The Problem of Cross-world Predication

Abstract

While standard first-order modal logic is quite powerful, it cannot express even very simple sentences like “I could have been taller than I actually am” or “Everyone could have been smarter than they actually are”. These are examples of cross-world predication, whereby objects in one world are related to (sometimes the same) objects in another world. Extending first-order modal logic to allow for cross-world predication in a motivated way has proven to be notoriously difficult. In this paper, I argue that the standard accounts of cross-world predication all leave something to be desired. I then propose an account of cross-world predication based on quantified hybrid logic and show how it overcomes the limitations of these previous accounts. I will conclude by discussing various philosophical consequences and applications of such an account.

Appendices

Appendices

1.1 A Two-Sorted Logic

In this section, we review the standard semantics for two-sorted first-order logic. We then state precisely in what sense the translation from Definition 5 from $\mathcal {L}^{\mathsf {1M}}$ to $\mathcal {L}^{\mathsf {2S}}$ is in fact accurate.

Throughout, we adopt the following convention:

Notation 16

If α ₁,…, α _n is any sequence (of variables, terms, objects, etc.), we may write “$\bar {\alpha }$” in place of “ α ₁,…, α _n ”. $\bar {\alpha }$ is assumed to be of the appropriate length in any given context. When f is some unary function, we may write “$f(\bar {\alpha })$” in place of “ f(α ₁),…, f(α _n )”. (Context will always distinguish between f(α ₁),…, f(α _n ) and f(α ₁,…, α _n ).) Where $\bar {\alpha }$ is a sequence, we’ll let $\mid \bar {\alpha }\mid$ be the length of $\bar {\alpha }$.

1.2 A.1 Models

First, we review standard two-sorted first-order models.

Definition 17 (Two-Sorted Models)

A $\mathcal {L}^{\mathsf {2S}}$ -model or two-sorted model is an ordered tuple $\mathfrak {M}$ = 〈W, D, V〉 where W and D are sets, and V is a function (the valuation function) such that:

• for each c ∈ C O N, V(c):W→D;

• for each P ^n/m∈P R E D ^n/m, V(P ^n/m)⊆D ⁿ×W ^m;

• V(E) ⊆ D×W;

• V(R) ⊆ W×W.

Usually, we are interested in $\mathcal {L}^{\mathsf {2S}}$-models which correspond to some $\mathcal {L}^{\mathsf {1M}}$-model.

Definition 18 (Model Correspondents)

Let $\mathcal {M}$ = 〈W, R, D, δ, I〉 be an $\mathcal {L}^{\mathsf {1M}}$-model. An $\mathcal {L}^{\mathsf {2S}}$ -correspondent of $\mathcal {M}$ is a $\mathcal {L}^{\mathsf {2S}}$-model $\mathfrak {M}$ = 〈W, D, V〉 such that:

• for all c ∈ C O N, V(c)(w) = I(c, w);

• for all P ⁿ∈P R E D ^n/1, V(P ⁿ) = {〈a ₁,…, a _n ;w〉∣〈a ₁,…, a _n 〉∈I(P ⁿ, w)};

• V(E) = {〈a;w〉∈D×W∣a ∈ δ(w)};

• V(R) = R.

An $\mathcal {L}^{\mathsf {2S}}$-correspondent is just an $\mathcal {L}^{2S}$-correspondent of some $\mathcal {L}^{\mathsf {1M}}$-model.

Notice in particular that Definition 18 doesn’t pick out a unique $\mathcal {L}^{\mathsf {2S}}$-correspondent for any given $\mathcal {L}^{\mathsf {1M}}$-model. This will be important below.

1.3 A.2 Semantics

Next, we review the standard semantics for two-sorted first-order logic.

Definition 19 (Two-Sorted Variable Assignment)

Let $\mathfrak {M}$ be an $\mathcal {L}^{\mathsf {2S}}$-model. A variable assignment for $\mathfrak {M}$ is a function assigning members of D to object variables, and members of W to state variables. The other definitions are as they are in Definition 2. For any $\mathcal {L}^{\mathsf {2S}}$-correspondent $\mathfrak {M}$ of $\mathcal {M}$, and any g on $\mathcal {M}$, we’ll say that g ^c for $\mathfrak {M}$ is a $\mathcal {L}^{\mathsf {2S}}$-correspondent for g if g(x) = g ^c(x) for all x ∈ V A R.

