Actuality and the a priori

Abstract

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.

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

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

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

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

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

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

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

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

• 9. $((\langle v,w\rangle \vec {c}\cong \langle y,z\rangle \vec {d})\wedge c\in \mathscr {D}_1)\Rightarrow (\exists d\in \mathscr {D}_2:\langle v,w\rangle \vec {c},c\cong \langle y,z\rangle \vec {d},d)$ (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.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,

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

Proof

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.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

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

and

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

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 ${\mathcal {M}}_2$.

Proof

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, $V(R,\langle w*,w*\rangle )\cap V(S,\langle w,w\rangle )=\varnothing$.

Proof

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 $\langle w,w\rangle$. $\square$