2.1 Language, consequence relations, and logics

Throughout this paper we work with the basic modal language $\mathcal {L}$ , given by:

$$ \begin{align*} \varphi := p\;|\;\neg\varphi\;|\;\varphi\land\varphi \;|\;\Box\varphi \end{align*} $$

p belongs to a fixed set Prop of propositional letters. Other connectives ( $\bot ,\top ,\rightarrow ,\ldots $ ) and modal operators ( $\Diamond ,\ldots $ ) are defined as usual. We may identify $\mathcal {L}$ with the set of $\mathcal {L}$ -formulas.

A consequence relation is a subset $\models $ of $\mathcal {P}(\mathcal {L})\times \mathcal {L}$ (assumed to have certain closure properties like reflexivity, monotonicity, etc.).

A modal logic is a set $\mathsf {L}\subseteq \mathcal {L}$ containing all classical propositional tautologies and closed under the rules of Modus Ponens ( $\varphi ,\varphi \rightarrow \psi \in \mathsf {L}$ implies $\psi \in \mathsf {L}$ ) and Uniform Substitution (if $\varphi \in \mathsf {L}$ , then $\varphi '\in \mathsf {L}$ , where $\varphi '$ is obtained by uniformly substituting formulas for propositional letters in $\varphi $ ). The letters $\mathsf {L},\mathsf {L}',\ldots $ will always stand for modal logics. $\mathsf {L}$ is normal if $\Box (p\rightarrow q)\rightarrow (\Box p \rightarrow \Box q)\in \mathsf {L}$ and $\mathsf {L}$ is closed under Necessitation ( $\varphi \in \mathsf {L}$ implies $\Box \varphi \in \mathsf {L}$ ).

We usually write $\models _{\mathsf {L}}\varphi $ instead of $\varphi \in \mathsf {L}$ . Each modal logic $\mathsf {L}$ has an associated consequence relation $\models _{\mathsf {L}}$ defined by

$$ \begin{align*} \Gamma\models_{\mathsf{L}}\varphi\; \mbox{ iff there is a finite } \Gamma_0\subseteq\Gamma \mbox{ such that } \bigwedge\Gamma_0\rightarrow\varphi\in\mathsf{L}. \end{align*} $$

With the understanding that $\bigwedge \emptyset =\top $ , we have that $\models _{\mathsf {L}}\varphi $ iff $\emptyset \models _{\mathsf {L}}\varphi $ , so there is no ambiguity in writing $\models _{\mathsf {L}}\varphi $ .

$\mathsf {K}$ is the smallest normal modal logic. If $\psi $ is a formula, $\mathsf {K}\psi $ is the smallest normal modal logic containing $\psi $ . With the usual naming of axioms, S4 = $\mathsf {K}\mathrm{T4}$ , S5 = $\mathsf {K}\mathrm{T4B} = \mathsf {K}\mathrm{T5}$ , etc.

2.2 Compositionality

Abstractly, an interpretation assigns a semantic value to each expression of a language. In the case of $\mathcal {L}$ , interpretable expressions are of two categories: formulas and operators ( $\neg $ , $\land $ , $\Box $ ). Usually, the interpretation of the operators is regarded as fixed, but here we are interested precisely in the range of possible interpretations of these, in particular of $\Box $ .

Compositionality relies on the fact that complex expressions are generated from atomic expressions by syntactic rules. Accordingly, only atomic expressions need to be interpreted; in the case of $\mathcal {L}$ , propositional letters and operators. Let a model be a pair $\mathcal {M}=(I,V)$ of an interpretation I of the operators and a valuation V of the propositional letters, where the range of V is a set of semantic values, and functions in I are in keeping with the syntactic type of the operators they interpret. We can then say that a compositional semantics for $\mathcal {L}$ is a binary function $[\kern-1.5pt[\cdot ]\kern-1.5pt]_\cdot $ , recursively assigning to each pair of a formula $\varphi $ and a model $\mathcal {M}$ a unique value $[\kern-1.5pt[ \varphi ]\kern-1.5pt]_{\mathcal {M}}$ , as follows:

(1) $$ \begin{align} \begin{aligned} &[\kern-1.5pt[\,p ]\kern-1.5pt]_{\mathcal{M}} = V(p) ,\\ &[\kern-1.5pt[\neg\varphi]\kern-1.5pt]_{\mathcal{M}} = I(\neg)([\kern-1.5pt[\varphi]\kern-1.5pt]_{\mathcal{M}}) ,\\ &[\kern-1.5pt[\varphi\land\psi]\kern-1.5pt]_{\mathcal{M}} = I(\land)([\kern-1.5pt[\varphi]\kern-1.5pt]_{\mathcal{M}},[\kern-1.5pt[\psi]\kern-1.5pt]_{\mathcal{M}}) ,\\ &[\kern-1.5pt[\Box\varphi]\kern-1.5pt]_{\mathcal{M}} = I(\Box)([\kern-1.5pt[\varphi]\kern-1.5pt]_{\mathcal{M}}) . \end{aligned} \end{align} $$

Thus, compositionality tells us that the interpretations of $\neg $ , $\land $ , and $\Box $ must be functions (of the appropriate arity) from formula values to formula values. Note that so far we have said nothing about what these values are.

2.3 Possible worlds semantics

Several compositional semantics have been proposed for $\mathcal {L}$ . As explained in Section 1, we focus on the general setting of possible worlds semantics, in which Kripke semantics has a privileged place. More precisely:

As usual, instead of writing $w \in [\kern-1.5pt[\varphi ]\kern-1.5pt]_{\mathcal {M}}$ , we often write

$$ \begin{align*} \mathcal{M},w\models\varphi. \end{align*} $$

So possible worlds semantics, as defined here, exactly captures one crucial feature of modal semantics: truth relativity. This need not involve any further structure among worlds; in particular, no relation of accessibility. But another characteristic feature is what we may call truth locality: the truth of $\varphi $ at w need not depend on the whole model, but only on the part of it that can be ‘seen’ from w. The idea is most directly implemented in Kripke semantics, a possible worlds semantics where models come equipped with an accessibility relation R, and the truth of a formula at w only depends on the part of the model that can be reached from w in successive steps via R.

However, truth locality has also been claimed for other forms of possible worlds semantics, such as topological semantics. In Section 4 we explore ways in which the notion of truth locality can be made precise, and to what extent it is characteristic of Kripke semantics.