Definition 20 (Two-Sorted Denotation)

Let τ ∈ T E R M ^2S, let $\mathfrak {M}$ be an $\mathcal {L}^{\mathsf {2S}}$-model, and let $g^{\mathsf {c}} \in \mathsf {VA}(\mathfrak {M})$. The denotation of τ at $\left \langle \mathfrak {M},g^{\mathsf {c}}\right \rangle$, ${[{\kern -2.3pt}[\tau ]{\kern -2.3pt}]}^{{\mathfrak {M}},g^{\mathsf {c}}}$, is defined as follows:

$$[{\kern-2.3pt}[\tau]{\kern-2.3pt}]^{\mathfrak{M},g^{\mathsf{c}}} = \left\{\begin{array}{ll} V(c)(g^{\mathsf{c}}(s)) & \text{if}\,\tau = c(s) \text{ where\,} c \in \mathsf{CON}\, \text{and}\, s \in \mathsf{SVAR} \\ g^{\mathsf{c}}(x) & \text{if}\, \tau = x \text{ where}\,x \in \mathsf{VAR}. \end{array}\right. $$

Definition 21 (Two-Sorted Satisfaction)

The two-sorted satisfaction relation, $\Vdash$, is defined recursively, for all $\mathcal {L}^{\mathsf {2S}}$-models $\mathfrak {M}$ = 〈W, D, V〉 and all variable assignments $g^{\mathsf {c}} \in \mathsf {VA}(\mathfrak {M})$:

$$\begin{array}{llll} \mathfrak{M},g^{\mathsf{c}} & \vDash P^{n/m}(\bar{\tau};\bar{s}) & \Leftrightarrow & \left\langle[{\kern-2.3pt}[\,\bar{\tau}\,]{\kern-2.3pt}]^{\mathfrak{M},g^{\mathsf{c}}};g^{\mathsf{c}}(\bar{s})\right\rangle \in V(P^{n/m}) \\ \mathfrak{M},g^{\mathsf{c}} & \vDash \tau_{1} = \tau_{2} & \Leftrightarrow & [{\kern-2.3pt}[\tau_{1}]{\kern-2.3pt}]^{\mathfrak{M},g^{\mathsf{c}}} = [{\kern-2.3pt}[\tau_{2}]{\kern-2.3pt}]^{\mathfrak{M},g^{\mathsf{c}}} \\ \mathfrak{M},g^{\mathsf{c}} & \vDash s_{1} = s_{2} & \Leftrightarrow & g^{\mathsf{c}}(s_{1}) = g^{\mathsf{c}}(s_{2}) \\ \mathfrak{M},g^{\mathsf{c}} & \vDash \mathsf{E}(\tau;s) & \Leftrightarrow & \left\langle [{\kern-2.3pt}[\tau]{\kern-2.3pt}]^{\mathfrak{M},g^{\mathsf{c}}};g^{\mathsf{c}}(s)\right\rangle \in V(\mathsf{E}) \\ \mathfrak{M},g^{\mathsf{c}} & \vDash \mathsf{R}(s_{1},s_{2}) & \Leftrightarrow & \left\langle g^{\mathsf{c}}(s_{1}),g^{\mathsf{c}}(s_{2})\right\rangle \in V(\mathsf{R}) \\ \mathfrak{M},g^{\mathsf{c}} & \vDash \neg\varphi & \Leftrightarrow & \mathfrak{M},g^{\mathsf{c}} \nvDash \varphi \\ \mathfrak{M},g^{\mathsf{c}} & \vDash \varphi \wedge \psi & \Leftrightarrow & \mathfrak{M},g^{\mathsf{c}} \vDash \varphi \text{ and}\; \mathcal{M},g \vDash \psi \\ \mathfrak{M},g^{\mathsf{c}} & \vDash \forall{x}\varphi & \Leftrightarrow & \forall{a \in D\colon} \mathfrak{M},(g^{\mathsf{c}})^{x}_{a} \vDash \varphi \\ \mathfrak{M},g^{\mathsf{c}} & \vDash \forall{s}\varphi & \Leftrightarrow & \forall{w \in W\colon} \mathfrak{M},(g^{\mathsf{c}})^{s}_{w} \vDash \varphi. \end{array} $$

1.4 A.3 Expressivity

Recall the translations from $\mathcal {L}^{\mathsf {1M}}$ into $\mathcal {L}^{\mathsf {2S}}$ from Definition 5. Given the definitions above, the following is easy to prove by induction on the complexity of $\mathcal {L}^{\mathsf {1M}}$-terms and $\mathcal {L}^{\mathsf {1M}}$-formulas.

