The Logic of Action and Control

Abstract

In this paper I propose and motivate a logic of the interdefined concepts of making true and control, understood as intensional propositional operators to be indexed to an agent. While bearing a resemblance to earlier logics in the tradition, the motivations, semantics, and object language theory differ on crucial points. Applying this logic to widespread formal theories of agency, I use it as a framework to argue against the ubiquitous assumption that the strongest actions or options available to a given agent must always be pairwise incompatible. The conclusion is that this assumption conflicts with failures of higher order control of agents over their degree or precision of control, failures exhibited by such imperfect agents as ourselves. I discuss models in this setting for understanding such imperfectly self-controlling agents. In an appendix, I prove several relevant results about the logic described, including soundness and completeness both for it and for certain natural extensions.

Appendix: A

We here prove relevant results for the logics discussed above. Proofs are largely adapted from analogous ones in [46] and [24].

1.1 A.1 The Logic C o n v

Our language ${\mathscr{L}}$ (or equivalently, its set of wff’s) is as usual that generated by a countable set At of sentential variables p,q,r,… and

$\mathsf {At} | \neg \varphi | \varphi \vee \psi | \varphi \wedge \psi | \varphi \rightarrow \psi | \Box \varphi$

Our logic itself, $\mathsf {Conv} \subset {\mathscr{L}}$, i.e. the φ such that ⊩_Convφ, is the set of wff’s generated by all instances of the following axiom schemas

All classical tautologies

T:

$\Box \varphi \rightarrow \varphi$

M:

$(\Box \varphi \wedge \Box \psi ) \rightarrow \Box (\varphi \wedge \psi )$

O:

$(\Box \varphi \wedge \Box \psi ) \rightarrow \Box (\varphi \vee \psi )$

Conv:

$\Box (\varphi \vee \psi \vee \chi ) \rightarrow \Box \varphi \rightarrow \Box (\varphi \vee \psi )$

with the following inference rules

M P :

if ⊩_Convφ and $\vdash _{\mathsf {Conv}} \varphi \rightarrow \psi$, ⊩_Convψ

R E :

if $\vdash _{\mathsf {Conv}} \psi \leftrightarrow \phi$, $\vdash _{\mathsf {Conv}} \Box \psi \leftrightarrow \Box \phi$

A model ${\mathscr{M}}$ of our logic is a triple 〈W,V, (S_min,S_max)〉, with W some nonempty set, $V : \mathsf {At} \rightarrow \mathcal {P}(W)$, and the possibly partial $(S_{\mathsf {min}}, S_{\mathsf {max}}) : W \hookrightarrow \mathcal {P}(W) \times \mathcal {P}(W)$ (or just S) such that (where defined) $\{ w \} \subseteq S_{\mathsf {min}}(w) \subseteq S_{\mathsf {max}}(w)$. The clauses of the interpretation function ⟦⟧ for the atoms and Boolean clauses are as usual, while $\llbracket \Box \varphi \rrbracket = \{ w \in W : S_{\mathsf {min}}(w) \subseteq \llbracket \varphi \rrbracket \subseteq S_{\mathsf {max}}(w)\}$. (Note that $w \notin \llbracket \Box \varphi \rrbracket$ when S is undefined on w, and that as components of a single function from W into the cartesian square of $\mathcal {P}(W)$ S_min and S_max are each defined iff the other is.) φ is valid in a model ${\mathscr{M}}$, i.e. $\vDash _{{\mathscr{M}}}\varphi$, iff $\llbracket \varphi \rrbracket _{{\mathscr{M}}} = W_{{\mathscr{M}}}$, and is valid simpliciter, i.e. $\vDash _{\mathsf {Conv}}\varphi$, iff for all models ${\mathscr{M}}$, $\vDash _{{\mathscr{M}}}\varphi$.

We now prove a lemma that will be useful in proving completeness.

Lemma A.1

If $\vdash _{\mathsf {Conv}} (\varphi \wedge \psi ) \rightarrow \chi$ and $\vdash _{\mathsf {Conv}}(\neg \varphi \wedge \neg \psi ) \rightarrow \neg \chi$, $\vdash _{\mathsf {Conv}} (\Box \varphi \wedge \Box \psi ) \rightarrow \Box \chi$.

Proof

Suppose the antecedent. Then by classical reasoning, we have $\vdash _{\mathsf {Conv}} (\varphi \wedge \psi ) \rightarrow \chi$ and $\vdash _{\mathsf {Conv}} \chi \rightarrow (\varphi \vee \psi )$. Now assume $\Box \varphi \wedge \Box \psi$. By M this gives us $\Box (\varphi \wedge \psi )$, and by O $\Box (\varphi \vee \psi )$. But then by Conv and RE we have $\Box \chi$, and discharging we have $\vdash _{\mathsf {Conv}} (\Box \varphi \wedge \Box \psi ) \rightarrow \Box \chi$. □

Corollary A.2

For any finite {φ₁,…,φ_n,ψ}, if $\vdash _{\mathsf {Conv}} (\varphi _{1} \wedge {\ldots } \wedge \varphi _{n}) \rightarrow \psi$ and $\vdash _{\mathsf {Conv}} (\neg \varphi _{1} \wedge {\ldots } \wedge \neg \varphi _{n}) \rightarrow \neg \psi$, $\vdash _{\mathsf {Conv}} (\Box \varphi _{1} \wedge {\ldots } \wedge \Box \varphi _{n}) \rightarrow \Box \psi$.

1.2 A.2 Soundness

It is easy to see that our semantics is sound.