2.4 Local interpretations

Fix a domain W and consider interpretations I of $\neg ,\land ,\Box $ as above over subsets of W.

Definition 2. I respects a consequence relation $\models $ if

$$ \begin{align*} \begin{aligned} &\Gamma\models\varphi \mbox{ implies that for all valuations } V \mbox{ over } W,\\ &\bigcap_{\psi\in\Gamma}[\kern-1.5pt[\psi]\kern-1.5pt]_{(W,I,V)} \subseteq [\kern-1.5pt[\varphi]\kern-1.5pt]_{(W,I,V)}. \end{aligned} \end{align*} $$

If this holds when $\models \;=\; \models _{\mathsf {L}}$ for some modal logic $\mathsf {L}$ , we also say that I respects $\mathsf {L}$ .

From now on we restrict attention to interpretations that respect some modal logic.Footnote ² Then we have the following categoricity result:

As a result, an interpretation I for $\mathcal {L}$ (over W) only needs to supply the function interpreting $\Box $ , so we can identify I with $I(\Box )$ . Let us make this official:

If $\mathcal {M}=(W,F,V)$ , the truth definition (1) now becomes:

(2) $$ \begin{align} \begin{aligned} & [\kern-1.5pt[\,p ]\kern-1.5pt]_{\mathcal{M}} = V(p) ,\\ & [\kern-1.5pt[\neg\varphi]\kern-1.5pt]_{\mathcal{M}} = W-[\kern-1.5pt[\varphi]\kern-1.5pt]_{\mathcal{M}} ,\\ & [\kern-1.5pt[\varphi\land\psi]\kern-1.5pt]_{\mathcal{M}} = [\kern-1.5pt[\varphi]\kern-1.5pt]_{\mathcal{M}}\cap[\kern-1.5pt[\psi]\kern-1.5pt]_{\mathcal{M}} ,\\ & [\kern-1.5pt[\Box\varphi]\kern-1.5pt]_{\mathcal{M}} = F([\kern-1.5pt[\varphi]\kern-1.5pt]_{\mathcal{M}}) . \end{aligned} \end{align} $$

Also, with the understanding that $\bigcap _{\psi \in \emptyset }[\kern-1.5pt[\psi ]\kern-1.5pt]_{\mathcal {M}} = W$ , we have:

2.5 Neighborhood frames

Local interpretations as defined above are in fact familiar objects: they are neighborhood frames. Usually, a neighborhood frame is given as $(W,N)$ , where $N\!:W\rightarrow \mathcal {P}(\mathcal {P}(W))$ . But $(W,N)$ can equally well be presented as $(W,m_N)$ where, for $X\subseteq W$ , $m_N(X) = \{w\in W\!: X\in N(w)\}$ . Essentially, these are two ways of describing the same object. In the present context, it is natural to start from the local interpretations $(W,F)$ , and define the corresponding families by

$$ \begin{align*} N_F(w) = \{X\subseteq W\!: w\in F(X)\}. \end{align*} $$

Sometimes the indexed families $N(w)$ of sets are easier to work with; we shall use both formats, whichever is most convenient for the purpose at hand.

The truth definition in neighborhood semantics is just as (2), or, if you will, (2) with the last clause expressed as follows:Footnote ⁴

(3) $$ \begin{align} [\kern-1.5pt[\Box\varphi]\kern-1.5pt]_{\mathcal{M}} = \{w\in W\!: [\kern-1.5pt[\varphi]\kern-1.5pt]_{\mathcal{M}} \in N_F(w)\}. \end{align} $$

Neighborhood semantics for modal logic was suggested in [Reference Montague and Klibansky15, Reference Scott and Lambert17], studied in [Reference Chellas4, Reference Segerberg18], and has recently been revived with a number of applications; see [Reference Pacuit16] for an extensive introduction. The main motivation for preferring neighborhood frames to relational Kripke frames has come from applications (such as deontic and epistemic logic) taken to require non-normal modal logics (logics in which not all theorems of $\mathsf {K}$ are provable). Also, neighborhood frames have been seen as interesting mathematical structures in themselves, for example, in connection with algebraic semantics (Section 2.8 below), and with topological interpretations of modal logic (see Section 4).

Interestingly, as we have just seen, there is a different and more direct reason for turning to neighborhood frames: the simple idea of using the possible worlds format as a compositional formal semantics for the basic modal language leads directly, via Theorem 3, to these structures as interpretations of $\Box $ . Thus, the following is not just a slogan but actually a result (recall Definition 1):Footnote ⁵

$$ \begin{align*} {possible\ worlds\ semantics\ =\ neighborhood\ semantics}. \end{align*} $$

This means that we can help ourselves to known facts and terminology from neighborhood semantics in what follows. For example, Corollary 5 simply says that $(W,F)$ respects $\mathsf {L}$ iff $\mathsf {L}$ is sound for $(W,F)$ (also written $(W,F)\models \mathsf {L}$ ).

The following fact is well-known.Footnote ⁶

2.6 Kripkean interpretations

Kripke semantics can be seen as a special case of neighborhood semantics. Recall that a Kripke frame is a pair $(W,R)$ where $R\subseteq W^2$ , and that, with $\mathcal {M} = (W,R,V)$ , the truth clause of boxed formulas in Kripke semantics is

$$ \begin{align*} \mathcal{M},w\models\Box\varphi\; \mbox{ iff for all } v \mbox{ such that } wRv, \mathcal{M},v\models\Box \phi, \end{align*} $$

or, equivalently,

$$ \begin{align*} [\kern-1.5pt[\Box\varphi]\kern-1.5pt]_{\mathcal{M}} = \{w\in W\!: R(w) \subseteq [\kern-1.5pt[\varphi]\kern-1.5pt]_{\mathcal{M}}\}, \end{align*} $$

where $R(w)$ is the set of R-successors of w.

We shall also need the following notion.

The next fact is well-known (except for the new terminology).

Thus, if $(W,F)$ is Kripkean, there is a unique accessibility relation, namely $Acc_F$ , with respect to which F encodes the standard truth clause for boxed formulas. Indeed, it is practically immediate that the following holds.

This is the precise sense in which Kripke semantics is a special kind of possible worlds/neighborhood semantics. Particular local interpretations of $\Box $ can sometimes be seen as giving particular meanings to “necessary.” For example, the function $F_{\mathrm{uni}}$ , defined by

$$ \begin{align*} F_{\text{uni}}(X) = \left\{\begin{array}{@{}ll} W & \text{if } X=W \\ \emptyset & \text{otherwise} \end{array} \right. \end{align*} $$

embodies logical or metaphysical necessity as truth in all possible worlds. (The index “uni” is because $Acc_{F_{\mathrm{uni}}}$ is the universal relation on W.)