Lemma 22 (Adequacy of Translation)

Let $\mathcal {M}$ be an $\mathcal {L}^{\mathsf {1M}}$ -model, $\mathfrak {M}$ an $\mathcal {L}^{\mathsf {2S}}$ -correspondent for $\mathcal {M}$ , w,v∈W, $g \in \mathsf {VA}(\mathcal {M})$ , g ^c an $\mathcal {L}^{\mathsf {2S}}$-of g for $\mathfrak {M}$, s,t∈S V A R, τ an $\mathcal {L}^{\mathsf {1M}}$-term, and φ an $\mathcal {L}_{\Pi }^{\mathsf {1M}}$ -formula.

(a):

$[{\kern -2.3pt}[\tau ]{\kern -2.3pt}]^{\mathcal {M},w,v,g} = [{\kern -2.3pt}[\mathsf {st}_{s,t}({\tau })]{\kern -2.3pt}]^{\mathfrak {M},(g^{\mathsf {c}})^{s,t}_{w,v}}$

(b):

$\mathcal {M},w,v,g \Vdash \varphi$ iff $\mathfrak {M},(g^{\mathsf {c}})^{s,t}_{w,v} \vDash \mathsf {ST}_{s,t}{(\varphi )}$.

One can now prove more rigorously that cross-world predication is in general not expressible in $\mathcal {L}^{\mathsf {1M}}$. First, let’s say what it means for a $\mathcal {L}^{\mathsf {1M}}$-formula to be expressible in $\mathcal {L}^{\mathsf {2S}}$:⁴⁷

Definition 23 (Expressivity)

An $\mathcal {L}^{\mathsf {1M}}$-formula $\varphi (\bar {x})$ expresses an $\mathcal {L}^{\mathsf {2S}}$-formula $\varphi ^{\mathsf {2S}}(\bar {x};s,t)$ if φ ^2S is equivalent (in $\mathcal {L}^{\mathsf {2S}}$) to S T _{s, t} (φ). Similarly for $\mathcal {L}_{Pi}^{\mathsf {1M}}$-formulas.

Now, recall that in Definition 18, no constraints are placed on how $\mathcal {L}^{\mathsf {2S}}$-correspondents are to interpret n/m-predicates where m>1. Thus, the extensions of the relevant 2/2-place predicates occurring in (17)–(19) could be anything—nothing about the $\mathcal {L}^{\mathsf {1M}}$-model will tell us what they must be in its $\mathcal {L}^{\mathsf {2S}}$-correspondents. Such arbitrariness makes it fairly easy to create two $\mathcal {L}^{\mathsf {2S}}$-correspondents of an $\mathcal {L}^{\mathsf {1M}}$-model which disagree on one of (17)–(19). But then it follows by Lemma 22 that none of (17)–(19) can be equivalent to the (possibilist or actualist) translation of a $\mathcal {L}^{\mathsf {1M}}$-formula. Thus:

Corollary 24 (Inexpressibility of Cross-World Predication)

There is no $\mathcal {L}_{\Pi }^{\mathsf {1M}}$ -formula that expresses any of ( 17 )–( 19 ).

The proof that cross-world quantification—in particular, (20)–(21)—are inexpressible requires more work. The more complicated proof via bisimulations is in Kocurek [22].

B Wehmeier’s Subjunctive Logic

In this section, we’ll briefly examine the framework of Wehmeier [41], which was also designed to solve the problems of cross-world predication and quantification, and explain how it relates to $\mathcal {L}^{\mathsf {H}}$.⁴⁸

The key idea behind Wehmeier’s proposal is the idea of mood. Essentially, Wehmeier suggests that what needs to be added to $\mathcal {L}^{\mathsf {1M}}$ to overcome these expressivity issues is not some new operators, but rather some way of distinguishing indiciative and subjunctive moods in the syntax. Wehmeier does this by introducing “mood markers” i, s, s ₁, s ₂, s ₃,… (i for “indicative”, s for “subjunctive”), which are basically state variables. Predicates will then be decorated with mood markers to indicate the worlds relevant for their evaluation, i.e., the world relative to which we calculate the extension of that predicate.

Formally, Wehmeier’s language $\mathcal {L}^{\mathsf {W}}$ is built as follows:

$$\begin{array}{@{}rcl@{}} \tau &::=& c \mid x \\ \varphi &::=& P^{n/t}(\tau_{1},\dots,\tau_{n}) \mid C^{n/t_{1},\dots,t_{n}}(\tau_{1},\dots,\tau_{n}) \mid \tau \,=\, \sigma \mid \neg\varphi \mid (\varphi \wedge \varphi) \mid {\Box_{k}^{t}}\varphi \mid \forall^{t}{x}\varphi \end{array} $$

where k ≥ 1, and t, t ₁,…, t _n are mood markers. In $\mathcal {L}^{\mathsf {W}}$, we distinguish two kinds of predicates: “ordinary” predicates P ⁿ that are only decorated with one mood marker, and “cross-world” predicates C ⁿ that are decorated with n-mood markers. While $\mathcal {L}^{\mathsf {H}}$ does not make this distinction, it could in principle by introducing a class of “ordinary” predicates are essentially insensitive to $\blacktriangleleft$-terms. We will keep this distinction to help ease the comparison betweeh $\mathcal {L}^{\mathsf {H}}$ and $\mathcal {L}^{\mathsf {W}}$.

