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We consider a natural-language sentence that cannot be formally represented in a first-order language for epistemic two-dimensional semantics. We also prove this claim in the “Appendix” section. It turns out, however, that the most natural ways to repair the expressive inadequacy of the first-order language render moot the original philosophical motivation of formalizing a priori knowability as necessity along the diagonal.

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  1. For more on this, and different characterizations of two-dimensionality, see Humberstone (2004).

  2. Such a framework proved to be very useful also for the purposes of investigating context-dependent terms (Montague 1968; Lewis 1970; Kaplan 1989), the pragmatics of assertion (Stalnaker 1978), as well as conceptual analysis and reductive explanation (Jackson 1998; Chalmers and Jackson 2001).

  3. Perhaps the main point can be generalized for other cases as well. What is important, as we shall make explicit, is a semantics considering necessity and actuality interpreted by the corresponding modal operators, as well as apriority taken as truth on the diagonal.

  4. Where \({\mathcal {M}}\) denotes a Kripke model and \(w*\) is the distinguished element of a set W, this can be done by rewriting \({\mathcal {M}},w\,\vDash\,p\) as \({\mathcal {M}},\langle w*,w\rangle \,\vDash\, p\).

  5. We purposefully avoid single quotation marks in order to not clutter the presentation. The context should make use-mention distinctions clear.

  6. The accessibility relations \({\mathcal {R}}_{\Box }\) and \({\mathcal {R}}_{\mathcal {D}}\) are just the ones in Fritz (2013, p. 1758), except that he also adds an accessibility relation for \({\mathcal {A}}\) formulas, which would be useful for us if we were investigating properties of the logic corresponding to how the frames are defined.

  7. Here we follow Holliday and Perry (2014) in their quantified two-dimensional semantics. Even though their semantics is defined to deal with the Hintikka–Kripke problem in the context of an epistemic logic, their rigidity condition is useful for our purposes. More on this below.

  8. About notation: Davies and Humberstone (1980, p. 4) write \({\mathcal {M}}\,\vDash\, ^v_w\varphi\), appending the superscripts and subscripts to the right side of the turnstile. We simply prefer having the world variables on the left side, which, in its present form, is intended as a two-dimensional version of \({\mathcal {M}},\langle v,w\rangle \,\vDash\, \varphi\).

  9. See, for example, Crossley and Humberstone (1977, pp. 14–15) and Hanson (2006, 2014) for a defense of general validity, and Zalta (1988), Nelson and Zalta (2012) for arguments in favour of local validity.

  10. This is also the case in Kocurek (2016).

  11. Local validity and general validity coincide for basic one-dimensional modal logics without actuality operators since there is no use for a distinguished element in the models. Similarly, there is no use for a distinguished point on the diagonal of a 2D frame since the actuality operator is not fixed to any particular point but to the first coordinate of every pair of possible worlds.

  12. Some terminological remarks are in order. It is usual to say ‘real-world’ instead of ‘local’ validity, in accordance with the terminology used in Crossley and Humberstone (1977). We find it odd, however, to call real-world validity truth at the distinguished pair in every model, otherwise we have no reason to prefer a different nomenclature.

  13. See p. 1611.

  14. We have adapted, of course, the semantic clause of \(\mathcal {F}\) for pairs of worlds.

  15. Davies (2004, p. 89) makes it clear that they did not intend to formalize anything like an epistemic logic, although the resulting system does give rise to a priori truths. Moreover, in response to Evans’ criticisms to the fixedly operator, he also considered the possibility of adding a primitive operator \(\mathcal {D}\) for diagonal necessity (cf. p. 92).

  16. Axiom \(\mathcal {F}\)6, for example, of \({\mathbf{S5}}\mathcal {AF}\), reads \(\Box \varphi \leftrightarrow \mathcal {FA}\varphi\) for \(\mathcal {A}\)-free \(\varphi\). See Davies and Humberstone (1980, p. 4).

  17. For more on this, see Sect. 3. Fritz (2013, p. 1761) makes similar observations.

  18. Both Restall and Fritz constrain their frames in such a way that there are at least as many horizontal as vertical worlds. See Restall (2012, p. 1618), and Fritz (2013, p. 1761). Without loss of generality, we can define 2D frames in a similar manner.

