c\ne \top$ of a complete lattice is completely meet-irreducible iff $c = \bigwedge S$ implies $c \in S$ for every $S\subseteq \mathbb {C}$; moreover, $c$ is completely meet-prime if $c\ne \top$ and, for every subset $S$ of the lattice, $c\ge \bigwedge S$ iff $c\ge s$ for some $s\in S$. An element $c\ne \top$ of a complete lattice is a co-atom if there is no element $y$ in the lattice such that $c<y< \top$.

If $c$ is an atom (resp. a co-atom), then $c$ is completely join-prime (resp. meet-prime), and if $c$ is completely join-prime (resp. meet-prime), then $c$ is completely join-irreducible (resp. meet-irreducible). If $\mathbb {C}$ is frame distributive (i.e. finite meets distribute over arbitrary joins) then the completely join-irreducible elements are completely join-prime, and if $\mathbb {C}$ is a complete Boolean lattice, then the completely join-prime elements are atoms. The collections of all completely join- and meet-irreducible elements of $\mathbb {C}$ are respectively denoted by $J^{\infty }(\mathbb {C})$ and $M^{\infty }(\mathbb {C})$.

Definition 36.6

A perfect lattice is a complete lattice $\mathbb {C}$ such that $J^{\infty }(\mathbb {C})$ join-generates $\mathbb {C}$ (i.e. every element of $\mathbb {C}$ is the join of elements in $J^{\infty }(\mathbb {C})$) and $M^{\infty }(\mathbb {C})$ meet-generates $\mathbb {C}$ (i.e. every element of $\mathbb {C}$ is the meet of elements in $M^{\infty }(\mathbb {C})$). A perfect distributive lattice is a perfect lattice such that $J^{\infty }(\mathbb {C})$ coincides with the set of all completely join-prime elements of $\mathbb {C}$ and $M^{\infty }(\mathbb {C})$ coincides with the set of all completely meet-prime elements of $\mathbb {C}$; a perfect Boolean lattice is a perfect lattice such that $J^{\infty }(\mathbb {C})$ coincides with the set of all the atoms of $\mathbb {C}$ (or $M^{\infty }(\mathbb {C})$ coincides with the set of all the co-atoms of $\mathbb {C}$).

Complete atomic modal algebras are those modal algebras $\mathbb {A}$ the lattice reducts of which is a perfect Boolean lattice and moreover, their $\Diamond$ operation preserves arbitrary joins, i.e. $\Diamond (\bigvee S) = \bigvee _{s\in S}\Diamond s$ for every $S\subseteq \mathbb {A}$. Discrete Stone duality between complete atomic modal algebras and their complete homomorphisms and Kripke frames and their bounded morphisms is defined on objects by mapping any Kripke frame $\mathcal {F} = (W, R)$ to its complex algebra $\mathcal {F}^+ = (\mathcal {P}(W), \langle R\rangle )$, where $\langle R\rangle X = R^{-1}[X] = \{w\in W : \exists x(x\in X\ \& \ wRx)\}$ for every $X\in \mathcal {P}(W)$, and every complete atomic modal algebra $\mathbb {A} = (\mathbb {B}, \Diamond )$ to its atom structure $\mathbb {A}_{+} = (J^{\infty }(\mathbb {B}), R)$, where $x R y$ iff $x\le \Diamond y$ for all atoms $x, y\in J^{\infty }(\mathbb {B})$. As a consequence of this duality, the Stone representation theorem holds for complete atomic modal algebras, which states that these can be equivalently characterized as the modal algebras each of which is isomorphic to the complex algebra of some Kripke frame.

Likewise, a Stone-type duality (extending the finite Birkhoff duality) holds between perfect distributive lattices and their complete homomorphisms and posets and monotone maps, which is defined on objects as follows: every poset $X$ is associated with the lattice $\mathcal {P}^{\uparrow }(X)$ of the upward-closed subsets of $X$, and every perfect lattice $\mathbb {C}$ is associated with $(J^{\infty }(\mathbb {C}),\ge )$ where $\ge$ is the reverse lattice order in $\mathbb {C}$, restricted to $J^{\infty }(\mathbb {C})$. As a consequence of this duality, perfect distributive lattices can be equivalently characterized (see e.g. [32]) as those lattices each of which is isomorphic to the lattice $\mathcal {P}^{\uparrow }(X)$ of the upward-closed subsets of some poset $X$.