The quantifiers are decorated with mood markers to determine the domain of quantification: $\phantom {\dot {i}\!}\forall ^{s_{n}}{x}\varphi$ says that every x that exists in w _n satisfies φ. Modal operators are both decorated by mood markers and subscriped with numbered indicies. The mood marker indicates world relative to which accessibility is determined, while the subscript indicates where we save the shifted world for reference: $\Box _{k}^{s_{n}}\varphi$ says that at every world v accessible to w _n , φ is true assuming we save v as the k ^th reference world.

With regards to models and semantics, Wehmeier essentially uses c w-models with a number of constraints. For one thing, he assumes that constants are rigid designators (so I(c, w) = I(c, v) for all w, v ∈ W) and that the extension of cross-world predicates are rigid in a similar sense (so I(C, w) = I(C, v) for all w, v ∈ W). We will follow Wehmeier and assume these constraints for ease of comparison to $\mathcal {L}^{\mathsf {H}}$, noting that they are not essential to any of the results that follow and may be dropped if so desired. In what follows, we’ll simply write “ I(c)” and “ I(C)” for brevity. He also assumes that R is universal (in which case, he can drop the mood marker decorating □) and that only binary predicates (in particular comparatives) are cross-world. However, we will not impose these additional constraints, again for ease of comparison to $\mathcal {L}^{\mathsf {H}}$.⁴⁹

The last major difference in Wehmeier’s models is that he defines the extension of ordinary predicates to be an ordered n-tuple of objects, not object-world pairs. In terms of c w-models, this means that the extension of predicates is insensitive to the world coordinates of object-world pairs. Thus, we can for our purposes define the class of models for Wehmeier as follows:

Definition 25 ( W-Models)

A c w-model $\mathcal {M}$ is a W -model if it meets the following two constraints:

• (i) for all c ∈ C O N and all w, v ∈ W, I(c, w) = I(c, v);

• (ii) for all ordinary predicates P, all w, v ₁,…, v _n ∈W, and all a ₁,…, a _n ∈D:

  $$\left\langle\left\langle a_{1},v_{1}\right\rangle,\dots,\left\langle a_{n},v_{n}\right\rangle\right\rangle \in I(P,w)\,\Leftrightarrow\,\left\langle\left\langle a_{1},w\right\rangle,\dots,\left\langle a_{n},w\right\rangle\right\rangle \in I(P,w); $$

• (iii) for all cross-world predicates C and all w, v ∈ W, I(C, w) = I(C, v).

The semantics relativizes satisfaction to indices of the form $\left \langle \mathcal {M},\bar {w},v,g\right \rangle$, where $\mathcal {M}$ is a W-model, and where $\bar {w} = w_{0},w_{1},w_{2},\dots$. The denotation of terms only needs to be relativized to a model and variable assignment, since the constants rigidly designate:

$${[{\kern-2.3pt}[\tau]{\kern-2.3pt}]}^{\mathcal{M},g} = \left\{\begin{array}{ll} I(c) & \text{if} \;\tau = c \in \mathsf{CON} \\ g(x) & \text{if} \;\tau = x \in \mathsf{VAR}. \end{array}\right. $$

Finally, the interesting semantic clauses are given as follows:

$$\begin{array}{@{}rcl@{}} &&\mathcal{M},\bar{w},v,g \Vdash_{\mathsf{W}} P^{t_{1}}(\tau_{1},\dots,\tau_{n}) \Leftrightarrow \big\langle\big\langle[{\kern-2.3pt}[\tau_{1}]{\kern-2.3pt}]^{\mathcal{M},g},u_{1}\big\rangle,\dots,\big\langle[{\kern-2.3pt}[\tau_{n}]{\kern-2.3pt}]^{\mathcal{M},g},u_{1}\big\rangle \big\rangle \in I(P,u_{1}) \\ &&\mathcal{M},\bar{w},v,g \Vdash_{\mathsf{W}} C^{t}(\tau_{1},\dots,\tau_{n})\hspace*{3.5pt}\Leftrightarrow \big\langle\big\langle[{\kern-2.3pt}[\tau_{1}]{\kern-2.3pt}]^{\mathcal{M},g},u_{1}\big\rangle,\dots,\big\langle[{\kern-2.3pt}[\tau_{n}]{\kern-2.3pt}]^{\mathcal{M},g},u_{n}\big\rangle\big\rangle \in I(C) \\ &&\mathcal{M},\bar{w},v,g \Vdash_{\mathsf{W}} \tau = \sigma \hspace*{3.10pc}\Leftrightarrow [{\kern-2.3pt}[\tau]{\kern-2.3pt}]^{\mathcal{M},g} = [{\kern-2.3pt}[\sigma]{\kern-2.3pt}]^{\mathcal{M},g} \\ &&\mathcal{M},\bar{w},v,g \Vdash_{\mathsf{W}} \Box_{k}^{t_{1}}\varphi \hspace*{3.6pc}\Leftrightarrow \forall{u \in R[u_{1}]\colon} \mathcal{M},\bar{w}[k \mapsto u],u,g \Vdash_{\mathsf{W}} \varphi\\ &&\mathcal{M},\bar{w},v,g \Vdash_{\mathsf{W}} \forall^{t_{1}}{x}\varphi \hspace*{3.25pc}\Leftrightarrow \forall{a \in \delta(u_{1})\colon} \mathcal{M},\bar{w},v,{g^{x}_{a}} \Vdash_{\mathsf{W}} \varphi \end{array} $$

where:

$$u_{m} = {\left\{\begin{array}{ll} w_{0} & \text{if} \;t_{m} = i \\ v & \text{if} \;t_{m} = s \\ w_{j} & \text{if} \;t_{m} = s_{j}. \end{array}\right.} $$

Thus, for instance, (Rich*) is formalized as:

$$ {\Box_{1}^{i}}{\Diamond_{2}^{s}}\forall^{s_{1}}{x}\left( \mathsf{Rich}^{s_{1}}(x) \rightarrow \mathsf{Poor}^{s}(x)\right). $$

(58)

With Wehmeier’s framework laid out, we now consider the question of whether $\mathcal {L}^{\mathsf {W}}$ is intertranslateable with $\mathcal {L}^{\mathsf {H}}$, at least over the class of W-models. First, notice that it’s easy to give a translation W 2 H(φ) from $\mathcal {L}^{\mathsf {W}}$ to $\mathcal {L}^{\mathsf {H}}$ (assume for simplicity that we replaced the mood marker i with s ₀):

$$\begin{array}{@{}rcl@{}} \mathsf{W2H}\left( P^{s}(\tau_{1},\dots,\tau_{n})\right) &=& P(\tau_{1},\dots,\tau_{n}) \\ \mathsf{W2H}\left( P^{t}(\tau_{1},\dots,\tau_{n})\right) &=& {@}_{t} P(\tau_{1},\dots,\tau_{n}) \\ \mathsf{W2H}\left( C^{t_{1},\dots,t_{n}}(\tau_{1},\dots,\tau_{n})\right) &=& C(\blacktriangleleft_{t_{1}}\tau_{1},\dots,\blacktriangleleft_{t_{n}}\tau_{n}) \\ \mathsf{W2H}\left( \tau = \sigma\right) &=& \tau = \sigma \\ \mathsf{W2H}\left( \neg\varphi\right) &=& \neg\mathsf{W2H}\left( \varphi\right) \\ \mathsf{W2H}\left( \varphi \wedge \psi\right) &=& \mathsf{W2H}\left( \varphi\right) \wedge \mathsf{W2H}\left( \psi\right) \\ \mathsf{W2H}\left( {\Box_{k}^{s}}\varphi\right) &=& \Box_{s_{k}}\mathsf{W2H}\left( \varphi\right) \\ \mathsf{W2H}\left( {\Box_{k}^{t}}\varphi\right) &=& {@}_{t}\Box_{s_{k}}\mathsf{W2H}\left( \varphi\right) \\ \mathsf{W2H}\left( \forall^{s}{x}\varphi\right) &=& \forall{x}\mathsf{W2H}\left( \varphi\right) \\ \mathsf{W2H}\left( \forall^{t}{x}\varphi\right) &=& \forall_{t}{x}\mathsf{W2H}\left( \varphi\right) \end{array} $$

where t≠s is a mood marker, and t ₁,…, t _n are mood markers (possibly including s). For example, the translation of (58) is:

$$ {@}_{s_{0}}\Box_{s_{1}}\Diamond_{s_{2}}\forall_{s_{1}}{x}\left( {@}_{s_{1}}\mathsf{Rich}(x) \rightarrow \mathsf{Poor}(x)\right) $$

(59)

which is essentially how we formalized (Rich*) in $\mathcal {L}^{\mathsf {H}}$, except the s ₂ in $\mathcal {L}^{\mathsf {H}}$ isn’t necessary.