However, as explained in Section 1, we are not thinking of Carnap’s problem for the meaning of $\Box $ as finding constraints that fix a unique interpretation, not even on a given universe. And the Kripkean local interpretation $(W,F_{\mathrm{uni}})$ is strikingly atypical, since the accessibility relation does not restrain the set of worlds accessible from any world. Rather, we are after constraints that force interpretations of $\Box $ in possible worlds semantics to behave just as in Kripke semantics. At this point, we are able to state a precise version of this question:

Since $\mathsf {K}$ is sound and complete for the class of all Kripke frames, respecting $\mathsf {K}$ will obviously be a minimal requirement. Indeed, for finite frames, it gives a complete answer.

So in the finite case, $\mathsf {K}$ forces the interpretations of $\neg $ , $\land $ , and $\Box $ to be the standard ones. However, this is very far from true for infinite frames. We have the following familiar negative result.

Thus, any answer to Carnap’s question for $\Box $ in general will involve extra semantic constraints.

2.7 Permutation invariance

Could permutation invariance clinch the answer to the local version of Carnap’s question, just as it did for the question about the meaning of $\forall $ in first-order logic? No: we show in this section that in the modal setting, it is much too restrictive.

A permutation $\pi $ of W lifts in the usual way to higher-type objects over W, in particular to binary relations on W, and to functions from $\mathcal {P}(W)$ to $\mathcal {P}(W)$ . If F is such a function, the function $\pi (F)$ is defined, for $X\subseteq W$ , by

$$ \begin{align*} \pi(F)(X) = \pi(F(\pi^{-1}(X))).^{7} \end{align*} $$

For example, $(W,F_{\mathrm{uni}})$ is permutation invariant. Here are the functions in three more permutation invariant local interpretations:

$$ \begin{align*} F_{\text{id}}(X) = X \hspace{29ex}(Acc_{F_{\text{id}}} = id_W), \end{align*} $$

$$ \begin{align*} F_{\text{emp}}(X) = W \hspace{26ex}(Acc_{F_{\text{emp}}} = \emptyset), \end{align*} $$

$$ \begin{align*} F_{\text{diff}}(X) = \left\{ \begin{array}{@{}ll} W & \text{if } X=W \\ \{w\} & \text{if } X=W-\{w\} \\ \emptyset & \text{otherwise} \end{array} \right. \hspace{2ex} (w\,Acc_{F_{\text{diff}}}\,w' \Leftrightarrow w\neq w'). \end{align*} $$

There isn’t much in the literature on the consequences of invariance constraints on modal operators, but two exceptions are [Reference van Benthem21] and [Reference MacFarlane13]. MacFarlane [Reference MacFarlane13] lists a number of permutation invariant functions from $\mathcal {P}(W)$ to $\mathcal {P}(W)$ , among them $F_{\mathrm{id}}$ and $F_{\mathrm{uni}}$ , as well as the following:

$$ \begin{align*} F_1(X) = \left\{ \begin{array}{@{}ll} W & \text{if } X\neq\emptyset, \\ \emptyset & \text{if } X=\emptyset, \end{array} \right. \end{align*} $$

$$ \begin{align*} F_2(X) = \left\{ \begin{array}{@{}ll} W & \text{if } |X|=5, \\ \emptyset & \text{otherwise.} \end{array} \right. \end{align*} $$

Note that $F_1$ and $F_2$ are not Kripkean: permutation invariance doesn’t enforce Kripkeanity.Footnote ⁸ Still, one may ask exactly which functions from $\mathcal {P}(W)$ to $\mathcal {P}(W)$ are permutation invariant. This is answered by a result in [Reference van Benthem21].Footnote ⁹ van Benthem shows that if F is a function from $\mathcal {P}(W)^n$ to $\mathcal {P}(W)$ , then F is permutation invariant if and only if, for all $X_1,\ldots ,X_n\subseteq W$ , $F(X_1,\ldots ,X_n)$ is a Boolean combination of $X_1,\ldots ,X_n$ . This implies in particular the following.

This shows how restrictive permutation invariance in itself is, as a constraint on operators from sets of worlds to sets of worlds. Very few of the permutation invariant operators are ‘reasonable’ interpretations of $\Box $ , and in particular, very few are Kripkean: we now show exactly which ones. First, a lemma, whose verification is left to the reader.

Proof. First, it is easily checked that these four local interpretations of $\Box $ are permutation invariant and Kripkean. For the converse, let $(W,F)$ be a permutation invariant Kripkean local interpretation. We claim:

$$ \begin{align*} Acc_{F} \mbox{ is permutation invariant}. \end{align*} $$

In other words, for every permutation $\pi $ of W,

$$ \begin{align*} w\,Acc_{F}\,v \:\Leftrightarrow\: \pi(w)\,Acc_{F}\, \pi(v). \end{align*} $$

To see this, note that we have $v \in \bigcap N_F(w) \Leftrightarrow \pi (v) \in \pi (\bigcap N_F(w)) = \bigcap N_F(\pi (w))$ , by Lemma 16. This proves the claim. But there are exactly four permutation invariant binary relations on any universe, namely, the universal relation, the empty relation, the identity relation, and the difference relation. Since we assumed that $(W,F)$ is Kripkean, i.e., that F is standard with respect to $Acc_{F}$ , the desired conclusion follows.□

These four local interpretations correspond to familiar logics. When $\mathcal {K}$ is a class of neighborhood frames (or a class of Kripke frames), define $Log(\mathcal {K})$ , the logic of $\mathcal {K}$ , as follows:

(4) $$ \begin{align} Log(\mathcal{K}) = \{\varphi\!: \forall \mathcal{F}\in\mathcal{K} \;\: \mathcal{F}\models\varphi\}. \end{align} $$

$Log(\mathcal {K})$ is always a modal logic. Now let $\mathcal {K}_{\mathrm{uni}}$ be the class of all neighborhood frames of the form $(W,F_{\mathrm{uni}})$ , and similarly for $\mathcal {K}_{\mathrm{id}}$ , $\mathcal {K}_{\mathrm{emp}}$ , and $\mathcal {K}_{\mathrm{diff}}$ . Then we have:

