modal logic

A *modal formula* is derived from the language of a logic $L$ inductively by: if $A$ is a formula of $L$, then $\box A$ is a modal formula. We denote the collection of modal formulas by $M$.

A *Kripke frame* aka. *modal frame* is a pair $(S,R)$ of a set $S$ and a binary relation $R\subset S\times S$. a relation $R\subseteq S\times S$ is equivalently a function $R[\cdot]:S\to P(S)$ sending $s$ to the set $R[s]$ of its successors.

A *Kripke model* is a triple $(S,R,\Vdash_R)$ where $(S,R)$ is a Kripke frame and a relation $\Vdash\subset S\times M$ (sometimes called *forcing relation*) subject to the following list of axioms:

(1) $\Vdash_R(s, \neg A)$ iff $(s,A)$ is not in $\Vdash_R$.

(2) $\Vdash_R(s,A\to B)$ iff $(s,A)$ is not in $\Vdash_R$ or $A,B\in \Vdash_R$

(3) $\Vdash_R(s,\Box A)$ iff $(t,A)\in \Vdash_R$ for all $t$ with $(s,t)\in R$

(Note that we used here the notation $\Vdash_R(s, A)$ to express the “propositional interpretation” of being a member of the relation.)

A formula $D$ is called to be *valid in a model* resp. *valid in a frame* resp. *valid in a class $F$ of frames* if $(s,A)\in \Vdash_R$ for all $s\in S$ resp. $(s,A)\in \Vdash_R$ for all $s\in S$ all relations $R\subseteq S\times S$ resp. $(s,A)\in \Vdash_R$ for all $(S,R)\in F$ for all$s\in S$ all relations $R\subseteq S\times S$.

A binary relation $R\subseteq S\times S$ is equivalently a function $R[\cdot]:S\to P(S)$ where $P$ denotes the (covariant) power set functor sending an $s\in S$ to the set of its $R$-successors $R[s]$ and a morphism $f\mapsto(X\mapsto \{f(x)|x\in X\})$. In other words: a frame is a coalgebra for the power set functor.

Generalizing this construction for relations of higher arity one obtains coalgebras for the functor $S\mapsto \product_{d\in t} P(S^{arity(d})$ for any modal similarity type $t$.

*Kripke models* one obtains as coalgebras of the functor $X\mapsto P(Prop)\times P(X)$ where $Prop$ is the set of propositional variables. To see this note that a *valuation* $V:Prop\to P(S)$ is equivalently a *$P(Prop)$-coloring of $S$*: a functor $K: s\mapso \{p\in Prop|s\in V(p)\}$.

The functors $P$ and $K$ from above are examples of Polynomial functors.

(…

(Definition 146, p.392) A *polynomial functor* is a $Set$-valued functor inductively defined by

$K::=I | C | K_0 + K_1 | K_0\times K_1 | K^D$

where $I$ denotes the identity functor, $C$ denotes the constant functor on an object $C$, $K_0 +K_1$ denotes the coproduct functor of two polynomial functors, and likewise for the product functor, $K^D$ denotes the exponent functor $X\mapsto K(X)^D$.

A *Kripke polynomial functor* is inductively defined by

$K::=I | C | K_0 + K_1 | K_0\times K_1 | K^D | P K$

where $P K$ denotes the composition of a polynomial functor $K$ with the power set functor $P$.

(…)

(p.392) Kripke frames are $PI$-coalgebras, and Kripke models are coalgebras for the functor $PProp\times PI$.

- Yde Venema, Algebras and Coalgebras, §6 (p.332-426) in Blackburn, van Benthem, Wolter, Handbook of modal logic, Elsevier, 2007.