Theorem 26 (Adequacy of W 2 H)

Let $\mathcal {M}$ be a W-model, w∈W, and $g \in \mathsf {VA}^{\mathsf {H}}(\mathcal {M})$. Then for any $\mathcal {L}^{\mathsf {W}}$-formula φ, we have that $\mathcal {M},g(s_{0}),g(s_{1}),g(s_{2}),\dots ,w,g \Vdash _{\mathsf {W}} \varphi$ iff $\mathcal {M},w,g \Vdash _{\mathsf {H}} \mathsf {W2H}(\varphi )$.

The reverse translation is not as straightforward. The issue we need to deal with is ↓, which allows one to reset a reference world at will. To deal with this, we need to first manipulate all $\mathcal {L}^{\mathsf {H}}$-formulas into a nice and manageable form.

Definition 27 (Nice $\mathcal {L}^{\mathsf {H}}$-formulas)

A $\mathcal {L}^{\mathsf {H}}$-formula φ is nice if φ is of the form ↓ s _n .ψ where:

• (i) every term has at most one occurrence of $\blacktriangleleft$

• (ii) s ₀ is never bound in φ (and thus, s ₀ and s _n are distinct)

• (iii) there is no occurrence of ↓ s _n . in ψ

• (iv) for all s ∈ S V A R where s is distinct from s _n , there is at most one occurrence of ↓ s. in ψ

• (v) for all s ∈ S V A R, if s has a free occurrence in ψ, then it does not also have a bound occurrence in ψ

• (vi) for all s ∈ S V A R, if ↓ s. occurs in ψ, then its single occurrence is prefixed by a □

• (vii) every occurrence of □ prefixes an occurrence of ↓ s. for some s ∈ S V A R

In other words, nice $\mathcal {L}^{\mathsf {H}}$-formulas are those such that

• (a) the state variables are nicely organized,

• (b) irrelevant stackings of $\blacktriangleleft$ are removed,

• (c) every □ is followed by exactly one unique ↓t., and

• (d) apart from the beginning of the formula, that’s the only place where ↓t. show up.

The following is easy to prove, but requires some tedious details and is simply a matter of reorganizing and rewriting variables appropriately.

Lemma 28 (Nice Normal Form)

Every $\mathcal {L}^{\mathsf {H}}$ -formula is equivalent to a nice $\mathcal {L}^{\mathsf {H}}$ -formula. Furthermore, there’s a recursive procedure for transforming each $\mathcal {L}^{\mathsf {H}}$ -formula into one that’s nice.

So to show that $\mathcal {L}^{\mathsf {H}}$ can be translated into $\mathcal {L}^{\mathsf {W}}$, it suffices to show that the nice $\mathcal {L}^{\mathsf {H}}$-formulas can be translated into $\mathcal {L}^{\mathsf {W}}$. The first step is to extract the “object” and “mood” parts of a given term as follows:

$$\begin{array}{@{}rcl@{}} \mathsf{ob}(\tau) &=& \left\{\begin{array}{ll} c & \text{if}\; \tau = c \in \mathsf{CON} \\ x & \text{if} \;\tau = x \in \mathsf{VAR} \\ \mathsf{ob}(\sigma) & \text{if}\; \tau = \blacktriangleleft_{t}\sigma \end{array}\right.\\ \mathsf{mo}(\tau)&=& \left\{\begin{array}{ll} s & \text{if}~ \tau \in \mathsf{CON} \cup \mathsf{VAR} \\ s_{k} & \text{if}~ \tau = \blacktriangleleft_{s_{k}}\sigma. \end{array}\right. \end{array} $$

Now, since every nice $\mathcal {L}^{\mathsf {H}}$-formula is of the form ↓ s _n .φ, we define a translation function H 2 W(φ) from $\mathcal {L}^{\mathsf {H}}$ to $\mathcal {L}^{\mathsf {W}}$ by induction on φ.