(5) $$ \begin{align} \begin{aligned} & Log(\mathcal{K}_{\text{uni}}) = \text{S}5 ,\\ & Log(\mathcal{K}_{\text{id}}) = \mathsf{K}(\Box p\leftrightarrow p) = \mathsf{L}_\circ ,\\ & Log(\mathcal{K}_{\text{emp}}) = \mathsf{K}(\Box\bot) = \mathsf{L}_\bullet ,\\ & Log(\mathcal{K}_{\text{diff}}) = \mathsf{K}\text{B}(p\land\Box p\rightarrow \Box\Box p) = {Else}. \end{aligned} \end{align} $$

Here $\mathsf {L}_\circ $ and $\mathsf {L}_\bullet $ are the two trivial normal modal logics: by a theorem of [Reference Makinson14], every consistent normal modal logic is a sublogic of one of these, and the only proper extension of each is the inconsistent modal logic $\mathcal {L}$ .Footnote ¹⁰ Else is the ‘logic of elsewhere’ from [Reference von Wright and Sosa24]: $\Box \varphi $ is true at w in this logic iff $\varphi $ is true everywhere else. (The axiomatization above is from [Reference Segerberg19].)

However, the main lesson to draw from the facts stated in this section, we think, is that permutation invariance is not an appropriate requirement for our local version of Carnap’s question. We are interested in constraints forcing a local interpretation to be Kripkean, which means that truth for boxed formulas is defined in a particular way from an accessibility relation. Permutation invariance drastically restricts the choice of accessibility relations, but we were looking for interpretations that are parametric in an arbitrary accessibility relation.

So why is permutation invariance appropriate for first-order logic? The answer seems to be that there we permute individuals in the domain which $\forall $ quantifies over, and permutation invariance (or rather isomorphism invariance) implements the reasonable idea that logical constants should be topic-neutral. In Kripke semantics for modal logic, on the other hand, permuting worlds can make a difference, when the accessibility relation is not one of the four relations above. ‘World neutrality’ makes sense for some notions of necessity, such as logical necessity, but not in general.

2.8 The algebraic perspective

The semantic frameworks mentioned so far can be seen as special cases of a more general algebraic semantics. Let a modal algebra be an algebra $\mathfrak {A} = (A,\lor ,\land ,\,^c,0,1,f)$ , where $(A,\lor ,\land, \,^c,0,1)$ is a Boolean algebra and f is a unary operation on A. $\mathfrak {A}$ is normal if f satisfies $f(a\land b) = f(a)\land f(b)$ and $f(1)=1$ .Footnote ¹¹ A valuation is now a map v from Prop to A, which is extended inductively to a map $\overline {v}$ from all formulas, with the obvious clauses for the Boolean operations, and with $\overline {v}(\Box \psi ) = f(\overline {v}(\psi ))$ . Any (normal) neighborhood frame $\mathcal {F}=(W,F)$ yields a (normal) modal algebra $\mathcal {F}^+ = (\mathcal {P}(W),\cup ,\cap ,\,^c,\emptyset ,W,F)$ , which is modally equivalent to $\mathcal {F}$ in the sense that for any valuation V on W (which is a valuation in the algebraic sense on $\mathcal {P}(W)$ ) we have, for all formulas $\varphi $ , $[\kern-1.5pt[\varphi ]\kern-1.5pt]_{(W,F,V)} = \overline {V}(\varphi )$ .

Algebras of the form $\mathcal {F}^+$ trivially have the following two properties, which we formulate for an arbitrary modal algebra $\mathfrak {A}$ .

1. (𝒞) completeness: if $X\subseteq A$ , then the join $\bigvee \!X$ is an element of A.

2. (𝒜) atomicity: any non-zero element is above an atom (a minimal non-zero element).

Now consider the property

1. (𝒱) complete multiplicativity: if $X\subseteq A$ and $\bigwedge \{f(a)\!: a\in X\}\in A$ , then

   • (*) $f(\bigwedge X) = \bigwedge \{f(a)\!: a\in X\}$ .

If $\mathfrak {A}$ satisfies $\mathcal {C}$ , then $\mathcal {V}$ just says that f distributes over arbitrary meets.

Properties $\mathcal {C}$ , $\mathcal {A}$ , and $\mathcal {V}$ figure in the duality between algebraic and model-theoretic semantics for modal logic, which goes back to [Reference Jónsson and Tarski11], in which Stone duality was generalized to Boolean algebras with operators and relational frames. Thomason [Reference Thomason20] and Goldblatt [Reference Goldblatt8] applied these techniques to modal logic. Adapting their results to arbitrary neighborhood frames, Došen [Reference Došen5] showed that the map $\mathcal {F}\mapsto \mathcal {F}^+$ , together with a map $\mathfrak {A} \mapsto \mathfrak {A}_+$ in the other direction, yields two contravariant functors that establish a dual equivalence between the category of (normal, Kripkean) neighborhood frames and bounded morphisms and the category of modal (normal, $\mathcal {C}\mathcal {A}\mathcal {V}$ -) $\mathcal {C}\mathcal {A}$ -algebras with complete homomorphisms.Footnote ¹²

Thus, Carnap’s question can be rephrased in algebraic terms: To what extent does respecting modal logics (perhaps in conjunction with other constraints) force a (normal) $\mathcal {C}\mathcal {A}$ -algebra to be completely multiplicative?

Complete multiplicativity (or equivalently, complete additivity) is studied in-depth in [Reference Holliday and Litak10]. The focus there is on modal completeness. A logic $\mathsf {L}$ is (sound and) complete relative to frames/algebras of a certain kind (Kripke/neighborhood frames, (normal) $\mathcal {C}\mathcal {A}$ -algebras, etc.) if there is a class $\mathcal {K}$ of such frames/algebras such that $\mathsf {L} = Log(\mathcal {K})$ . Holliday and Litak show (among many other things) that there are modal logics incomplete with respect to any class of normal modal $\mathcal {V}$ -algebras.