Theorem A.3

If ⊩_Convφ, $\vDash _{\mathsf {Conv}}\varphi$

Proof

We omit as routine the proofs of the validity of classical tautologies and MP. The validity of T is straightforward from the stipulation that (when defined) $\{ w \} \subseteq S_{\mathsf {min}}(w)$. The validity of M is straightforward from the fact that for $X, Y \supseteq S_{\mathsf {min}}(w)$, $X \cap Y \supseteq S_{\mathsf {min}}(w)$. The validity of O is straightforward from the fact that for $X, Y \subseteq S_{\mathsf {max}}(w)$, $X \cup Y \subseteq S_{\mathsf {max}}(w)$. The validity of Conv is straightforward from the fact that, for any X,Y,Z: $X \subseteq (X \cup Y) \subseteq (X \cup Y \cup Z)$. By induction, for any φ,ψ provably equivalent and for which our theorem holds, in all ${\mathscr{M}}$, ⟦φ⟧ = ⟦ψ⟧, validating RE by the intensionality of the clause for $\Box$. □

1.3 A.3 Completeness

Let a consistent set Γ be a set of wff’s in ${\mathscr{L}}$ such that for no $\varphi _{1} {\ldots } \varphi _{n} \in {\mathscr{L}}$ (for finite n) do we have ⊩_Conv¬(φ₁ ∧… ∧ φ_n). Let a maximal consistent set Γ be a consistent set such that for all $\varphi \in {\mathscr{L}}$ either φ ∈Γ or ¬φ ∈Γ. The proof that for any consistent set Γ there is a maximal consistent set ${\Gamma }^{\prime }$ such that ${\Gamma } \subseteq {\Gamma }^{\prime }$ is as usual. We denote the set of maximal consistent sets in $\mathcal {P}({\mathscr{L}})$ as ${\mathscr{L}}_{\mathsf {M}}$.

We define the canonical model ${\mathscr{M}}_{\mathsf {canon}}$, as per the above, as the triple 〈W,V, (S_min,S_max)〉 such that

• $W_{{\mathscr{M}}_{\mathsf {canon}}} = {\mathscr{L}}_{\mathsf {M}}$

• For p ∈At, $V_{{\mathscr{M}}_{\mathsf {canon}}}(p) = \{ w \in W_{{\mathscr{M}}_{\mathsf {canon}}} : p \in w \}$

• $S_{\mathsf {min}{\mathscr{M}}_{\mathsf {canon}}}(w^{\prime }) = \{ w \in W_{{\mathscr{M}}_{\mathsf {canon}}} :$ for all $\Box \varphi \in w^{\prime }, \varphi \in w \}$ unless no $\Box \varphi \in w^{\prime }$, in which case undefined

• $S_{\mathsf {max}{\mathscr{M}}_{\mathsf {canon}}}(w^{\prime }) = \{ w \in W_{{\mathscr{M}}_{\mathsf {canon}}} :$ for some $\Box \varphi \in w^{\prime }, \varphi \in w\}$ unless no $\Box \varphi \in w^{\prime }$, in which case undefined

Lemma A.4

Let Γ be some consistent set of wff’s of the form $\Box \varphi$, ${\Gamma }_{-\Box }$ be the set obtained by exchanging φ for each $\Box \varphi \in {\Gamma }$, and $\neg {\Gamma }_{-\Box }$ the set obtained by negating each member of ${\Gamma }_{-\Box }$. Then if, for arbitrary $\neg \Box \psi$, we have ${\Gamma } \cup \{\neg \Box \psi \}$ consistent, we have either ${\Gamma }_{-\Box } \cup \{\neg \psi \}$ consistent or $\neg {\Gamma }_{-\Box } \cup \{\psi \}$ consistent.

Proof

Suppose that ${\Gamma } \cup \{\neg \Box \psi \}$ is but neither ${\Gamma }_{-\Box } \cup \{\neg \psi \}$ nor $\neg {\Gamma }_{-\Box } \cup \{ \psi \}$ also is consistent. Then there must be some finite $\{ \gamma _{1}, \ldots , \gamma _{n} \} \subseteq {\Gamma }_{-\Box }$ where ⊩_Conv¬(γ₁ ∧… ∧ γ_n ∧¬ψ) and some (not necessarily disjoint) $\{ \gamma _{n+1}, \ldots , \gamma _{m} \} \subseteq {\Gamma }_{-\Box }$ where ⊩_Conv¬(¬γ_n+ 1 ∧… ∧¬γ_m ∧ ψ). Indeed, by the monotonocity of inconsistency, we may simplify by saying we have both ⊩_Conv¬(γ₁ ∧… ∧ γ_m ∧¬ψ) and ⊩_Conv¬(¬γ₁ ∧… ∧¬γ_m ∧ ψ).

By classical reasoning and the first conjunct, we have $\vdash _{\mathsf {Conv}} (\gamma _{1} \wedge {\ldots } \wedge \gamma _{m}) \rightarrow \psi$. By classical reasoning and the second conjunct, we have $\vdash _{\mathsf {Conv}} (\neg \gamma _{1} \wedge {\ldots } \wedge \neg \gamma _{m}) \rightarrow \neg \psi$. But since by supposition ${\Gamma } \cup \{\neg \Box \psi \}$ is consistent, we have $\nvdash _{\mathsf {Conv}} (\Box \gamma _{1} \wedge {\ldots } \wedge \Box \gamma _{m}) \rightarrow \Box \psi$. But by Corollary A.2, this is impossible. □

Theorem A.5

$w \in \llbracket \varphi \rrbracket _{{\mathscr{M}}_{\mathsf {canon}}}$ iff φ ∈ w

Proof

The proof proceeds by induction. It is straightforward to see that it holds in the base step for all p ∈At, as the interpretation function ⟦⟧ in that case simply reduces to the assignment function V, which in turn by construction reduces in ${\mathscr{M}}_{\mathsf {canon}}$ to ∈. We omit as routine the induction steps for the Boolean connectives, leaving only our operator $\Box$.