  19. See, for example, Chalmers and Rabern (2014, p. 212), although in the same paper he recognizes difficulties for the semantics of the apriority operator caused by the nesting problem.

  20. Proof: Suppose \({\mathcal {M}}\,\vDash\, x=y\) for an assignment V. By the unrestricted semantics of identity terms, \(V(x)=V(y)\). Let \(\langle w,w\rangle\) be any pair of possible worlds in W such that \(\langle w*,w*\rangle \mathcal {R}_{\mathcal {D}}\langle w,w\rangle\). Then, given \(V(x)=V(y)\), it follows that \({\mathcal {M}}^w_w\,\vDash\, x=y\), whence \({\mathcal {M}}\,\vDash\, \mathcal {D}x=y\), therefore \({\mathcal {M}}\,\vDash\, x=y\supset \mathcal {D}x=y\). An analogous result is available for Davis and Humberstone’s \({\mathbf{S5}}\mathcal {AF}\), the only difference being that it involves the compound operator \(\mathcal {FA}\) rather than \(\mathcal {D}\).

  21. We thereby assimilate proper names to individual constants in \(\mathcal {L}\). This follows closely the presentation in Holliday and Perry (2014, §4.5).

  22. Cf. Chalmers (2004, pp. 158–162; 2014, p. 212).

  23. The same strategy was employed in tense logics by Kamp (1971) for the “now” operator.

  24. In fact, in \({\mathbf{S5}}\mathcal {A}\) there is no need at all of evaluating formulas with respect to a pair of worlds.

  25. This is similar to how the sentence appears in Lampert (manuscript). The only difference being that (3c) was formulated in terms of deep possibility, the dual of deep necessity. Admittedly, not much of a natural-language statement, since the notion of deep necessity seems to be even more philosophically loaded in comparison to apriority. Yet, in a language intended to formalize it, this is exactly the kind of sentence expected to be expressible.

  26. Chalmers (2004, p. 219) motivates conceivability as the dual of a priori knowability. However, in Chalmers (2011) he argues that this is problematic.

  27. Notice that a rendering similar to (4) is available for Davies and Humberstone’s \({\mathbf{S5}}\mathcal {AF}\), the only difference being that \(\mathcal {FA}\) occupies the place of \(\mathcal {D}\).

  28. Hodes (1984, p. 25, Theorem 15) proved that a sentence resembling (1), but restricted to a single predicate letter, is not representable in S5, although similar inexpressibility results were previously conjectured by Hazen (1976). Wehmeier (2001) offers an elegant simplification of Hodes’ argument. More recently, Kocurek (2016) presents a thorough investigation of several inexpressibility results using bisimulations.

  29. See Lampert (manuscript).

  30. This is also acknowledged by Humberstone. See his (1982, fn. 16).

  31. This contrasts with a purely rhetorical use of ‘actually’, as pointed out by Crossley and Humberstone (1977, p. 11).

  32. A similar proposal can be found in Forbes (1989, pp. 93–102). Nonetheless, the first time a proposal like this one appeared in the literature seems to be in Humberstone (1982, p. 2), with the only difference that he used explicit quantification over sets.

  33. This interpretation comes from Boolos (1984).

  34. We should point out that Bricker’s formalization depends on his assumption that \(y\prec xx\) may hold at a world at which the objects assigned to the plural variables do not exist (see p. 387). We run along with Bricker on this, but only to illustrate how plural quantifiers do not solve the problem for the two-dimensional case even on such controversial grounds.

  35. The move from de dicto to de re knowledge ascriptions has been defended by Soames (2005) on the basis of the following exportation principle—understood relative to an assignment function and pair of possible worlds:


    For any name n and predicate F, if ‘a knows/believes that n is F’ is true, then ‘a knows/believes that [x / n] is F’ is true.