Carnap’s question, on the other hand, concerns the role of $\mathcal {V}$ for modal correspondence. Define

(6) $$ \begin{align} \mathit{Fr}_{\text{nbd}}(\mathsf{L}) = \{(W,F)\!: (W,F)\models\mathsf{L}\}. \end{align} $$

A class $\mathcal {K}$ of frames corresponds to a modal logic $\mathsf {L}$ if $\mathcal {K} = \mathit {Fr}_{\mathrm{nbd}}(\mathsf {L})$ ; similarly for classes of algebras. We have seen (Fact 13) that the class $\mathcal {K}\mathcal {R}_{\mathrm{nbd}}$ of all Kripkean neighborhood frames corresponds to no modal logic; in other words, it is modally undefinable. Dually, while among $\mathcal {C}\mathcal {A}$ -algebras, the class of normal $\mathcal {C}\mathcal {A}$ -algebras corresponds to the logic $\mathsf {K}$ , the class of $\mathcal {C}\mathcal {A}\mathcal {V}$ -algebras is modally undefinable.

(In)completeness and (un)definability are two sides of the same coin, as is seen from the familiar (antitone) Galois connection:

(GC) $$ \begin{align} \mathcal{K} \subseteq \mathit{Fr}_{\text{nbd}}(\mathsf{L}) \;\Leftrightarrow\; \mathsf{L} \subseteq Log(\mathcal{K}) \end{align} $$

(similarly for other kinds of frames or algebras). Although the undefinability of $\mathcal {K}\mathcal {R}_{\mathrm{nbd}}$ (or the class of $\mathcal {C}\mathcal {A}\mathcal {V}$ -algebras) is an easy and well-known fact, whereas the incompleteness of certain modal logics relative to normal $\mathcal {V}$ -algebras established in [Reference Holliday and Litak10] solves a long-standing open problem, the algebraic formulation focusing on the role of complete multiplicativity provides an interesting perspective on Carnap’s question about the meaning of $\Box $ .Footnote ¹³

2.9 Global interpretations

The function F in a local interpretation/neighborhood frame $(W,F)$ literally interprets $\Box $ . The accessibility relation R in a Kripke frame $(W,R)$ , on the other hand, is a parameter in the interpretation of $\Box $ . Each choice of R yields a local Kripkean interpretation function, which we will call $F_R$ : for $X\subseteq W$ , define

(7) $$ \begin{align} F_R(X) = \{w\in W\!: R(w)\subseteq X\}. \end{align} $$

The class of frames of the form $(W,F_R)$ for $R\subseteq W^2$ , i.e., the class of Kripkean local interpretations, can be seen as the standard global/parametric interpretation. But in general possible worlds semantics, as defined here, there are no accessibility relations in the background. So we shall simply say that a (global) interpretation is any class of neighborhood frames.

A bit of terminology may be helpful. Let us name the above map from Kripke frames to neighborhood frames, as well as a map in the opposite direction:

(8) $$ \begin{align} \begin{aligned} &\mbox{a.}\ \ nbd(W,R) = (W,F_R);\\ &\mbox{b.}\ \ kr(W,F) = (W,Acc_F). \end{aligned} \end{align} $$

Then, with $\mathcal {K}\mathcal {R}$ as the class of all Kripke frames, we have, since $Acc_{F_R} = R$ and, when $(W,F)$ is Kripkean, $F = F_{Acc_F}$ :

We have:

(9) $$ \begin{align} \mathcal{K} \mbox{ respects } \mathsf{L}\; \mbox{ iff } \;\mathsf{L}\subseteq Log(\mathcal{K})\; \mbox{ iff } \;\mathcal{K} \subseteq \mathit{Fr}_{\text{nbd}}(\mathsf{L}) . \end{align} $$

The global version of Carnap’s question is thus to what extent respecting modal consequence relations, perhaps in conjunction with invariance constraints, force an interpretation to be Kripkean, and in particular when it is forced to be the standard interpretation. We can now restate Theorem 12 and Fact 13 as follows.

Clearly, a meaning of $\Box $ cannot be identified with a single local interpretation/neighborhood frame—not even over a given universe. This is clear already in case of Kripke semantics, where the accessibility relation is a parameter. Global interpretations as defined above are better candidates for such meanings.

4.1 Invariance under generated subframes

To answer the first question, we define a notion of generated subframe for arbitrary neighborhood frames, which specializes to the usual notion for topological frames and for Kripke frames.Footnote ¹⁶ Starting with the families of sets format: given $(W,N)$ and $A\subseteq W$ , the subframe

$$ \begin{align*} (A,N_A) \end{align*} $$

is defined by letting $N_A(w) = \{X\cap A\!: X\in N(w)\}$ , for $w\in A$ .Footnote ¹⁷

Definition 27. $(A,N_A)$ is a generated subframe of $(W,N)$ , in symbols, $(A,N_A) \subseteq _g (W,N)$ , if:

(12) $$ \begin{align} \mbox{For all } X\subseteq W \mbox{ and all } w\in A, \;X\in N(w)\; \mbox{ iff } \;X\cap A\in N_A(w) . \end{align} $$

If we use the functional format instead, one readily checks that given $(W,F)$ and $A\subseteq W$ , condition (12) becomes:

(13) $$ \begin{align} \mbox{For all } X\subseteq W, \;F(X) \cap A = F(X\cap A) \cap A. \end{align} $$

Define the function $F_A$ from $\mathcal {P}(A)$ to $\mathcal {P}(A)$ , for $X\subseteq A$ , by

$$ \begin{align*} F_A(X) = F(X)\cap A. \end{align*} $$

Then, when (12), or (13), holds, $(A,F_A)$ is the functional version of $(A,N_A)$ : for $X\subseteq A$ and $w\in A$ ,

$$ \begin{align*} w\in F_A(X)\; \mbox{ iff } X\in N_A(w). \end{align*} $$

We omit the straightforward verification of the next fact. In (i), the generated subframe relation $(W',R') \subseteq _g (W,R)$ between Kripke frames is the usual one.

Next, we have:

Proof. Easy induction on $\varphi $ . Let us check the case $\varphi = \Box \psi $ . We have, with $\mathcal {M} = (W,F,V)$ and $\mathcal {M}_A = (A,F_A,V_A)$ :

$$ \begin{align*} [\kern-1.5pt[\Box\psi]\kern-1.5pt]_{\mathcal{M}} \cap A & = F([\kern-1.5pt[\psi]\kern-1.5pt]_{\mathcal{M}}) \cap A\\ & = F([\kern-1.5pt[\psi]\kern-1.5pt]_{\mathcal{M}} \cap A) \cap A \;\;\;\; \text{(by (13))}\\ & = F_A([\kern-1.5pt[\psi]\kern-1.5pt]_{\mathcal{M}} \cap A) \\ & = F_A([\kern-1.5pt[\psi]\kern-1.5pt]_{\mathcal{M}_A}) \;\;\;\;\;\;\;\;\;\;\;\;\;\; \text{(ind.\ hyp.)}\\ & = [\kern-1.5pt[\Box\psi]\kern-1.5pt]_{\mathcal{M}_A}. \end{align*} $$

□

Thus, we see that invariance under generated subframes in Kripke semantics and in topological semantics, in particular Fact 26, are special cases of Proposition 29. In other words, locality in this sense is not specific to these special forms of possible worlds semantics, but holds in general. If we want to use truth locality to single out the standard interpretation, we must find a stronger notion.