We first prove right to left. Suppose $\Box \varphi \in w$. Now by hypothesis we have $w^{\prime } \in \llbracket \varphi \rrbracket _{{\mathscr{M}}_{\mathsf {canon}}}$ iff $\varphi \in w^{\prime }$; by construction we then have for all $w^{\prime } \in S_{\mathsf {min}{{\mathscr{M}}_{\mathsf {canon}}}}(w)$ that $\varphi \in w^{\prime }$, and thus $\llbracket \varphi \rrbracket _{{\mathscr{M}}_{\mathsf {canon}}} \supseteq S_{\mathsf {min}{\mathscr{M}}_{\mathsf {canon}}}(w)$. Moreover, by construction of $S_{\mathsf {max}{\mathscr{M}}_{\mathsf {canon}}}$ and by the inductive hypothesis, $\llbracket \varphi \rrbracket _{{\mathscr{M}}_{\mathsf {canon}}} \subseteq S_{\mathsf {max}{\mathscr{M}}_{\mathsf {canon}}}(w)$. So by construction of ⟦⟧ we have $w \in \llbracket \Box \varphi \rrbracket _{{\mathscr{M}}_{\mathsf {canon}}}$.

Now suppose $\Box \varphi \notin w$, i.e. (by maximality) $\neg \Box \varphi \in w$. Either $S_{{\mathscr{M}}_{\mathsf {canon}}}(w)$ is defined or not. If not, then by stipulation $w \notin \llbracket \Box \varphi \rrbracket _{{\mathscr{M}}_{\mathsf {canon}}}$. Now let Γ be $\{\Box \psi : \Box \psi \in w\}$; then, if $S_{{\mathscr{M}}_{\mathsf {canon}}}(w)$ is defined, by Lemma A.4 and consistency of $w \supset {\Gamma } \cup \{\neg \Box \varphi \}$ we have either ${\Gamma }_{-\Box } \cup \{\neg \varphi \}$ consistent or $\neg {\Gamma }_{-\Box } \cup \{\varphi \}$ consistent. But then (as the domain ${\mathscr{L}}_{\mathsf {M}}$ includes all maximal consistent sets, and all consistent sets can be extended to some $v \in {\mathscr{L}}_{\mathsf {M}}$) there must exist either some $w^{\prime } \supset {\Gamma }_{-\Box } \cup \{\neg \varphi \}$ or $w^{\prime \prime } \supset \neg {\Gamma }_{-\Box } \cup \{\varphi \}$. If the former, then by the inductive hypothesis, the consistency of $w^{\prime }$, and construction of $S_{\mathsf {min}{\mathscr{M}}_{\mathsf {canon}}}$ we have $w^{\prime } \in (S_{\mathsf {min}{\mathscr{M}}_{\mathsf {canon}}}(w) \setminus \llbracket \varphi \rrbracket )$, and thus $w \notin \llbracket \Box \varphi \rrbracket _{{\mathscr{M}}_{\mathsf {canon}}}$. If the latter, then by the inductive hypothesis, the consistency of $w^{\prime \prime }$, and construction of $S_{\mathsf {max}{\mathscr{M}}_{\mathsf {canon}}}$ we have $w^{\prime \prime } \in (\llbracket \varphi \rrbracket \setminus S_{\mathsf {max}{\mathscr{M}}_{\mathsf {canon}}}(w))$, and thus $w \notin \llbracket \Box \varphi \rrbracket _{{\mathscr{M}}_{\mathsf {canon}}}$. □

Corollary A.6

$\vDash _{M_{\mathsf {canon}}}\varphi$ iff ⊩_Convφ

Proof

Suppose ⊩_Convφ. Then we have {¬φ} inconsistent, and thus by maximality of $w \in W_{{\mathscr{M}}_{\mathsf {canon}}}$ each such w must contain φ, which by the above means $\vDash _{M_{\mathsf {canon}}}\varphi$.

Suppose $\nvdash _{\mathsf {Conv}}\varphi$. Then we have {¬φ} consistent, meaning some $w \in W_{{\mathscr{M}}_{\mathsf {canon}}} = {\mathscr{L}}_{\mathsf {M}}$ must extend it, so by the consistency of $w \in W_{{\mathscr{M}}_{\mathsf {canon}}}$ this w must not contain φ, which by the above means $\nvDash _{M_{\mathsf {canon}}}\varphi$. □

By familiar reasoning, proving completeness for our logic now only requires, in light of Corollary A.6, that the frame $\langle W_{{\mathscr{M}}_{\mathsf {canon}}}, (S_{\mathsf {min}{\mathscr{M}}_{\mathsf {canon}}}, S_{\mathsf {max}{\mathscr{M}}_{\mathsf {canon}}}) \rangle$ satisfies the requirements for a model frame given in the first section. (That it satisfies the requirements for its base valuation function $V_{{\mathscr{M}}_{\mathsf {canon}}}$ is immediate.)

Corollary A.7

If $\vDash _{\mathsf {Conv}}\varphi$, ⊩_Convφ

Proof

The conditions for a frame are just that, for all w with S(w) defined, a) w ∈ S_min(w) and b) $S_{\mathsf {min}}(w) \subseteq S_{\mathsf {max}}(w)$.

$S_{{\mathscr{M}}_{\mathsf {canon}}}(w)$, by construction, is defined only for w with some $\Box \varphi \in w$. Let w be such a world. Now by T and maximality, for all $\Box \varphi \in w$, also φ ∈ w, and so by construction of $S_{\mathsf {min}{\mathscr{M}}_{\mathsf {canon}}}(w)$ we have $w \in S_{\mathsf {min}{\mathscr{M}}_{\mathsf {canon}}}(w)$.