$$\begin{array}{@{}rcl@{}} \mathsf{H2W}\left( P(\tau_{1},\dots,\tau_{n})\right) \,&=&\, P^{s}(\mathsf{ob}(\tau_{1}),\dots,\mathsf{ob}(\tau_{n})) \\ \mathsf{H2W}\left( C(\tau_{1},\dots,\tau_{n})\right) \,&=&\, C^{\mathsf{mo}(\tau_{1}),\dots,\mathsf{mo}(\tau_{n})}(\mathsf{ob}(\tau_{1}),\dots,\mathsf{ob}(\tau_{n})) \\ \mathsf{H2W}\left( \tau = \sigma\right) \,&=&\, \mathsf{ob}(\tau) = \mathsf{ob}(\sigma) \\ \mathsf{H2W}\left( \mathsf{E}(\tau)\right) \,&=&\, \exists^{s}{y}(y = \mathsf{ob}(\tau)) \\ \mathsf{H2W}\left( \neg\varphi\right) \,&=&\, \neg\mathsf{H2W}\left( \varphi\right) \\ \mathsf{H2W}\left( \varphi \wedge \psi\right) \,&=&\, \mathsf{H2W}\left( \varphi\right) \wedge \mathsf{H2W}\left( \psi\right) \\ \mathsf{H2W}\left( \Box_{s_{k}}\varphi\right) \,&=&\, {\Box_{k}^{s}}\mathsf{H2W}\left( \varphi\right) \\ \mathsf{H2W}\left( {@}_{s_{k}}\varphi\right) \,&=&\, \mathsf{H2W}\left( \varphi\right)[{s/s_{k}}] \\ \mathsf{H2W}\left( \forall{x}\varphi\right) \,&=&\, \forall_{s}{x}\mathsf{H2W}\left( \varphi\right) \end{array} $$

where y does not occur in τ and H 2 W(φ)[s/t] is the result of replacing every instance of s that’s not within the scope of a modal with t.⁵⁰

Theorem 29 (Adequacy of H 2 W)

Let $\mathcal {M}$ be a W-model, w ∈ W, and $g \in \mathsf {VA}^{\mathsf {H}}(\mathcal {M})$. Then for any nice $\mathcal {L}^{\mathsf {H}}$-formula ↓s_n.φ, we have that:

$$\mathcal{M},w,g \Vdash_{\mathsf{H}} \downarrow s_{n}.\varphi \Leftrightarrow \mathcal{M},g^{s_{n}}_{w}(s_{0}),g^{s_{n}}_{w}(s_{1}),g^{s_{n}}_{w}(s_{2}),\dots,w,g^{s_{n}}_{w} \Vdash_{\mathsf{W}} \mathsf{H2W}\left( \varphi\right). $$

Thus, in a very strong sense, we can think of $\mathcal {L}^{\mathsf {H}}$ as a generalization of Wehmeier’s original framework that lifts various restrictions he placed on the models and the syntax.

C Characterization of Cross-World Formulas

The goal of this section is to prove that Theorem 15, viz., that every non-cross-world formula of the form φ(x ₁,…, x _n ;t ₁,…, t _m ) is equivalent to a boolean combination of $\mathcal {L}^{\mathsf {H}}$-formulas that are either of the form ${@}_{t_{i}}\psi$ or of the form θ, where ψ and θ are $\mathcal {L}^{\mathsf {1M}-}$-formulas.

Definition 30 (Cross-world)

An $\mathcal {L}^{\mathsf {H}}$-formula is explicitly non-cross-world if it neither contains an instance of $\blacktriangleleft$ nor an object quantifier scoping over an instance of @. Thus, non-cross-world formulas are those that are equivalent to some explicitly non-cross-world formula.

Definition 31 (Isolation)

An isolated atom is any $\mathcal {L}^{\mathsf {H}}$-formula either of the form @ _t ψ or of the form θ where ψ and θ are $\mathcal {L}^{\mathsf {1M}-}$-formulas. An $\mathcal {L}^{\mathsf {H}}$-formula φ is in isolated form if it is a boolean combination of isolated atoms.

Clearly, every $\mathcal {L}^{\mathsf {H}}$-formula in isolated form is (explicitly) non-cross-world.

Theorem 32 (Non-Cross-World is Isolation)

Every non-cross-world $\mathcal {L}^{\mathsf {H}}$ -formula is equivalent to a $\mathcal {L}^{\mathsf {H}}$ -formula in isolated form.

Theorem 32 is just a more concise statement of Theorem 15.

Proof

It suffices to show the claim for explicitly non-cross-world $\mathcal {L}^{\mathsf {H}}$-formulas φ. We proceed by induction. Clearly this holds for atomics and boolean combinations of non-cross-world formulas. Furthermore, if ∀x ψ is an explicitly non-cross-world formula, then ψ must not contain @, and hence ∀x ψ is already an $\mathcal {L}^{\mathsf {1M}-}$-formula (and so a fortiori in isolated form). So the interesting cases are the modals.

