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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.

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Notes

  1. Originally from von Stechow [38, p. 35].

  2. I borrow the terminology from Wehmeier [41].

  3. From Button [8, p. 245].

  4. I’m grateful to Mike Deigan for drawing my attention to these kinds of sentences.

  5. For instance, Pollock [32, pp. 6–7], though Pollock ultimately endorses a different view of counteridenticals in the book (pp. 101–103).

  6. From Cresswell [10, p. 34]. The sentence is often stated as: “Everyone who’s actually rich could have been poor.” I find the rendering above less ambiguous.

  7. There are several proofs that first-order modal logic without @ can’t express (Rich), regardless of whether we use possibilist or actualist quantifiers [19, 40]. A proof that adding @ doesn’t help (at least for the language with just actualist quantifiers) can be found in Hodes [18]; a more modern proof using bisimulations can be found in Kocurek [22].

  8. The proof of this can also be found in Kocurek [22].

  9. One could add λ-abstraction, in the spirit of Fitting and Mendelsohn [14, Chps. 9-10], but doing so doesn’t increase the expressive power enough to solve the problems of cross-world predication and quantification. All λ-abstraction does is essentially allow for “rigidification” of non-rigid terms, which doesn’t help much. However, combining λ-abstraction with more powerful logics can help. For instance, see Fitting [13] for an approach that combines λ-abstraction with the approach in Section 5 for sentences like (Tall) without quantifiers. Alternatively, with higher-order λ-abstraction, one can greatly increase the expressive power of the language by encoding worlds into the object language directly [29, 43, 44].

  10. We’ll use “ ;” to separate object terms and state variables. Also, we’ll use “ P n” in place of “ P n/1”.

  11. We won’t really need formulas of the form t 1 = t 2 in this section, but we include them for the sake of generality. Such \(\mathcal {L}^{\mathsf {2S}}\)-formulas will be discussed in Section 8.

  12. The formalizations assume possibilist object quantifiers, but the actualist readings could be obtained by relativizing the object quantifiers to E appropriately. We also assume (just for simplicity) that we always start truth evaluation at diagonal points of evaluation, whereby the world considered as actual is the world of evaluation. That way, we don’t need to treat separate readings of the sentences, where the free state variables either pick out the starting world of evaluation or the world considered as actual.

  13. See e.g., Blackburn et al. [4].

  14. Normally, we would want (or need) to add @ in front of all our formalizations to ensure that the shifted worlds are accessible from the actual world, so that the formalization can more accurately capture the meaning of the sentence in context. But as noted in footnote 12, since we’re assuming our starting point of evaluation is a diagonal one, we may drop @ without loss.

  15. The most well-known defendents of this approach are von Stechow [38] and Cresswell [10, Chp. 5]. Fitting [13] also defends a version of this approach in a language with λ-abstraction instead of quantifiers.

  16. For instance, Priest [33, p. 123] and Fitting [13, p. 4] argue this.

  17. See Partee [31], Cresswell [10], Stone [39], Kratzer [23], King [21], Schlenker [36], and Schaffer [35] for a discussion.

  18. Several authors have gone so far as to use the term “cross-world comparatives” as opposed to “cross-world predication”, e.g., Lewis [25, p. 436] and Cantwell [9]. This is misleading, though, since many cross-world predications don’t involve comparatives—e.g., “I could have sat between where Russ and Matt are actually sitting” has no comparative in it. But such examples still seem to involve some kind of comparison (in this case, comparing locations), and are still formalizable on the degree approach.

  19. Sider [37, p. 26] has argued that sentences like “Some American philosopher admires some ancient Greek philosopher” express cross-time predications of a similar sort that can’t be expressed in first-order temporal logic. The force of this example, however, depends on how we set up first-order temporal logic. If we allow “tenseless” quantifiers ( π and Σ), and if we allow extensions of predicates to contain non-existents, then the sentence can easily be formalized as ΣxΣy(A m P h i l(x)∧P G r e e k P h i l(x)∧A d m i r e(x, y)). (In defense of Sider, he raised this point in a discussion on presentism, which probably would not allow for either of these in their preferred temporal logic.) By contrast, (Cherish) is problematic even allowing for tenseless quantifiers and non-existent objects in extensions of predicates.