Since $S_{{\mathscr{M}}_{\mathsf {canon}}}(w)$ is defined only when there exists such a $\Box \varphi \in w$, the set of $w^{\prime } \in W_{{\mathscr{M}}_{\mathsf {canon}}}$ with some $\psi \in w^{\prime }$ such that $\Box \psi \in w$ extends the set of those containing all such ψ, i.e. $S_{\mathsf {min}{\mathscr{M}}_{\mathsf {canon}}}(w) \subseteq S_{\mathsf {max}.{\mathscr{M}}_{\mathsf {canon}}}(w)$. □

Corollary A.8

$\vDash _{\mathsf {Conv}}\varphi$ iff ⊩_Convφ

1.4 The Logics C o n v 4, C o n v5^∗, and C o n v 45^∗

The logics $\mathsf {Conv4}, \mathsf {Conv5^{*}}, \mathsf {Conv45^{*}} \subset {\mathscr{L}}$ are those generated by the inference rules and axiom schemas of Conv (mutatis mutandis) plus each of the following and both, respectively.

4 :

$\Box \varphi \rightarrow \Box \Box \varphi$

5^∗:

$(\Box \varphi \wedge \neg \Box \psi ) \rightarrow \Box (\varphi \wedge \neg \Box \psi )$

The models of Conv4 and Conv5^∗ are just like those of Conv, with the added stipulation that, for the first, $S_{\mathsf {min}}(w^{\prime }) \subseteq S_{\mathsf {min}}(w)$ and $S_{\mathsf {max}}(w^{\prime }) \supseteq S_{\mathsf {max}}(w)$ for all $w^{\prime } \in S_{\mathsf {min}}(w)$ where S(w) is defined; for the second, $S_{\mathsf {min}}(w^{\prime }) \supseteq S_{\mathsf {min}}(w)$ and $S_{\mathsf {max}}(w^{\prime }) \subseteq S_{\mathsf {max}}(w)$ for all $w^{\prime } \in S_{\mathsf {min}}(w)$ where $S(w), S(w^{\prime })$ are defined. Their canonical models are constructed just like that of Conv, save for maximal consistency in the respective logics replacing maximal ⊩_Conv-consistency.

Soundness is routine.

Theorem A.9

If ⊩_Conv4φ, $\vDash _{\mathsf {Conv4}} \varphi$

Proof

To extend our soundness proof for Conv, observe first that $w \in \llbracket \Box \Box \varphi \rrbracket$ iff for each $w^{\prime } \in S_{\mathsf {min}}(w)$, $w^{\prime } \in \llbracket \Box \varphi \rrbracket$, and for no $w^{\prime \prime } \notin S_{\mathsf {max}}(w)$ is $w^{\prime \prime } \in \llbracket \Box \varphi \rrbracket$. Now clearly, if $w \in \llbracket \Box \varphi \rrbracket$, by the at most restriction of $S_{\mathsf {min}}(w^{\prime })$ and at most expansion of $S_{\mathsf {max}}(w^{\prime })$ for $w^{\prime } \in S_{\mathsf {min}}(w)$, $S_{\mathsf {min}}(w^{\prime }) \subseteq \llbracket \varphi \rrbracket \subseteq S_{\mathsf {max}}(w^{\prime })$ (i.e. $w^{\prime } \in \llbracket \Box \varphi \rrbracket$). Moreover, if $w \in \llbracket \Box \varphi \rrbracket$, $S_{\mathsf {min}}(w) \subseteq \llbracket \varphi \rrbracket \subseteq S_{\mathsf {max}}(w)$, so for no $w^{\prime \prime } \notin S_{\mathsf {max}}(w)$ do we have $w^{\prime \prime } \in \llbracket \varphi \rrbracket$, much less $w^{\prime \prime } \in \llbracket \Box \varphi \rrbracket$. So we have $w \in \llbracket \Box \Box \varphi \rrbracket$ if $w \in \llbracket \Box \varphi \rrbracket$, i.e. we have 4 valid. □

Theorem A.10

If $\vdash _{\mathsf {Conv5^{*}}} \varphi$, $\vDash _{\mathsf {Conv5^{*}}} \varphi$

Proof

To see that 5^∗ is valid, we will show that, whenever $w \in \llbracket \Box \varphi \wedge \neg \Box \psi \rrbracket$, $w \in \llbracket \Box (\varphi \wedge \neg \Box \psi ) \rrbracket$. Consider that $w \in \llbracket \Box (\varphi \wedge \neg \Box \psi ) \rrbracket$ iff for all $w^{\prime } \in S_{\mathsf {min}}$, $w^{\prime } \in \llbracket \varphi \rrbracket$ but not $w^{\prime } \in \llbracket \Box \psi \rrbracket$ (i.e. not $S_{\mathsf {min}}(w^{\prime }) \subseteq \llbracket \psi \rrbracket \subseteq S_{\mathsf {max}}(w^{\prime })$), and for all $w^{\prime \prime } \notin S_{\mathsf {max}}(w)$, either $w^{\prime \prime } \notin \llbracket \varphi \rrbracket$ or $w^{\prime \prime } \in \llbracket \Box \psi \rrbracket$. But since for such $w^{\prime } \in S_{\mathsf {min}}$, $S_{\mathsf {min}}(w^{\prime }) \supseteq S_{\mathsf {min}}(w)$ and $S_{\mathsf {max}}(w^{\prime }) \subseteq S_{\mathsf {max}}(w)$ (where $S(w^{\prime })$ is defined), if we have $S_{\mathsf {min}}(w) \subseteq \llbracket \varphi \rrbracket \subseteq S_{\mathsf {max}}(w)$ but not $S_{\mathsf {min}}(w) \subseteq \llbracket \psi \rrbracket \subseteq S_{\mathsf {max}}(w)$ (i.e. if $w \in \llbracket \Box \varphi \wedge \neg \Box \psi \rrbracket$), we have the latter also for (defined) $S(w^{\prime })$ (so that $w^{\prime } \notin \llbracket \Box \psi \rrbracket$) and we have $w^{\prime } \in \llbracket \varphi \rrbracket$ trivially, as $w^{\prime } \in S_{\mathsf {min}}(w) \subseteq \llbracket \varphi \rrbracket$. (Where S is not defined, $w^{\prime } \notin \llbracket \Box \psi \rrbracket$ trivially.) And, since if $w \in \llbracket \Box \varphi \wedge \neg \Box \psi \rrbracket$ we have $\llbracket \varphi \rrbracket \subseteq S_{\mathsf {max}}(w)$, for no such $w^{\prime \prime } \notin S_{\mathsf {max}}(w)$ do we have $w^{\prime \prime } \in \llbracket \varphi \rrbracket$. □