As was mentioned early on, just in the same way in which the duality between complete atomic Boolean algebras and sets can be expanded to a duality between complete atomic modal algebras and Kripke frames, the duality between perfect distributive lattices and posets can be expanded to a duality between perfect DLOs and posets endowed with arrays of relations, each of which dualizes one additional operation in the usual way, i.e., $n$-ary operations give rise to $n+1$-ary relations, and the assignments between operations and relations are defined as in the classical setting. We are not going to report on this duality in full detail (we refer e.g. to [22, 33, 47]), but we limit ourselves to mention that, for instance, the DLOs endowed with four unary operators as in (36.5) are dual to the relational structures $\mathcal {F}=(W, \le , R_\Diamond , R_\Box , R_\lhd , R_\rhd )$ such that $(W, \le )$ is a nonempty poset, $R_\Diamond , R_\Box , R_\lhd , R_\rhd$ are binary relations on $W$ and the following inclusions hold:

$$\begin{aligned} {{\ge }\circ {R_\Diamond }\circ {\ge }}\ \subseteq \ {{R_\Diamond }}&\quad \quad {{\le }\circ {R_\rhd }\circ {\ge }}\ \subseteq \ {{R_\rhd }}\\ {{\le }\circ {R_\Box }\circ {\le }}\ \subseteq \ {{R_\Box }}&\quad \quad {{\ge }\circ {R_\lhd }\circ {\le }}\ \subseteq \ {{R_\lhd }}. \end{aligned}$$

The complex algebra of any such relational structure $\mathcal {F}$ (cf. [33, Sect. 2.3]) is

$$ \mathcal {F}^+ = (\mathcal {P}^{\uparrow }(W), \cup , \cap , \varnothing , W, \langle R_\Diamond \rangle , [R_\Box ], \langle R_\lhd ], [R_\rhd \rangle ),$$

where, for every $X\subseteq W$,

$$\begin{aligned}{}[R_\Box ]{X}&:= \{w\in W\ |\ R_{\Box }[w]\subseteq X\} = (R_\Box ^{-1}[X^c])^c\\ \langle R_\Diamond \rangle X&:= \{w\in W\ |\ R_{\Diamond }[w]\cap X \ne \varnothing \} = R_\Diamond ^{-1}[X] \\ [R_\rhd \rangle X&:= \{w\in W\ |\ R_{\rhd }[w]\subseteq X^c\} = (R_\rhd ^{-1}[X])^c\\ \langle R_\lhd ] {X}&:= \{w\in W\ |\ R_{\lhd }[w]\cap X^c \ne \varnothing \} = R_\lhd ^{-1}[X^c]. \end{aligned}$$

Here $(\cdot )^c$ denotes the complement relative to $W$, while $R[x] = \{w\mid w\in W \text{ and } x Rw\}$ and $R^{-1}[x] = \{w\mid w\in W \text{ and } w Rx\}$. Moreover, $R[X] = \bigcup \{R[x]\mid x\in X\}$ and $R^{-1}[X] = \bigcup \{R^{-1}[x]\mid x\in X\}$.

1.2 Adjunction and Residuation

Let $P$ and $Q$ be partial orders. The maps $f: P\rightarrow Q$ and $g: Q\rightarrow P$ form an adjoint pair (notation: $f\dashv g$) iff for every $x\in P$ and $y\in Q$, $f(x)\le y\ \text{ iff } \ x\le g(y).$ Whenever $f\dashv g$, $f$ is the left adjoint of $g$ and $g$ is the right adjoint of $f$. Adjoint maps are order-preserving. If a map admits a left (resp. right) adjoint, the adjoint is unique and can be computed pointwise from the map itself and the order.