• Necessity.  φ = □ψ. Since φ is a non-cross-world formula, ψ must be too. By inductive hypothesis, suppose ψ is in isolated form. Using standard rewrite rules (and since @ _s and ¬ commute), WLOG, we can suppose ψ is of the form:

  $$\psi = \bigwedge_{i = 1}^{k}\left( {@}_{{t_{1}^{i}}}{\alpha_{1}^{i}} \vee {\cdots} \vee {@}_{t_{n_{i}}^{i}}\alpha_{n_{i}}^{i} \vee \beta^{i}\right) $$

  where each α and β is an $\mathcal {L}^{\mathsf {1M}-}$-formula. Since □ distributes over conjunction, it suffices to check that formulas of the form:

  $$\Box\left( {@}_{t_{1}}\alpha_{1} \vee {\cdots} \vee {@}_{t_{n}}\alpha_{n} \vee \beta\right) $$

  where α ₁,…, α _n , β are $\mathcal {L}^{\mathsf {1M}-}$-formulas, can be written as a boolean combination of isolated atoms. But it’s easy to check that this is equivalent to:

  $${@}_{t_{1}}\alpha_{1} \vee {\cdots} \vee {@}_{t_{n}}\alpha_{n} \vee \Box\beta, $$

  which is a disjunction of isolated atoms. ✓

• Actuality.  φ = @ _s ψ. Again, WLOG, write ψ as:

  $$\psi = \bigwedge_{i = 1}^{k}\left( {@}_{{t_{1}^{i}}}{\alpha_{1}^{i}} \vee {\cdots} \vee {@}_{t_{n_{i}}^{i}}\alpha_{n_{i}}^{i} \vee \beta^{i}\right) $$

  It’s easy to check that @ _s ψ is equivalent to:

  $$\bigwedge_{i = 1}^{k}{@}_{s}\left( {@}_{{t_{1}^{i}}}{\alpha_{1}^{i}} \vee {\cdots} \vee {@}_{t_{n_{i}}^{i}}\alpha_{n_{i}}^{i} \vee \beta^{i}\right) $$

  which is equivalent to:

  $$\bigwedge_{i = 1}^{k}\left( {@}_{{t_{1}^{i}}}{\alpha_{1}^{i}} \vee {\cdots} \vee {@}_{t_{n_{i}}^{i}}\alpha_{n_{i}}^{i} \vee {@}_{s}\beta^{i}\right) $$

  which is in isolated form. ✓

• Saving.  φ = ↓ s.ψ. Again, WLOG, write ψ as:

  $$\psi = \bigwedge_{i = 1}^{k}\left( {@}_{{t_{1}^{i}}}{\alpha_{1}^{i}} \vee {\cdots} \vee {@}_{t_{n_{i}}^{i}}\alpha_{n_{i}}^{i} \vee @_{s}{\gamma_{1}^{i}} \vee {\cdots} \vee @_{s}\gamma_{m_{i}}^{i} \vee \beta^{i}\right) $$

  where also each γ is an $\mathcal {L}^{\mathsf {1M}-}$-formula, and where none of ${t_{1}^{i}},\dots ,t_{n_{i}}^{i}$ are s. Then ↓ s.ψ is equivalent to:

  $$\bigwedge_{i = 1}^{k}\left( {@}_{{t_{1}^{i}}}{\alpha_{1}^{i}} \vee {\cdots} \vee {@}_{t_{n_{i}}^{i}}\alpha_{n_{i}}^{i} \vee {\gamma_{1}^{i}} \vee {\cdots} \vee \gamma_{m_{i}}^{i} \vee \beta^{i}\right) $$

  which is in isolated form.

□

It follows that, up to equivalence, the non-cross-world $\mathcal {L}^{\mathsf {H}}$-formulas are exactly the $\mathcal {L}^{\mathsf {H}}$-formulas in isolated form. Notice that in the proof above, once we’ve rewritten φ into isolated form, no bound state variables are left: only the free state variables of φ remain after being transformed into isolated form. In particular, if φ doesn’t contain any free state variables, then the result of this procedure will be to rewrite φ as an $\mathcal {L}^{\mathsf {1M}-}$-formula. As a result, Theorem 32 shows that all state-closed non-cross-world $\mathcal {L}^{\mathsf {H}}$-formulas are equivalent to $\mathcal {L}^{\mathsf {1M}-}$-formulas. Hence, up to equivalence, the state-closed non-cross-world $\mathcal {L}^{\mathsf {H}}$-formulas are exactly the $\mathcal {L}^{\mathsf {1M}-}$-formulas.