Completeness is similarly routine.

Theorem A.11

If $\vDash _{\mathsf {Conv4}} \varphi$, ⊩_Conv4φ

Proof

By our earlier result, it will suffice to show, where ${\mathscr{M}}^{\prime }_{\mathsf {canon}}$ is the canonical model of Conv4, that for $w \in W_{{\mathscr{M}}^{\prime }_{\mathsf {canon}}}$ with $S_{{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w)$ defined, $S_{\mathsf {min}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w^{\prime }) \subseteq S_{\mathsf {min}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w)$ and $S_{\mathsf {max}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w^{\prime }) \supseteq S_{\mathsf {max}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w)$ for all $w^{\prime } \in S_{\mathsf {min}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w)$.

Suppose there is some $w^{\prime } \in S_{\mathsf {min}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w)$ and $v \in S_{\mathsf {min}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w^{\prime }), \notin S_{\mathsf {min}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w)$ (i.e. $S_{\mathsf {min}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w^{\prime }) \nsubseteq S_{\mathsf {min}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w)$). Then by construction, there is some $\Box \varphi \in w, \notin w^{\prime }$. But by 4 this is impossible, since for any $\Box \varphi \in w$, $\Box \Box \varphi \in w$, and so $\Box \varphi \in w^{\prime }$. (For this same reason, $S_{{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w^{\prime })$ is always defined when $S_{{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w)$ is defined and $w^{\prime } \in S_{\mathsf {min}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w)$.)

Suppose next there is some $w^{\prime } \in S_{\mathsf {min}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w)$ and $v \in S_{\mathsf {max}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w), \notin S_{\mathsf {min}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w^{\prime })$ (i.e. $S_{\mathsf {max}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w^{\prime }) \nsupseteq S_{\mathsf {max}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w)$). Then again by construction, there is some $\Box \varphi \in w, \notin w^{\prime }$, which we have just seen to be impossible. □

Theorem A.12

If $\vDash _{\mathsf {Conv5^{*}}} \varphi$, $\vdash _{\mathsf {Conv5^{*}}} \varphi$

Proof

Again it will suffice to prove that for $w \in W_{{\mathscr{M}}^{\prime }_{\mathsf {canon}}}$ with $S_{{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w)$ defined, $S_{\mathsf {min}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w^{\prime }) \supseteq S_{\mathsf {min}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w)$ and $S_{\mathsf {max}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w^{\prime }) \subseteq S_{\mathsf {max}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w)$ for all $w^{\prime } \in S_{\mathsf {min}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w)$ with some $\Box \psi \in w^{\prime }$ (where ${\mathscr{M}}^{\prime }_{\mathsf {canon}}$ is now the canonical model of Conv5^∗).

Suppose there is some $w^{\prime } \in S_{\mathsf {min}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w)$ (with $S_{{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w^{\prime })$ defined) and $v \in S_{\mathsf {min}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w), \notin S_{\mathsf {min}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w^{\prime })$ (i.e. $S_{\mathsf {min}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w^{\prime }) \nsupseteq S_{\mathsf {min}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w)$). Then by construction (and definedness of $S_{{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w^{\prime })$), there is some $\Box \varphi \in w^{\prime }, \notin w$. Now since $S_{{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w)$ is defined, there exists $\Box \psi \in w$, with ψ≠φ. We then by maximality have $\Box \psi \wedge \neg \Box \varphi \in w$. But then we have by 5^∗ that $\Box (\psi \wedge \neg \Box \varphi ) \in w$, and so by construction and maximality $\psi \wedge \neg \Box \varphi , \neg \Box \varphi \in w^{\prime }$, which is by consistency of $w^{\prime }$ impossible.

Suppose next there is some $w^{\prime } \in S_{\mathsf {min}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w)$ and $v \in S_{\mathsf {max}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w^{\prime }), \notin S_{\mathsf {min}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w)$ (i.e. $S_{\mathsf {max}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w^{\prime }) \nsubseteq S_{\mathsf {max}{\mathscr{M}}^{\prime }_{\mathsf {canon}}}(w)$). Then again by construction, there is some $\Box \varphi \in w^{\prime }, \notin w$, which we have just seen to be impossible. □

Corollary A.13

Conv4 and Conv5^∗ are independent.

Proof

For a model of Conv5^∗ invalidating 4, let W = {0, 1}, S_min(0) = S_max(0) = W, S(1) be undefined. For a model of Conv4 invalidating 5^∗, let W = {0, 1}, S_min(0) = S_max(0) = S_max(1) = W, S_min(1) = {1}. For one satisfying both let W = {0}, S_min(0) = S_max(0) = W. □

Corollary A.14

Conv45^∗ is sound and complete on the class of frames where for any w with S(w) defined, S(v) = S(u) where v,u ∈ S_min(w).