  20. Based on a sentence from Button [8, p. 246].

  21. A similar argument is made by Lewis [26, p. 13].

  22. E.g., Milne [30], Cantwell [9], and Forbes [16].

  23. The following formalization is, for the most part, based on Forbes [16] and Wehmeier [41]. Milne [30] and Cantwell [9] both incorporate degrees into their solutions by having a unary degree function symbol deg in the language. Then cross-world predications involve terms of the form dthat deg(τ). For instance, (Tall) just becomes (deg(me)> Taller dthat deg(me)). The function approach as presented here, by contrast, does not use degrees, as they’re not needed to solve the problem of cross-world predication; nor are they helpful in avoiding the objections that follow.

  24. This is a slightly more general definition than that of Wehmeier [41, p. 109].

  25. For a justification of this choice, see footnote 41 in Section 8.

  26. One issue is whether the final “could be” in “A polar bear could be bigger than a grizzly bear could be” should be interpreted relative to the actual world or the shifted world (in other words, whether we should instead replace □ in (28) with @□). If that “could be” should be interpreted relative to the actual world, then (Polar) is expressible on the function approach (many thanks to an anonymous reviewer for pointing this out). For our purposes, we only need the existence of the interpretation where “could be” is interpreted relative to the shifted world. This interpretation can be forced on a temporal reading, as in: “One day, there will be a polar bear that’s bigger than any grizzly bear will ever be from then on.”

  27. This appears, e.g., in Lewis [25, p. 437], although Lewis uses “ ‡” instead of “ ”.

  28. Here’s a similar example from Kratzer [24] using different modalities: “Whenever it snowed, some local person dreamed that it snowed more than it actually did, and that the local weather channel erroneously reported that it had snowed less, but still more than it snowed in reality.”

  29. From Boolos [5, p. 432].

  30. Unlike in ordinary hybrid logic, I haven’t allowed for state variables to count as well-formed formulas. This is because such additional expressive power isn’t necessary here. See footnote 41 in Section 8.

  31. While these abbreviations may make the hybrid solution look much like the two-sorted language, there are subtle but important differences between the two approaches. This will be discussed in Section 8.

  32. After completing this paper, it came to my attention that Yanovich [42] has explicitly used quantified hybrid logic, in its familiar form, for addressing the problem of cross-world quantification. The problem of cross-world predication is not addressed in his paper however.

  33. Such limitations are discussed in Hazen [17, p. 39].

  34. Where F is the “it will be the case that” operator, and P is the “it was the case that” operator.

  35. From Butterfield and Stirling [7].

  36. 36 Such possibilities could be situations as in Kratzer [24], though they need not be cashed out in that particular way.

  37. From Cumming [11, p. 529].

  38. Here, the use of \(\blacktriangleleft _{s}\) as opposed to \(\blacktriangleleft \) might be necessary, since there are powerful arguments that doxastic and epsitemic modal operators shift the actual world, e.g., Rabinowicz and Segerberg [34].

  39. This approach is similar to the ones taken by, e.g., Aloni [1] and Holliday and Perry [20]. These authors could be seen as the kind of approach sketched above, except with explicit quantification over “roles”, akin to the degree approach.

  40. It also bans other kinds of atomics if constants are non-rigid designators, and if there are no possibilist quantifiers. For instance, \(\mathcal {L}^{\mathsf {H}}\) can’t be built from atomics of the form P(c(s);t, t ) where s and t are distinct. It is possible to state precisely which atomic formulas are allowed, but it won’t be necessary to go into the details here.