    Soames takes the above principle to be “intuitively compelling,” (p. 261) which seems to be the only support offered in favour of (E). Yet, some two-dimensionalists—Chalmers, in particular—deny that (E) has any plausibility whatsoever. See Chalmers (2011, pp. 630–633) for counterexamples to (E) based on knowledge attributions. A more recent defense of de re a priori knowledge can nevertheless be found in Dorr (2011).

  36. Ditto for the corresponding 2D Barcan formula.

  37. For example, see Humberstone (1982, p. 13).

  38. See, for example, Uzquiano (2011, p. 225) and Williamson (2003, pp. 456–457).

  39. It would be the same if we were quantifying over sets rather than plurally.

  40. See Uzquiano (2011, p. 225).

  41. This is also defended by Bricker (1989, p. 387) and Forbes (1985, p. 109).

  42. See also Williamson (2013, p. 248) for similar remarks.

  43. For recent discussion on this, see Uzquiano (2011), Williamson (2003, 2010, 2013, pp. 241–254), and Linnebo (2016).

  44. The proof is similar to the necessity of identity. For suppose that \({\mathcal {M}}\,\vDash\, y\prec xx\) for an assignment V. Where \(\langle w,w\rangle\) is any pair of possible worlds in W such that \(\langle w*,w*\rangle \mathcal {R}_{\mathcal {D}}\langle w,w\rangle\), since \(V(y)\in V(xx)\), it follows that \({\mathcal {M}}^w_w\,\vDash\, y\prec xx\), whence \({\mathcal {M}}\,\vDash\, \mathcal {D}y\prec xx\). The other direction is obvious, and the argument for \((\mathcal {D}\nprec )\) is analogous.

  45. Another possible move is to use full quantification over sets, rather than pluralities. This is the strategy adopted by Meyer (2013) in order to eliminate the actuality operator in quantified modal logic. However, the same issues arise concerning what is validated by the models, regardless of whether we assume sets or pluralities in the semantics.

  46. Again, this inference rule is valid for any sentence unless @ is in the language.

  47. See, for instance, van Benthem (2010) and, more recently, Urquhart (2015).

  48. The models are constructed resembling the ones in Wehmeier (2001), where he proves a similar result for a first-order modal language.


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I am deeply grateful to I-Sen Chen, Alex Kocurek, Hanti Lin, Harrison Smith-Jaoudi, Shawn Standefer, and Lloyd Humberstone for comments on different versions of this paper. Thanks also to Adam Sennet, Greg Ray, Ted Shear, G. J. Mattey, Rachel Boddy, Tyrus Fisher, Rohan French, and Greg Restall for many helpful conversations and suggestions. This paper has been presented at the Philosophy Department of the University of California, Davis; the Davis Logic, Language, Epistemology, and Mathematics Working Group; and the Logic Seminar at the University of Melbourne, Australia. I want to express my gratitude to the audiences for many insightful questions and suggestions. Finally, I am thankful to Andrew Parisi for encouraging me to write this paper, as well as to an anonymous referee for comments and advice.

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Correspondence to Fabio Lampert.

Appendix: Proof of the expressive incompleteness of \(\mathcal {L}\)

Appendix: Proof of the expressive incompleteness of \(\mathcal {L}\)

Since bisimulation implies model equivalence, we show that there is a bisimulation between two constant domain 2D-centered models differing with respect to sentence (3). But first some notation is in order. Let \(\vec {c}\) denote a sequence of objects or terms of \(\mathcal {L}\), where its length is denoted by \(\Vert \vec {c}\Vert\). The nth-member of \(\vec {c}\) is denoted by \(\vec {c}_n\). Furthermore, let \(\varphi |\vec {x}|\) be a formula whose free variables are all in \(\vec {x}\), and whenever \(\Vert \vec {c}\Vert =\Vert \vec {x}\Vert\), let \(\varphi |\vec {c}|\) be the result of appropriately substituting constants in \(\vec {c}\) for variables in \(\vec {x}\). Also, we use \(\langle v,w\rangle\) for members of \(W_1\) and \(\langle y,z\rangle\) for members of \(W_2\), as well as c for members of \({\mathscr {D}}_1\) and d for members of \(\mathscr {D}_2\). We omit any mention of \({\mathcal {M}}\) in the evaluation clauses by writing \(^v_w\,\vDash_1\varphi\) whenever \(\varphi\) holds at \(\langle v,w\rangle\) in \({\mathcal {M}}_1\)—similarly for \({\mathcal {M}}_2\).