Proof

This is straightforward from the preceding proofs. □

Corollary A.15

Among models in which, for all w with S(w) defined, $S(w^{\prime })$ is defined for all $w^{\prime } \in S_{\mathsf {min}}(w)$, the Conv5^∗ models are the Conv45^∗ models.

Proof

Take any such model of Conv5^∗ and any w with S(w) defined. For any $w^{\prime } \in S_{\mathsf {min}}(w)$, $S_{\mathsf {min}}(w^{\prime })$ (which we know to be defined) must contain w, as $w \in S_{\mathsf {min}}(w) \subseteq S_{\mathsf {min}}(w^{\prime })$. So, by the defining property of Conv5^∗ models, $S_{\mathsf {min}}(w) \subseteq S_{\mathsf {min}}(w^{\prime })$ (as $w^{\prime } \in S_{\mathsf {min}}(w)$) and $S_{\mathsf {min}}(w^{\prime }) \subseteq S_{\mathsf {min}}(w)$ (as $w \in S_{\mathsf {min}}(w^{\prime })$), i.e. $S_{\mathsf {min}}(w) = S_{\mathsf {min}}(w^{\prime })$; similarly, $S_{\mathsf {max}}(w) \supseteq S_{\mathsf {max}}(w^{\prime })$ and $S_{\mathsf {max}}(w^{\prime }) \supseteq S_{\mathsf {max}}(w)$, i.e. $S_{\mathsf {max}}(w) = S_{\mathsf {min}}(w^{\prime })$. Thus, for all $w^{\prime } \in S_{\mathsf {min}}(w)$, $S(w) = S(w^{\prime })$.

The reverse direction is trivial. □

We can now prove the correspondence between Conv45^∗ and the principle attributed to Lewis in Section 3.1.

Fact A.16

Take an arbitrary sandwich frame 〈W,S〉. Let $\mathcal {N}(p \subseteq W)$ be $\{ w \in W : S_{\mathsf {min}}(w) \subseteq p \subseteq S_{\mathsf {max}}(w)\}$. Let O_w be the intersection of all p ∋ w such that for some $q \subseteq W$ either $p = \mathcal {N}(q)$ or $p = W \setminus \mathcal {N}(q)$.

If for all w (with S(w) defined), $S_{\mathsf {min}}(w) \subseteq O_{w} \subseteq S_{\mathsf {max}}(w)$, then for all v and u ∈ S_min(v), S(v) = S(u), and vice versa.

Proof

Left to right: Clearly, given the antecedent, for all w (with S(w) defined), O_w = S_min(w), as by definition $O_{w} \subseteq \mathcal {N}(S_{\mathsf {min}}(w)) \subseteq S_{\mathsf {min}}(w)$. It is now easy to see that for any v ∈ O_w, neither is there u ∈ O_v,∉O_w nor $u^{\prime } \in O_{w}, \notin O_{v}$. For if the former, then , which is prohibited by construction of O as long as v ∈ O_w = S_min(w); and if the latter, either or no $\mathcal {N}(p) \ni v$ at all, both of which are prohibited by construction of O as long as v ∈ O_w = S_min(w), since $w \notin \mathcal {N}(p)$ iff $w \in (W \setminus \mathcal {N}(p))$. Therefore O_w = O_v, and thus S_min(w) = S_min(v).

Moreover, whenever O_w = S_min(w) = O_v = S_min(v), S_max(w) = S_max(v), as otherwise w,v will be distinguished by some p in that $w \in , v\notin \mathcal {N}(p)$ (or vice versa). But again by definition of O, this is impossible. So S(w) = S(v).

Right to left: Suppose for all w (with S(w) defined) and v ∈ S_min(w), S(w) = S(v). Then $w \in \mathcal {N}(p)$ iff $v \in \mathcal {N}(p)$, so O_w = O_v. So by v ∈ O_v for all v ∈ S_min(w), $S_{\mathsf {min}}(w) \subseteq O_{w}$. But $O_{w} \subseteq \mathcal {N}(S_{\mathsf {max}}(w)) \subseteq S_{\mathsf {max}}(w)$. So $S_{\mathsf {min}}(w) \subseteq O_{w} \subseteq S_{\mathsf {max}}(w)$. □

1.5 A.5 The Logic C o n v +

Let the language ${\mathscr{L}}_{\boxdot }$ be ${\mathscr{L}}$ closed under the introduction of formulas $\ulcorner \boxdot \varphi \urcorner$. Let the logic $\mathsf {Conv+} \subset {\mathscr{L}}_{\boxdot }$ be that generated by the axiom schemas and inference rules of Conv (mutatis mutandis), together with the additional axiom schemas

$\mathsf {K_{\boxdot }}$ :

$\boxdot (\varphi \rightarrow \psi ) \rightarrow (\boxdot \varphi \rightarrow \boxdot \psi )$

$\mathsf {T_{\boxdot }}$ :

$\boxdot \varphi \rightarrow \varphi$

P S :

$(\boxdot \varphi \wedge \Box \psi ) \rightarrow \Box (\varphi \wedge \psi )$

and the additional inference rule

N e c :

if ⊩_Conv+φ, $\vdash _{\mathsf {Conv+}} \boxdot \varphi$

A model ${\mathscr{M}}^{+}$ of Conv+ is just like a model of Conv, together with an extra parameter $R : W \rightarrow \mathcal {P}(W)$; R obeys the further constraints that for all w ∈ W, $S_{\mathsf {min}}(w) \subseteq R(w)$ (where defined), and w ∈ R(w). As usual, $\llbracket \boxdot \varphi \rrbracket _{{\mathscr{M}}}^{+} = \{ w \in W : R(w) \subseteq \llbracket \varphi \rrbracket _{{\mathscr{M}}}^{+} \}$. The canonical model ${\mathscr{M}}^{+}_{\mathsf {canon}}$ of Conv+ is constructed as for Conv, with the obvious extensions for $R_{{\mathscr{M}}^{+}_{\mathsf {canon}}}$ (particularly that $w^{\prime } \in R_{{\mathscr{M}}^{+}_{\mathsf {canon}}}(w)$ iff $\varphi \in w^{\prime }$ for all $\Box \varphi \in w$).