4.2 Rooted subframes

In Kripke semantics, but not, for example, in topological semantics, there is always a smallest generated subset containing a given point w. Is this a unique feature of Kripkean frames? We show in this subsection that the answer is Yes, among topological frames, but not in general.

Definition 30. If $(W,F)$ is a neighborhood frame and $w\in W$ , define the rooted subframe $(W,F)[w] = (W',F_{W'})$ of $(W,F)$ , where

$$ \begin{align*} W' = \bigcap\{X\subseteq W\!: w\in X \text{ and } (X,F_X) \subseteq_g (W,F)\}. \end{align*} $$

It follows that when $(W,F)$ is Kripkean, so F is standard with respect to a relation R, then $(W,F)[w] = nbd((W,R)[w])$ , where $(W,R)[w]$ is the usual Kripke subframe of $(W,R)$ generated by w. But in general, rooted subframes need not be generated subframes, even when $(W,F)$ is normal. One may check, for example, that this fails at 0 in a frame based on the natural numbers where the family $N(n)$ is the set of sets containing n for $n \neq 0$ , and $N(0)$ is the set of co-finite sets containing 0.

We can use this criterion as one notion of locality.

Thus, Kripkean frames are strongly local. Here is a partial converse.

Proof. Suppose $(W,F)$ is strongly local, where $F= int_\tau $ for some topology $\tau $ . By Fact 28(ii), for $X\subseteq W$ ,

(a) $$ \begin{align} (X,F_X) \subseteq_g (W,F) \mbox{ iff } X \mbox{ is open}. \end{align} $$

It follows that every set in $N_F(w)$ includes an open set in $N_F(w)$ (namely, its interior). Now take any $w\in W$ , and let $A^w = \bigcap \{X\subseteq W\!: w\in X\ \mathrm{and } X\ \mathrm{is open}\}$ . We have:

(b) $$ \begin{align} A^w \subseteq\: \bigcap N_F(w). \end{align} $$

To see this, take $v\in A^w$ and let X be any set in $N_F(w)$ ; we must show $v\in X$ . Let Y be an open subset Y of X in $N_F(w)$ . Then $w\in F(Y) = int_\tau (Y) \subseteq Y$ , so $w\in Y$ and therefore, by the definition of $A^w$ , $v\in Y\subseteq X$ , and (b) is proved. Next, we claim that for $X\subseteq W$ ,

(c) $$ \begin{align} X\subseteq W \mbox{ is open iff } X \subseteq F(X). \end{align} $$

To prove (c), note that by (a), X is open iff for all $Y\subseteq W$ , $F(Y)\cap X = F(X\cap Y)\cap X$ . With $Y=W$ we get, since $F(W)=int_\tau (W) = W$ , that $X = F(X)\cap X$ , that is, $X\subseteq F(X)$ . For the other direction, recall that $int_\tau (X\cap Y)=int_\tau (X)\cap int_\tau (Y)$ . So if $X\subseteq F(X)$ , then, for any $Y\subseteq W$ , $F(X\cap Y)\cap X = F(X) \cap F(Y) \cap X = F(Y) \cap X$ , so X is open. This proves (c).Footnote ¹⁸ Finally,

(d) $$ \begin{align} A^w \in N_F(w). \end{align} $$

This is because $w\in A^w$ by definition, and $A^w$ is open by (a) and the assumption of strong locality, so we have $A^w \subseteq F(A^w)$ by (c), and thus $w\in F(A^w)$ , i.e., $A^w \in N_F(w)$ . So (d) holds. But then, since $N_F(w)$ is a filter, it follows by (b) that $\bigcap N_F(w) \in N_F(w)$ , which means that $N_F(w)$ is principal. Since w was arbitrary, this implies as before that $(W,F)$ is Kripkean.□

Say that a global interpretation $\mathcal {K}$ is strongly local if each $(W,F)\in \mathcal {K}$ is strongly local. The following corollary is immediate.

$(W,F)[w]$ is a reasonable notion of point-generated subframe, provided $(W,F)$ is strongly local.Footnote ¹⁹ But the next example shows that strong locality is not in general sufficient to enforce the standard interpretation, even for normal interpretations.

Example 36. Let $W = \Bbb {N}$ , and define $(\Bbb {N},F)$ via the families $N_F(n)$ as follows:

$$ \begin{align*} N_F(n) &= \{X\subseteq\Bbb{N}\!: n+1\in X\}, \text{ for } n>0 ,\\ N_F(0) &= \{X\subseteq\Bbb{N}\!: 1\in X \text{ and } X \text{ is co-finite}\} . \end{align*} $$

Each $N_F(n)$ is a filter, so $(\Bbb {N},F)$ respects $\mathsf {K}$ , but $N_F(0)$ is non-principal, so $(\Bbb {N},F)$ is not Kripkean. The potential accessibility relation is the successor relation: $Acc_F = \{(n,n+1)\!: n\in \Bbb {N}\}$ . Using the fact observed in note 18, one checks that the domain of each $(\Bbb {N},F)[n]$ is $\{k\!: k\geq n\}$ . In particular, $(\Bbb {N},F)[0] = (\Bbb {N},F)$ . It is now easy to verify that $(\Bbb {N},F)$ is strongly local.

5.1 Filter frames

In order to do so, it will prove handy to take yet another perspective on our general semantics for the modal operator. Invariance under Kripkean bisimulation shall require that all that matters to the action of $(W,F)$ be its Kripkean component, the potential accessibility relation hidden in F. The idea can be made more vivid by actually decomposing F into two parts, that accessibility relation on the one hand and a filtering component on the other, corresponding to the special twist that neighborhood semantics adds to Kripke semantics.