Next we define the general notion of a world-object bisimulation for constant domain 2D-centered models:

Definition 5.1

(World-Object Bisimulation) Let \(\mathcal {N}_1=\langle W _1,\langle w*,w*\rangle ,\mathcal {R}_{\Box 1},\mathcal {R}_{\mathcal {D}1},\mathscr {D}_1,V_1\rangle\) and \(\mathcal {N}_2=\langle W _2,\langle v*,v*\rangle ,\mathcal {R}_{\Box 2},\mathcal {R}_{\mathcal {D}2},\mathscr {D}_2,V_2\rangle\) be two constant domain 2D-centered models. A world-object bisimulation between \(\mathcal {N}_1\) and \(\mathcal {N}_2\) is a non-empty relation \(\cong \subseteq (W_1\times \mathscr {D}_1)\times (W_2\times \mathscr {D}_2)\) such that \(\langle w*,w*\rangle \vec {c}\cong \langle w*,w*\rangle \vec {d}\), satisfying the following conditions:

  1. 1.

    \((\langle v,w\rangle \vec {c}\cong \langle y,z\rangle \vec {d})\Rightarrow (^v_w\,\vDash_1 P^n_i\vec {c}\, \Leftrightarrow \, ^y_z\,\vDash_2 P^n_i\vec {d})\), for every predicate symbol \(P^n_i\) (the atomic condition);

  2. 2.

    \((\langle v,w\rangle \vec {c}\cong \langle y,z\rangle \vec {d})\Rightarrow \forall n,m\le \Vert \vec {c}\Vert (^v_w\,\vDash_1 \vec {c}_n=\vec {c}_m\Leftrightarrow \, ^y_z\,\vDash_2\vec {d}_n=\vec {d}_m)\) (the identity condition);

  3. 3.

    \(((\langle v,w\rangle \vec {c}\cong \langle y,z\rangle \vec {d})\wedge \langle y,z\rangle \mathcal {R}_{\Box 2}\langle y,z'\rangle )\Rightarrow (\exists \langle v,w'\rangle \in W_1:\langle v,w\rangle \mathcal {R}_{\Box 1}\langle v,w'\rangle \wedge (\langle v,w'\rangle \vec {c}\cong \langle y,z'\rangle \vec {d}))\) (the \(\mathcal {R}_{\Box }\)-forth condition);

  4. 4.

    \(((\langle v,w\rangle \vec {c}\cong \langle y,z\rangle \vec {d})\wedge \langle v,w\rangle \mathcal {R}_{\Box 1}\langle v,w'\rangle )\Rightarrow (\exists \langle y,z'\rangle \in W_2:\langle y,z\rangle \mathcal {R}_{\Box 2}\langle y,z'\rangle \wedge (\langle v,w'\rangle \vec {c}\cong \langle y,z'\rangle \vec {d}))\) (the \(\mathcal {R}_{\Box }\)-back condition);

  5. 5.

    \((\langle v,w\rangle \vec {c}\cong \langle y,z\rangle \vec {d})\Rightarrow (\langle v,v\rangle \vec {c}\cong \langle y,y\rangle \vec {d})\) (the actuality condition);

  6. 6.

    \(((\langle v,w\rangle \vec {c}\cong \langle y,z\rangle \vec {d})\wedge \langle y,z\rangle \mathcal {R}_{\mathcal {D}2}\langle y',y'\rangle )\Rightarrow (\exists \langle v',v'\rangle \in W_1:\langle v,w\rangle \mathcal {R}_{\mathcal {D}1}\langle v',v'\rangle \wedge (\langle v',v'\rangle \vec {c}\cong \langle y',y'\rangle \vec {d}))\) (the \(\mathcal {R}_{\mathcal {D}}\)-forth condition);

  7. 7.