It is easy to see the logic is sound.

Theorem A.17

If ⊩_Conv+φ, $\vDash _{\mathsf {Conv+}} \varphi$

Proof

The only distinctive case to show is for PS. By the stated constraint on R and S_min, whenever $w \in \llbracket \boxdot \varphi \rrbracket , \in \llbracket \Box \psi \rrbracket$, $S_{\mathsf {min}}(w) \subseteq \llbracket \psi \rrbracket \subseteq S_{\mathsf {max}}(w)$ (by construction of $\llbracket \Box \psi \rrbracket$) and $S_{\mathsf {min}}(w) \subseteq \llbracket \varphi \rrbracket$ (by the above constraint), and thus $S_{\mathsf {min}}(w) \subseteq \llbracket \varphi \rrbracket \cap \llbracket \psi \rrbracket = \llbracket \varphi \wedge \psi \rrbracket \subseteq \llbracket \psi \rrbracket \subseteq S_{\mathsf {max}}(w)$, and so by construction of $\llbracket \Box (\varphi \wedge \psi ) \rrbracket$, $w \in \llbracket \Box (\varphi \wedge \psi )\rrbracket$. Whence the conclusion follows trivially. □

The bulk of the completeness proof can rely on those given above.

Theorem A.18

If $\vDash _{\mathsf {Conv+}} \varphi$, ⊩_Conv+φ

Proof

It is easy to check that the proofs for Theorem A.5 and Corollary A.6 can be extended to proofs of their analogues for Conv+. Thus all that is required is a proof that our canonical model ${\mathscr{M}}^{+}_{\mathsf {canon}}$ satisfies the criteria for a model.

For the constraints on R by itself the proofs are well known. We thus need only demonstrate that, when defined, $S_{\mathsf {min} {\mathscr{M}}^{+}_{\mathsf {canon}}}(w) \subseteq R_{{\mathscr{M}}^{+}_{\mathsf {canon}}}(w)$. Suppose there is some $w^{\prime } \in S_{\mathsf {min} {\mathscr{M}}^{+}_{\mathsf {canon}}}(w), \notin R_{{\mathscr{M}}^{+}_{\mathsf {canon}}}(w)$ (where these are defined). Then by construction there are $\varphi \in w^{\prime }, \psi \notin w^{\prime }$ (that is, by maximality, $\varphi , \neg \psi \in w^{\prime }$) such that $\Box \varphi , \boxdot \psi \in w$. But then we do not have (by consistency of $w^{\prime }$) $\varphi \wedge \psi \in w^{\prime }$, and so by construction of $S_{\mathsf {min} {\mathscr{M}}^{+}_{\mathsf {canon}}}$ we do not have $\Box (\varphi \wedge \psi ) \in w$, which contradicts the hypothesis given PS (and maximality). □

Say that a model is partitional just in case, for all w, for all $v, v^{\prime } \in R(w)$, S_min(v) and $S_{\mathsf {min}}(v^{\prime })$ are (when defined) either identical or pairwise disjoint.

Theorem A.19

No formulas define the partitional models.

Proof

It will suffice to show that, for two models ${\mathscr{M}}, {\mathscr{M}}^{\prime }$ with ${\mathscr{M}}$ partitional but not ${\mathscr{M}}^{\prime }$, the φ valid on ${\mathscr{M}}$ are just those valid on ${\mathscr{M}}^{\prime }$.

Let $W_{{\mathscr{M}}} = \{ w_{0}, w_{1} \}$; $S_{\mathsf {min} {\mathscr{M}}}(w_{0}) = S_{\mathsf {max} {\mathscr{M}}}(w_{0}) = \{w_{0}\}$; $S_{{\mathscr{M}}}(w_{1})$ undefined; and for all atomic p, $V_{{\mathscr{M}}}(p) = \{w_{0}\}$. Let $W_{{\mathscr{M}}}^{\prime } = \{ v_{0}, v_{1}, v_{2} \}$; $S_{\mathsf {min} {\mathscr{M}}^{\prime }}(v_{0}) = S_{\mathsf {max} {\mathscr{M}}^{\prime }}(v_{0}) = \{v_{0},v_{1}\}$, $S_{{\mathscr{M}}}^{\prime }(v_{1}) = (\{v_{1}\}, \{v_{0},v_{1}\})$; $S_{{\mathscr{M}}}^{\prime }(v_{2})$ undefined; and for all p, $V_{{\mathscr{M}}}^{\prime }(p) = \{v_{0},v_{1}\}$. For both, R is the universal relation (the constant function to W). Clearly, ${\mathscr{M}}$ but not ${\mathscr{M}}^{\prime }$ is partitional, so it remains to show $\vDash _{{\mathscr{M}}} \varphi$ iff $\vDash _{{\mathscr{M}}^{\prime }} \varphi$.