Consider interpreting $\Box $ over a set of worlds W by means of a pair $(\mathit {FF},R)$ where for every w, $\mathit {FF}(w)$ is a free filter over W, and R is an accessibility relation in the usual sense. Recall that a filter is free if the intersection of all its members is empty. Such a frame $(W,\mathit {FF},R)$ interprets $\Box $ by means of the following semantic clause, for $\mathcal {M} = (W,\mathit {FF},R,V)$ :

(14) $$ \begin{align} \mathcal{M},w\models\Box\varphi\; \mbox{ iff }\;R(w)\subseteq[\kern-1.5pt[\varphi]\kern-1.5pt]_{\mathcal{M}} \text{ and } [\kern-1.5pt[\varphi]\kern-1.5pt]_{\mathcal{M}} \in \mathit{FF}(w). \end{align} $$

The clauses for atomic formulas, negations, and conjunctions are as usual. We shall call such frames filter interpretations:Footnote ²¹ formulas of the form $\Box \varphi $ deemed true at a world w are those such that $\varphi $ is true at worlds reachable from w, just as for standard Kripkean interpretations, and $\varphi $ satisfies the condition imposed by a local filter; this is the neighborhood twist. On this view, the filter part of the frame has filtering as its only business. Because the information about the accessibility relation is encoded separately, local filters are now required to be free, that is, not to encapsulate a requirement to include any fixed subset of W.

There is a natural correspondence between filter frames $(W,\mathit {FF},R)$ and the usual normal neighborhood frames $(W,N)$ . To get from a filter interpretation $(W,\mathit {FF},R)$ to its associated neighborhood frame $(W,N)=(W,\mathit {FF},R)^*$ , one simply needs to reconstruct the neighborhood $N(w)$ at every point, by setting, for each $w \in W$ , $N(w) = \mathit {FF}(w) +R(w)$ , where, for $\mathcal {A} \subseteq \mathcal {P}(W)$ and $B \subseteq W$ , $\mathcal {A}+B$ is $\{X \cup B\!: X \in \mathcal {A}\}$ . In the other direction, to get from a neighborhood frame $(W,N)$ to a filter interpretation $(W,\mathit {FF},R)=(W,N)^{o}$ , one needs, at each $w \in W$ , to divide $N(w)$ into two components, an underlying free filter and worlds accessible at w. The latter are simply defined by setting $R(w) = \bigcap N(w)$ , so that R is nothing but $Acc_F$ . To get the former, we define $\mathit {FF}(w)$ as $N(w)$ ‘set free’, by letting $\mathit {FF}(w) = N(w) - \bigcap N(w)$ , where $N(w) - \bigcap N(w) = \{ A \subseteq W\!: \mathrm{there is }B \in N(w)\ \mathrm{ such that } B - \bigcap N(w) \subseteq A \} $ .Footnote ²² Shifting in this way from neighborhood semantics to filter models and vice versa preserves truth of modal formulas: it is simply a matter of perspective:

Proof. For (i), we need to prove (a) that $(W,\mathit {FF},R)^{*}$ is a normal frame, and (b) that, for a given valuation V and world w, it makes the same formulas true as $(W,\mathit {FF},R)$ .

(i.a) We check that $\mathit {FF}(w)+R(w)$ is a filter. First, it is closed under finite intersections. Let $X \cup R(w)$ and $Y \cup R(w)$ be two sets in $\mathit {FF}(w)+R(w)$ with $X,Y \in \mathit {FF}(w)$ . Because $\mathit {FF}(w)$ is a filter, $X \cap Y$ is in $\mathit {FF}(w)$ , hence $(X \cap Y) \cup R(w)$ is in $\mathit {FF}(w) + R(w)$ . Second, it is upward closed. Let $X \cup R(w)$ be a set in $\mathit {FF}(w)+R(w)$ and $X \cup R(w) \subseteq Y$ . Y may be written as $(Y - (R(w) - X)) \cup R(w)$ . Again, because $\mathit {FF}(w)$ is a filter, $(Y - (R(w) - X)) \supseteq X$ guarantees that $Y - (R(w) - X)$ is in $\mathit {FF}(w)$ , hence $(Y - (R(w) - X)) \cup R(w)$ is in $\mathit {FF}(w) + R(w)$ .

(i.b) Given the semantic clause for filter interpretations, it is enough to show that for all $A\subseteq W$ , $R(w)\subseteq A$ and $A \in \mathit {FF}(w)$ iff $A \in \mathit {FF}(w)+R(w)$ . The direction from left to right is immediate by definition of $\mathit {FF}(w)+R(w)$ . From right to left, if $A \in \mathit {FF}(w) + R(w)$ , $A = X \cup R(w)$ for some $X \in \mathit {FF}(w)$ , and since $\mathit {FF}(w)$ is upward closed, $X \cup R(w)$ is also in $\mathit {FF}(w)$ .

For (ii), we must show (a) that $(W,N)^{o}$ is a filter frame, (b) that, for a given valuation and world, it makes the same formulas true as $(W,N)$ , and (c) that transforming it back into a neighborhood frame yields $(W,N)$ itself.

(ii.a) We check that $N(w) - \bigcap N(w)$ is a free filter. First, it is a filter. It is sufficient to show that $N(w)- \bigcap N(w)$ is closed under finite intersections. Take $A,B \in N(w) - \bigcap N(w)$ . So there are $A',B' \in N(w)$ such that $A' - \bigcap N(w)\subseteq A$ and $B' - \bigcap N(w)\subseteq B$ . By normality, $A' \cap B' \in N(w)$ , and $(A' \cap B') - \bigcap N(w) \subseteq A \cap B$ , so $A\cap B \in N(w) - \bigcap N(w)$ . Second, $N(w) - \bigcap N(w)$ is free. If not, there would be a $w' \in W$ such that $w' \in \bigcap \mathit {FF}(w)$ . Thus, $w' \in \bigcap (N(w)- \bigcap N(w))$ , and it follows that $w' \in \bigcap N(w)$ . But then, since $W\in N(w)$ and we have $w'\not \in W-\bigcap N(w) \in \mathit {FF}(w)$ , we get that $w' \not \in \bigcap \mathit {FF}(w)$ , a contradiction.

(ii.b) Given the semantic clause for filter models, we need to show that $A \in N(w)\;$ iff $\;\bigcap N(w) \subseteq A$ and $A \in N(w) - \bigcap N(w)$ . The direction from left to right is immediate: if $A \in N(w)$ , then $\bigcap N(w) \subseteq A$ and $A-\bigcap N(w) \subseteq A$ , so $A \in N(w) - \bigcap N(w)$ . The direction from right to left is simple as well. Since $A \in N(w) - \bigcap N(w)$ , A is a superset of some set of the form $A' - \bigcap N(w)$ with $A' \in N(w)$ . But since $\bigcap N(w) \subseteq A$ , A is also a superset of $A'$ , hence A is in $N(w)$ by upward closure again.