    \(((\langle v,w\rangle \vec {c}\cong \langle y,z\rangle \vec {d})\wedge \langle v,w\rangle \mathcal {R}_{\mathcal {D}1}\langle v',v'\rangle )\Rightarrow (\exists \langle y',y'\rangle \in W_2:\langle y,z\rangle \mathcal {R}_{\mathcal {D}2}\langle y,y'\rangle \wedge (\langle v',v'\rangle \vec {c}\cong \langle y',y'\rangle \vec {d}))\) (the \(\mathcal {R}_{\mathcal {D}}\)-back condition);

  8. 8.

    \(((\langle v,w\rangle \vec {c}\cong \langle y,z\rangle \vec {d})\wedge d\in \mathscr {D}_2)\Rightarrow (\exists c\in \mathscr {D}_1:\langle v,w\rangle \vec {c},c\cong \langle y,z\rangle \vec {d},d)\) (the quantifier-forth condition);

  9. 9.

    \(((\langle v,w\rangle \vec {c}\cong \langle y,z\rangle \vec {d})\wedge c\in \mathscr {D}_1)\Rightarrow (\exists d\in \mathscr {D}_2:\langle v,w\rangle \vec {c},c\cong \langle y,z\rangle \vec {d},d)\) (the quantifier-back condition);

Conditions 1, 3, and 4 are standard for both atomic and modal formulas, although they appear here under a two-dimensional framework. Similarly, 8 and 9 are usual for constant domain quantifiers.Footnote 47 One can find conditions 2 and 5 for (unrestricted) identity and actuality in Kocurek (2016), which also presents conditions for the fixedly operator. Our conditions 6 and 7 for \(\mathcal {R}_{\mathcal {D}}\) are defined in order to handle the \(\mathcal {D}\) operator.

Lemma 5.1

(Invariance) For any constant domain 2D-centered models \(\mathcal {N}_1\) and \(\mathcal {N}_2\), if \(\cong\) is a world-object bisimulation between \(\mathcal {N}_1\) and \(\mathcal {N}_2\), \(\langle v,w\rangle \in W_1\), \(\langle y,z\rangle \in W_2\), and \(\varphi |\vec {x}|\) is a formula,

$$\begin{aligned} (\langle v,w\rangle \vec {c}\cong \langle y,z\rangle \vec {d})\Rightarrow ( ^v_w\,\vDash_1\varphi |\vec {c}|\Leftrightarrow \, ^y_z\,\vDash_2\varphi |\vec {d}|). \end{aligned}$$


By induction on \(\varphi |\vec {x}|\). The atomic cases including identity hold by construction given conditions 1 and 2, while the truth-functional cases are straightforward.

Let \(\varphi |\vec {x}|\) be \(\Box \varphi |\vec {x}|\). Suppose that \(\langle v,w\rangle \vec {c}\cong \langle y,z\rangle \vec {d}\) and \(^v_w\,\vDash_1\Box \varphi |\vec {c}|\). If \(\langle y,z\rangle \mathcal {R}_{\Box 2}\langle y,z'\rangle\), then \(\exists \langle v,w'\rangle \in W_1\) such that \(\langle v,w\rangle \mathcal {R}_{\Box 1}\langle v,w'\rangle\) and \(\langle v,w'\rangle \vec {c}\cong \langle y,z'\rangle \vec {d}\), by condition 3. Thus, \(^v_{w'}\,\vDash_1\varphi |\vec {c}|\). By the induction hypothesis, \(^y_{z'}\,\vDash_2\varphi |\vec {d}|\). Therefore, \(^y_z\,\vDash_2\Box \varphi |\vec {d}|\). The other direction is analogous given condition 6.

Let \(\varphi |\vec {x}|\) be \(\mathcal {A}\varphi |\vec {x}|\). Suppose that \(\langle v,w\rangle \vec {c}\cong \langle y,z\rangle \vec {d}\) and \(^v_w\,\vDash _1\mathcal {A}\varphi |\vec {c}|\). Thus, \(^v_v\,\vDash_1\varphi |\vec {c}|\), by the semantics of \(\mathcal {A}\). By condition 5, \(\langle v,v\rangle \vec {c}\cong \langle y,y\rangle \vec {d}\), whence \(^y_y\,\vDash_2\varphi |\vec {d}|\), by the induction hypothesis. Therefore, \(^y_z\,\vDash_2\mathcal {A}\varphi |\vec {d}|\), by the semantics of \(\mathcal {A}\). The other direction is analogous.