  41. The difference is even smaller if we either allow state variables as formulas, as is ordinarily done in hybrid logic, or if we define an identity relation ≡ such that \(\mathcal {M},w,g \Vdash \tau \equiv \sigma \) iff \([{\kern -2.3pt}[\tau ]{\kern -2.3pt}]^{\mathcal {M},w,g} = [{\kern -2.3pt}[\sigma ]{\kern -2.3pt}]^{\mathcal {M},w,g}\). If either of these are added to the syntax, then we can express R(s, t) either as @ s t or \(@_{s}\Diamond (c \equiv \blacktriangleleft _{t} c)\). We can also express s = t either as @ s t or \(@_{s}(c \equiv \blacktriangleleft _{t} c)\). This is one reason why we use = and why we don’t allow state variables to also be formulas, apart from the fact that we don’t need the extra expressive power of the expanded language to express the sentences we’re interested in.

  42. Whether it’s transitive depends on how one understands fictional discourse. If R L o t R is understood linking worlds compatible with The Lord of the Rings as written in this world, then it’s probably transitive. If R L o t R is understood as linking worlds compatible with The Lord of the Rings as written in the world of evaluation, then it’s not, as some fictional worlds compatible with The Lord of the Rings won’t even contain such a fiction.

    The latter theory seems more plausible, since it seems possible that Bilbo never existed in The Lord of the Rings (if Tolkien had decided to write the story differently, for example). But if that’s so, then two different worlds of evaluation might disagree on which worlds are compatible with The Lord of the Rings. Hence, R L o t R would have to be interpreted as compatability with The Lord of the Rings as it’s written in the world of evaluation, not this world. But I won’t settle the matter here due to space constraints.

  43. However, if we add any global modality, then we again lose R-boundedness. Thanks to Balder ten Cate for pointing this out.

  44. Of course, this needs to be qualified. Here’s a potential counterexample: “The authors of Principia could have written more clearly than they actually did”. This sentence cannot be expressed in \(\mathcal {L}^{\mathsf {H}}\), yet clearly is cross-world. But the reason this sentence is not expressible is not because of the cross-world part of the sentence, but because it involves plurals; and this is a problem even for the two-sorted language, our most expressive language thus far. So really, what should be said is this: to refute the thesis, one would need to find a sentence that can be formalized into the two-sorted language that isn’t expressible in \(\mathcal {L}^{\mathsf {H}}\) but still appears to be a cross-world sentence.

  45. A version of this theorem was proven in Yanovich [42] (p. 84, Prop. 3) for just cross-world quantificational sentences without free state variables. The result came to my attention only after completing this paper.

  46. \(\mathcal {L}^{\mathsf {H}}\) does add some new state-open non-cross-world formulas (that’s why the theorem doesn’t just say that the non-cross-world fragment of \(\mathcal {L}^{\mathsf {H}}\) is \(\mathcal {L}^{\mathsf {1M}-}\) full stop). But as far as minimality is concerned, this shouldn’t be troubling for two reasons. First, over diagonal indices—which, in the hybrid setting, means indices of the form \(\left \langle \mathcal {M},w,g\right \rangle \) where g(s) = w for all sS V A R—the restriction to state-closed \(\mathcal {L}^{\mathsf {H}}\)-formulas can be dropped. That is, every non-cross-world formulas is equivalent over diagonal indices to an \(\mathcal {L}^{\mathsf {1M}-}\)-formula. This is just because if φ is an \(\mathcal {L}^{\mathsf {1M}-}\)-formula, then \({@}_{t_{i}}\varphi \) is equivalent over diagonal indices to φ. Second, even over non-diagonal indices, \(\mathcal {L}^{\mathsf {H}}\) doesn’t introduce a wholly new kind of non-cross-world formulas as previous approaches did. After all, just consider the case where our non-cross-world formula φ only contains our special state variable s 0 that by convention picks out the world considered as actual. Then φ will be equivalent to a boolean combination of non-cross-world formulas in \(\mathcal {L}^{\mathsf {1M}}\) (with @). In general, the new kind of non-cross-world formulas that \(\mathcal {L}^{\mathsf {H}}\) introduces simply assert that old kinds of non-cross-world formulas hold elsewhere. By contrast, the new non-cross-world formulas in the degree approach or the two-sorted language will be of a wholly different sort not found in anything like \(\mathcal {L}^{\mathsf {1M}}\).