First we prove by induction on complexity that w₁ ∈⟦φ⟧ iff v₂ ∈⟦φ⟧. For the atomic base case, this is given. For the Boolean connectives, this follows by classicality of the logic. Moreover, as S(w₁) and S(v₂) are undefined, the step for $\Box \varphi$ is likewise trivial.

Next, w₀ ∈⟦φ⟧ iff v₁ ∈⟦φ⟧. Again, the base step and Boolean inductive step are trivial. For $\Box \varphi$, if w₀∉⟦φ⟧ and v₁∉⟦φ⟧, it follows by T that $v_{1} \notin \llbracket \Box \varphi \rrbracket$ and $w_{0} \notin \llbracket \Box \varphi \rrbracket$; if w₀ ∈⟦φ⟧ and v₁ ∈⟦φ⟧, either v₂ ∈⟦φ⟧ and (thus, by the above) w₁ ∈⟦φ⟧ or not. If so, $v_{1} \notin \llbracket \Box \varphi \rrbracket$ and $w_{0} \notin \llbracket \Box \varphi \rrbracket$, since v₂∉S_max(v₁) and w₁∉S_max(w₀). If not, then $S_{\mathsf {min}}(v_{1}) \subseteq \llbracket \varphi \rrbracket \subseteq S_{\mathsf {max}}(v_{1})$ (meaning $v_{1} \in \llbracket \Box \varphi \rrbracket$) and S_min(w₀) = ⟦φ⟧ = S_max(w₀) (meaning $w_{0} \in \llbracket \Box \varphi \rrbracket$).

Finally, w₀ ∈⟦φ⟧ iff v₀ ∈⟦φ⟧. Base and Boolean steps are trivial. For $\Box \varphi$, by the inductive hypothesis we have w₀ ∈⟦φ⟧ iff v₀ ∈⟦φ⟧. As (by T) the result is trivial if v₀∉⟦φ⟧ and w₀∉⟦φ⟧, suppose v₀ ∈⟦φ⟧ and w₀ ∈⟦φ⟧. Again, either v₂ ∈⟦φ⟧ (and thus w₁ ∈⟦φ⟧) or not. In the former case, the conclusion follows as before. Otherwise, by the preceding paragraph, we have w₀ ∈⟦φ⟧ and v₁ ∈⟦φ⟧, so v₀ ∈⟦φ⟧ and v₁ ∈⟦φ⟧, and thus, as S_min(v₀) = S_max(v₀) = {v₀,v₁} and v₂∉⟦φ⟧, $v_{0} \in \llbracket \Box \varphi \rrbracket$. But (since we are assuming w₁∉⟦φ⟧), as S_min(w₀) = S_max(w₀) = {w₀} = ⟦φ⟧, also $w_{0} \in \llbracket \Box \varphi \rrbracket$.

So the $\varphi \in {\mathscr{L}}$ true at w₀ are just those true at v₀,v₁, and the $\varphi \in {\mathscr{L}}$ true at w₁ are just those true at v₂ extending this to $\varphi \in {\mathscr{L}}_{\boxdot }$ is straightforward. The desired result–$\vDash _{{\mathscr{M}}} \varphi$ iff $\vDash _{{\mathscr{M}}^{\prime }} \varphi$–is now immediate. □

We now prove Fact 3.3.

Proof

Suppose our model is partitional and that (3 - 5) and (C_i) are true (at w ∈ W), where i is the least number such that $\Box (p \wedge q_{i})$ and $\neg \Box (p \wedge q_{i-1})$ are true. As (by (4)) $S_{\mathsf {min}}(w) \subseteq \llbracket q_{500,000} \rrbracket$ and $S_{\mathsf {min}}(w) \subseteq R(w)$, so (by (3) and PS) for all $w^{\prime } \in S_{\mathsf {min}}(w)$, $S_{\mathsf {min}}(w^{\prime })$ is defined and $S_{\mathsf {min}}(w^{\prime }) \subseteq \llbracket p \rrbracket \subseteq S_{\mathsf {max}}(w^{\prime })$. By partitionality, $S_{\mathsf {min}}(w) = S_{\mathsf {min}}(w^{\prime })$. We thus have for all $\Box (\varphi \wedge p)$ true at w (i.e. such that $S_{\mathsf {min}}(w) \subseteq \llbracket \varphi \rrbracket \cap \llbracket p \rrbracket \subseteq \llbracket p \rrbracket \subseteq S_{\mathsf {max}}(w)$, with S_min defined and fixed throughout S_min(w)) that $\llbracket \Box (\varphi \wedge p) \rrbracket \supseteq S_{\mathsf {min}}(w)$, and thus $S_{\mathsf {min}}(w) \subseteq \llbracket \Box (p \wedge q_{i}) \rrbracket \subseteq \llbracket p \rrbracket \subseteq S_{\mathsf {max}}(w)$, and therefore we have $\Box \Box (p \wedge q_{i})$ true (at w).

Since we also have $\neg \Box (p \wedge q_{i-1})$ true (at w), we by similar reasoning have $\Box (p \wedge q_{i-1})$ true at no $w^{\prime } \in S_{\mathsf {min}}(w)$, and thus $S_{\mathsf {min}}(w) \subseteq \llbracket p \rrbracket \cap \llbracket \neg \Box (p \wedge q_{i-1}) \rrbracket \subseteq \llbracket p \rrbracket \subseteq S_{\mathsf {max}}(w)$, and thus $\Box (p \wedge \neg \Box (p \wedge q_{i-1}))$ true (at w).

We by these and M therefore have $\Box (\Box (p \wedge q_{i}) \wedge p \wedge \neg \Box (p \wedge q_{i-1}))$, or equivalently $\Box (\Box (p \wedge q_{i}) \wedge \neg \Box (p \wedge q_{i-1}))$, true (at w), which contradicts (C_i). □