Let \(\varphi |\vec {x}|\) be \(\mathcal {D}\varphi |\vec {x}|\). Suppose that \(\langle v,w\rangle \vec {c}\cong \langle y,z\rangle \vec {d}\) and \(^v_w\,\vDash_1\mathcal {D}\varphi |\vec {c}|\). If \(\langle y,z\rangle \mathcal {R}_{\mathcal {D}2}\langle y',y'\rangle\), then \(\exists \langle v',v'\rangle \in W_1\) such that \(\langle v,w\rangle \mathcal {R}_{\mathcal {D}1}\langle v',v'\rangle\) and \(\langle v',v'\rangle \vec {c}\cong \langle y',y'\rangle \vec {d}\), by condition 6. Thus, \(^{v'}_{v'}\,\vDash_1\varphi |\vec {c}|\). By the induction hypothesis, \(^{y'}_{y'}\,\vDash_2\varphi |\vec {d}|\). Therefore, \(^y_z\,\vDash_2\mathcal {D}\varphi |\vec {d}|\). The other direction is analogous given condition 7.

Let \(\varphi |\vec {x}|\) be \(\exists y\varphi |\vec {x},y|\). Suppose that \(\langle v,w\rangle \vec {c}\cong \langle y,z\rangle \vec {d}\) and \(^v_w\,\vDash_1\exists y\varphi |\vec {c},y|\). Thus, \(\exists c\in \mathscr {D}_1\) such that \(^v_w\,\vDash_1\varphi |\vec {c},c|\). By condition 9, \(\exists d\in \mathscr {D}_2\) such that \(\langle v,w\rangle \vec {c},c\cong \langle y,z\rangle \vec {d},d\). By the induction hypothesis, \(^y_z\,\vDash_2\varphi |\vec {d},d|\), whence \(^y_z\,\vDash_2\exists y\varphi |\vec {d},y|\). The other direction is analogous given condition 8. \(\square\)

We define our two models as follows.Footnote 48 Let \({\mathcal {M}}_1=\langle W _1,\langle w*,w*\rangle ,\mathcal {R}_{\Box 1},\mathcal {R}_{\mathcal {D}1},\mathscr {D}_1,V_1\rangle\) be such that \(W_1\) is the set of all pairs of subsets of \(\mathbb {N}\) that are both infinite and coinfinite containing at least one odd number, \(\langle w*,w*\rangle\) is the pair \(\langle 2\mathbb {N}+1,2\mathbb {N}+1\rangle\) of sets of odd numbers, \(\mathscr {D}_1\) is \(\mathbb {N}\), for each \(\langle i,j\rangle \in W_1\) let \(V_1(R,\langle i,j\rangle )=V_1(S,\langle i,j\rangle )=i\cup j\), and let the accessibility relations be defined just as in Definition 1.2. The extension of every other predicate symbol is empty. On the other hand, our second model, \({\mathcal {M}}_2=\langle W _2,\langle w*,w*\rangle ,\mathcal {R}_{\Box 2},\mathcal {R}_{\mathcal {D}2},\mathscr {D}_2,V_2\rangle\), is just like \({\mathcal {M}}_1\) except for \(W_2=W_1\cup \{\langle 2\mathbb {N},2\mathbb {N}\rangle \}\), where \(2\mathbb {N}\) is the set of even numbers, and \(V_2(R,\langle 2\mathbb {N},2\mathbb {N}\rangle )=V_2(S,\langle 2\mathbb {N},2\mathbb {N}\rangle )=2\mathbb {N}\cup 2\mathbb {N}\). No other predicate has an extension defined on \(\langle 2\mathbb {N},2\mathbb {N}\rangle\).