(ii.c) We need to show that for any $X \subseteq W$ , $X \in (N(w) - \bigcap N(w)) + \bigcap N(w)$ iff $X \in N(w)$ . By definition, we have $X\in (N(w) - \bigcap N(w)) + \bigcap N(w)$ iff

(*) $$ \begin{align} \exists Y,Z\subseteq W[Z\in N(w) \;\&\; Z-\bigcap N(w)\subseteq Y \;\&\: X=Y\cup \bigcap N(w)]. \end{align} $$

From left to right, (*) entails that $Z \subseteq X$ , so $X\in N(w)$ by closure under supersets. From right to left we can take $Y=Z=X$ , since $X\in N(w)$ implies $\bigcap N(w)\subseteq X$ .□

What do Kripkean frames amount to in the world of filter frames? $(W,\mathit {FF},R)$ will semantically behave like a Kripke frame when there is no filtering, that is when the second conjunct $[\kern-1.5pt[\varphi ]\kern-1.5pt]_{\mathcal {M}} \in \mathit {FF}(w)$ in the interpretation of $\Box $ by clause (14) is entailed by the first conjunct. More precisely, say that a filter frame $(W,\mathit {FF},R)$ is Kripkean when its neighborhood match $(W,\mathit {FF},R)^*$ is. Then we have the following.

Special cases of Kripkean filter frames are those of the form $(W,c_{\mathcal {P}(W)},R)$ where $c_{\mathcal {P}(W)}: W \rightarrow \mathcal {P}(\mathcal {P}(W))$ is the constant function defined by $c_{\mathcal {P}(W)}(w)=\mathcal {P}(W)$ . One might have expected $(W,\mathit {FF},R)$ to be Kripkean only if $\mathit {FF} = c_{\mathcal {P}(W)}$ , but this is not mandatory, since the filtering only matters for propositions true at reachable worlds. Say that two frames $(W,\mathit {FF},R)$ and $(W,\mathit {FF}',R)$ are R-equivalent if, whenever $w \in W$ and $R(w) \subseteq X\subseteq W$ , we have $X \in \mathit {FF}(w)$ if and only if $X \in \mathit {FF}'(w)$ .

Given a logic $\mathsf {L}$ , we can look at all the filter frames which make it valid, and set $\mathit {Fr}_{\mathrm{ff}}(\mathsf {L}) = \{ (W,\mathit {FF},R) : (W,\mathit {FF},R) \models \mathsf {L}\}$ in keeping with the previous notation for $\mathit {Fr}_{\mathrm{nbd}}(\mathsf {L})$ . One gets correspondence results between axioms of modal logic and conditions on $\mathit {FF}$ and R, in the spirit of traditional correspondence results for Kripke frames which relate axioms and conditions on R. Here are a few examples:

Thus, some correspondence results for filter models are simple rewritings of classical correspondences, as witnessed by the T and 4 axioms. But sometimes they involve a congenial mix of conditions on $\mathit {FF}$ and R, as illustrated by D. It would be interesting to find other axioms with the same behavior as D, and also to characterize the class of modal axioms for which correspondence smoothly transfers as it does for T and 4.

5.2 Bisimulation invariance

A Kripkean bisimulation shall simply be a bisimulation in the usual sense between the underlying Kripkean structures of two models. In the context of filter models $(W,\mathit {FF},R,V)$ , where this underlying Kripkean structure is made explicit as R, the definition is straightforward:

The locality of modal logic with respect to Kripke models is often framed in terms of invariance under bisimulation: bisimilar worlds satisfy the same modal formulas, so that only the local features of the structure of the Kripke models, as captured by the zigzag clauses (ii) and (iii) in the definition of a bisimulation, matter to modal satisfaction. How much of Kripkeanity does invariance under bisimulation give us in the present context? In general, k-bisimilarity between filter frames does not guarantee modal equivalence, since the filtering part of those frames may differ widely. Is there a sense in which Kripkean frames are the k-bisimulation invariant part of our general frames?

First, invariance under k-bisimulation is a property of classes of frames. Just as in the usual case, it amounts to requiring that bisimilar models based on frames in the class are modally equivalent:

For a given logic $\mathsf {L}$ , our target is the subclass of $\mathit {Fr}_{\mathrm{ff}}(\mathsf {L})$ consisting only of Kripkean filter frames, which we label $\mathit {Fr}_{\mathrm{kff}}(\mathsf {L})$ . $\mathit {Fr}_{\mathrm{kff}}(\mathsf {L})$ is invariant under k-bisimulation, but in general, there will be more than one maximal invariant subclass of $\mathit {Fr}_{\mathrm{ff}}(\mathsf {L})$ . Taking $\mathsf {K}$ for $\mathsf {L}$ , one may simply pick a handful of non k-bisimilar non-Kripkean frames, and grow those frames into a maximal k-bisimulation invariant class which will not be comparable with $\mathit {Fr}_{\mathrm{kff}}(\mathsf {K})$ . One needs to assume that the Kripkean frames are also there, or invariance cannot do the job. We shall phrase this as a closure condition on classes of frames.

We are now able to characterize the Kripkean interpretations of a normal modal logic:

Kripkean logics as discussed in Section 3 can be readily characterized on the same basis:

If $\mathit {Fr}_{\mathrm{ff}}(\mathsf {L})$ were always invariant under k-extension, one could infer from Theorem 44 that a modal logic is Kripkean iff it simply is invariant under k-bisimulation. But as a case in point, $\mathit {Fr}_{\mathrm{ff}}(\mathsf {K}\mathrm{D})$ is not closed under k-extension. By Fact 40, D is valid on non-serial frames on condition that worlds without successors come with proper filters, so that closure under k-extension fails.

When does it hold? We end by formulating this question, which as far as we know is open, in terms of ordinary neighborhood frames rather than filter frames:

The logics $\mathsf {K}$ , $\mathsf {K}$ T, $\mathsf {K}4$ (hence S4), and $\mathsf {K}$ 5 have this property, as does any normal logic extending $\mathsf {K}$ B, by Proposition 23. But it fails, for example, for $\mathsf {K}$ D and $\mathsf {K}$ 45D.