Proposition 36.3

• 1. Right adjoints (resp. left adjoints) between complete lattices are exactly the completely meet-preserving (resp. join-preserving) maps;

• 2. right (resp. left) adjoints on powerset algebras $\mathcal {P}(W)$ are exactly the maps defined by assignments of type $X\mapsto [R]X = (R^{-1}[X^c])^c$ (resp. $X\mapsto \langle R\rangle X = R^{-1}[X]$) for some binary relation $R$ on $W$.

• 3. For any binary relation $R$ on $W$, the left adjoint of $[R]$ is the map $\langle R^{-1}\rangle$, defined by the assignment $X\mapsto R[X]$.

Proof

1. See [26, Proposition 7.34].

2. For a left adjoint $f:\mathcal {P}(W)\longrightarrow \mathcal {P}(W)$, define $R$ as follows: for every $x,z\in W$, $x R z$ iff $x\in f(\{z\}).$ For a right adjoint $g:\mathcal {P}(W)\longrightarrow \mathcal {P}(W)$, define $R$ as follows: for every $x,z\in W$, $x R z$ iff $x\not \in g(W\setminus \{z\}).$

The notion of adjunction can be made parametric and generalized to $n$-ary maps in a component-wise fashion: an $n$-ary map $f: P^n \rightarrow P$ on a poset $P$ is residuated if there exists a collection of maps $\{g_i: P^n\rightarrow P\mid 1\le i \le n\}$ s.t. for every $1\le i\le n$ and for all $x_1,\ldots , x_n, y\in P$,

$$ f(x_1,\ldots , x_n)\le y\quad \mathrm{{iff}}\quad x_i\le g_i(x_1,\ldots , x_{i-1},y,x_{i+1},\ldots , x_n). $$

The map $g_i$ is the $i$-th residual of $f$. Residuated maps are order preserving in each coordinate, and for each $1\le i\le n$, the residual $g_i$ is order-preserving in its $i$th coordinate and order-reversing in all other coordinates. The facts stated in the following example and proposition are well known in the literature in their binary instance (cf. [31, Sect. 3.1.3]):

Example 36.4

For every $(n+1)$-ary relation $\mathcal {S}$ on $W$ and every $(X_1,\ldots , X_n)\in \mathcal {P}(W)^n$, let

$$\mathcal {S}[X_1,\ldots , X_n]:= \{y\in W\mid \exists x_1\cdots \exists x_n[\bigwedge _{i = 1}^n x_i\in X_i\ \wedge \ \mathcal {S}(x_1,\ldots , x_n, y)]\}.$$

The $n$-ary operation on $\mathcal {P}(W)$ defined by the assignment $(X_1,\ldots , X_n)\mapsto \mathcal {S}[X_1, \ldots , X_n]$ is residuated and its $i$-th residual is the map $g_i: \mathcal {P}(W)^n\rightarrow \mathcal {P}(W)$ which maps every $n$-tuple $(X_1,\ldots ,X_{i-1}, Y, X_{i+1},\ldots , X_n)$ to the set $\{w\in W\mid \alpha _{\mathcal {S}}^i(w)\},$ where $\alpha _{\mathcal {S}}^i(w)$ is the following first-order formula:

$$ \forall x_1\cdots \forall y\cdots \forall x_n[(\bigwedge _{k\in \mathbf {n}_i} x_k\in X_k\ \& \ \mathcal {S}(x_1,\ldots , w,\ldots , x_n, y))\Rightarrow y\in Y],$$

and moreover $\mathbf {n}_i = \{1,\ldots , n\}\setminus \{i\}$.

Proposition 36.4

If $f: P^n\rightarrow P$ is residuated and $\{g_i: P^n\rightarrow P\mid 1\le i\le n\}$ is the collection of its residuals, then:

• 1. if $P$ is a complete lattice, then $f$ preserves arbitrary joins in each coordinate;

• 2. if $P$ is a powerset algebra, $f$ coincides with the map defined by the assignment $\mathcal {S}[X_1,\ldots , X_n]$ as in Example 36.4, for some $(n+1)$-ary relation $\mathcal {S}$ on $W$.