In order to prove that \({\mathcal {M}}_1\) and \(\mathcal {M}_2\) are bisimilar, let \(\langle v,w\rangle \in W_1\) and \(\langle v',w'\rangle \in W_2\) be corresponding pairs of worlds in the models \({\mathcal {M}}_1\) and \(\mathcal {M}_2\). For any \(\langle v,w\rangle \in W_1\) and \(\langle v',w'\rangle \in W_2-\{\langle 2\mathbb {N},2\mathbb {N}\rangle \}\), say that the mapping \(\rho\) from \(W_1\) onto \(W_2-\{\langle 2\mathbb {N},2\mathbb {N}\rangle \}\) is an isomorphism between \({\mathcal {M}}_1\) and the submodel of \({\mathcal {M}}_2\) defined on \(W_2-\{\langle 2\mathbb {N},2\mathbb {N}\rangle \}\) such that \(\rho (c)=d\), in which case for \(\langle v,w\rangle \in W_1\) and \(\langle v',w'\rangle \in W_2\) other than \(\langle 2\mathbb {N},2\mathbb {N}\rangle\), and for any non-empty predicate symbol \(P^n_i\), we have

$$\begin{aligned} (c\in V_1(P^n_i,\langle v,w\rangle ))\, \Leftrightarrow \, (\rho (c)\in V_2(P^n_i,\langle v',w'\rangle )). \end{aligned}$$

Moreover, we define the following relations between \({\mathcal {M}}_1\) and \(\mathcal {M}_2\). For any \(\langle v,w\rangle \in W_1\) and \(\langle v',w'\rangle \in W_2\) other than \(\langle 2\mathbb {N},2\mathbb {N}\rangle\), set

$$\begin{aligned} (\langle v,w\rangle \vec {c}\cong \langle v',w'\rangle \vec {d})\Leftrightarrow (\rho (\vec {c})=\vec {d}). \end{aligned}$$

For the actuality condition, let

$$\begin{aligned} (\langle v,w\rangle \vec {c}\cong \langle v',w'\rangle \vec {d})\Leftrightarrow (\langle v,v\rangle \vec {c}\cong \langle v',v'\rangle \rho (\vec {c})). \end{aligned}$$

In the case of the extra pair of worlds \(\langle 2\mathbb {N},2\mathbb {N}\rangle \in W_2\), we define

$$\begin{aligned} (\langle v,v\rangle \vec {c}\cong \langle 2\mathbb {N},2\mathbb {N}\rangle \vec {d})\Leftrightarrow \forall n\le \Vert \vec {c}\Vert (\vec {c}_n\in (V_1(S,\langle v,v\rangle ))\, \Leftrightarrow \, (\vec {d}_n\in V_2(S,\langle 2\mathbb {N},2\mathbb {N}\rangle )), \end{aligned}$$


$$\begin{aligned} (\langle v,v\rangle \vec {c}\cong \langle 2\mathbb {N},2\mathbb {N}\rangle \vec {d})\Leftrightarrow \forall n\le \Vert \vec {c}\Vert (\vec {c}_n\in (V_1(R,\langle v,v\rangle ))\, \Leftrightarrow \, (\vec {d}_n\in V_2(R,\langle 2\mathbb {N},2\mathbb {N}\rangle )). \end{aligned}$$

Finally, set

$$\begin{aligned} (\langle v,w\rangle \vec {c}\cong \langle 2\mathbb {N},2\mathbb {N}\rangle \vec {d})\Leftrightarrow (\langle v,v\rangle \vec {c}\cong \langle 2\mathbb {N},2\mathbb {N}\rangle \vec {d}). \end{aligned}$$

Lemma 5.2

\(\cong\) is a bisimulation between \({\mathcal {M}}_1\) and \({\mathcal {M}}_2\).


Conditions 1, 2, and 5 hold by construction. Since \(\langle 2\mathbb {N},2\mathbb {N}\rangle\) is not \(\mathcal {R}_{\Box }\)-accessible with respect to any pair of worlds, conditions 3 and 4 are easily seen to be met as well.