  47. This is not the most general definition of expressivity, but for our purposes, we only need this particular instance of the more general definition.

  48. I again allow myself the flexibility of re-writing Wehmeier’s notation, for the sake of continuity.

  49. Wehmeier has indicated in personal communication that something like the proposal given here is the proposal he would adopt were he to drop the various semantic constraints made in Wehmeier [41].

  50. In the case of \({\Box ^{s}_{k}}\), we do not count the s here as being within its own scope, so such an instance of s would also be replaced by t if its not in the scope of other modals. So for example, \(\mathsf {H2W}\left ({@}_{s_{k}}\Box _{s_{n}}\varphi \right ) = \mathsf {H2W}\left (\Box _{s_{n}}\varphi \right )[s/s_{k}] = (\Box _{s_{n}}^{s}\mathsf {H2W}\left (\varphi \right ))[s/s_{k}] = \Box _{n}^{s_{k}}\mathsf {H2W}\left (\varphi \right )[s/s_{k}]\).

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Acknowledgments

Many thanks to Johan van Benthem, Russell Buehler, Balder ten Cate, Sophie Dandelet, Melissa Fusco, Wesley Holliday, Grace Paterson, Justin Vlasits, Kai Wehmeier, and Seth Yalcin for all their helpful comments and suggestions. A version of this paper was presented at the Berkeley-Stanford Circle in Logic and Philosophy in October 2014. A version of this paper was also presented at a C-ALPHA-sponsored talk at UC Irvine in March 2015 and at the Logic Seminar at Stanford in May 2015. I am very grateful for all the helpful comments and discussion from these talks.

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Appendices

Appendices

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 \(\bar {\alpha }\).

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 cC O N, V(c):WD;

  • for each P n/mP 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 cC O N, V(c)(w) = I(c, w);

  • for all P nP 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×Waδ(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.

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 xV A R.

Definition 20 (Two-Sorted Denotation)

Let τT 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 \(g^{\mathsf {c}} \in \mathsf {VA}(\mathfrak {M})\):

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

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 \(\mathcal {L}^{\mathsf {2S}}\):Footnote 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 \(\mathcal {L}^{\mathsf {H}}\).Footnote 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, vW) and that the extension of cross-world predicates are rigid in a similar sense (so I(C, w) = I(C, v) for all w, vW). 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 \(\mathcal {L}^{\mathsf {H}}\).Footnote 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:

  1. (i)

    for all cC O N and all w, vW, I(c, w) = I(c, v);

  2. (ii)

    for all ordinary predicates P, all w, v 1,…, v n W, and all a 1,…, a n D:

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

    for all cross-world predicates C and all w, vW, 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 ts 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:

  1. (i)

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

  2. (ii)

    s 0 is never bound in φ (and thus, s 0 and s n are distinct)

  3. (iii)

    there is no occurrence of s n . in ψ

  4. (iv)

    for all sS V A R where s is distinct from s n , there is at most one occurrence of s. in ψ

  5. (v)

    for all sS V A R, if s has a free occurrence in ψ, then it does not also have a bound occurrence in ψ

  6. (vi)

    for all sS V A R, if s. occurs in ψ, then its single occurrence is prefixed by a □

  7. (vii)

    every occurrence of □ prefixes an occurrence of s. for some sS V A R

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

  1. (a)

    the state variables are nicely organized,

  2. (b)

    irrelevant stackings of \(\blacktriangleleft \) are removed,

  3. (c)

    every □ is followed by exactly one unique t., and

  4. (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.Footnote 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 ↓sn.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.

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Kocurek, A.W. The Problem of Cross-world Predication. J Philos Logic 45, 697–742 (2016). https://doi.org/10.1007/s10992-015-9389-z

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Keywords

  • Cross-world predication
  • Cross-world quantification
  • Expressive power
  • First-order modal logic
  • Hybrid logic