For conditions 6 and 7, the only cases we need to consider involve \(\langle 2\mathbb {N},2\mathbb {N}\rangle\). Suppose \(\langle v,w\rangle \vec {c}\cong \langle v',w'\rangle \vec {d}\) and that \(\langle v',w'\rangle \mathcal {R}_{\mathcal {D}2}\langle 2\mathbb {N},2\mathbb {N}\rangle\). Since \(\vec {c}\) contains a single element, we have only two cases. If \(\vec {c}\) is even, choose a pair \(\langle v,w\rangle \mathcal {R}_{\mathcal {D}1}\langle y,y\rangle\) such that \(\vec {c}\in V_1(S,\langle y,y\rangle )\). By the definition of \({\mathcal {M}}_1\), we know that there will be such a pair, in which case we have \(\langle y,y\rangle \vec {c}\cong \langle 2\mathbb {N},2\mathbb {N}\rangle \vec {d}\), by construction. If \(\vec {c}\) is odd, choose a pair \(\langle v,w\rangle \mathcal {R}_{\mathcal {D}1}\langle z,z\rangle\) such that \(\vec {c}\notin V_1(S,\langle z,z\rangle )\), whence \(\vec {d}\notin V_2(S,\langle 2\mathbb {N},2\mathbb {N}\rangle )\). Therefore, \(\langle z,z\rangle \vec {c}\cong \langle 2\mathbb {N},2\mathbb {N}\rangle \vec {d}\). Condition 7 is analogous.

With respect to conditions 8 and 9, again, we only need to check the cases involving the extra pair \(\langle 2\mathbb {N},2\mathbb {N}\rangle\). Suppose that \(\langle v,w\rangle \vec {c}\cong \langle 2\mathbb {N},2\mathbb {N}\rangle \vec {d}\) and that \(c\in \mathscr {D}_1\). We only have cases involving the predicates S and R, since all the other predicates are empty. If \(c\in V_1(S,\langle v,v \rangle )\), then \(\langle v,w\rangle \vec {c},c\cong \langle 2\mathbb {N},2\mathbb {N}\rangle \vec {d},d\), by construction, since \(V_2(S,\langle 2\mathbb {N},2\mathbb {N}\rangle )\) is not empty. If \(c\notin V_1(S,\langle v,v \rangle )\), then choose any \(d\in V_2(S,\langle w*,w*\rangle )\), in which case \(d\notin V_2(S,\langle 2\mathbb {N},2\mathbb {N}\rangle )\), and then we have \(\langle v,w\rangle \vec {c},c\cong \langle 2\mathbb {N},2\mathbb {N}\rangle \vec {d},d\). The argument for R is very similar. Condition 8 is analogous. \(\square\)

Theorem 5.1

There is no sentence \(\varphi\) of \(\mathcal {L}\) such that for every constant domain 2D-centered model \({\mathcal {M}}=\langle W ,\langle w*,w*\rangle ,\mathcal {R}_{\Box },\mathcal {R}_{\mathcal {D}},\mathscr {D},V\rangle\), \({\mathcal {M}}\,\vDash\, \varphi\) if and only if there is a pair of possible worlds \(\langle w,w\rangle \in W\), such that for two predicate symbols R and S, \(V(R,\langle w*,w*\rangle )\cap V(S,\langle w,w\rangle )=\varnothing\).


By construction of the models, in \({\mathcal {M}}_2\) there is a pair \(\langle 2\mathbb {N},2\mathbb {N}\rangle \in W_2\) such that \(V_2(R,\langle w*,w*\rangle )\cap V_2(S,\langle 2\mathbb {N},2{\mathbb {N}}\rangle )=\varnothing\), but in \({\mathcal {M}}_1\) there is no pair of worlds \(\langle w,w\rangle \in W_1\) such that \(V(R,\langle w*,w*\rangle )\cap V(S,\langle w,w\rangle )=\varnothing\), since \(V_1(R,\langle w*,w*\rangle )\) contains only odd numbers, and for any \(\langle w,w\rangle \in W_1\), \(V_1(S,\langle w,w\rangle )\) contains at least one odd number for each coordinate of the pair \(\langle w,w\rangle\). \(\square\)

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Lampert, F. Actuality and the a priori. Philos Stud 175, 809–830 (2018).

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