The Logic of Framing Effects

Abstract

Framing effects concern the having of different attitudes towards logically or necessarily equivalent contents. Framing is of crucial importance for cognitive science, behavioral economics, decision theory, and the social sciences at large. We model a typical kind of framing, grounded in (i) the structural distinction between beliefs activated in working memory and beliefs left inactive in long term memory, and (ii) the topic- or subject matter-sensitivity of belief: a feature of propositional attitudes which is attracting growing research attention. We introduce a class of models featuring (i) and (ii) to represent, and reason about, agents whose belief states can be subject to framing effects. We axiomatize a logic which we prove to be sound and complete with respect to the class.

Appendix A: Proofs

1.1 A.1 Proof of Lemma 1

The proof follows by induction on the structure of φ, where cases for the propositional variables, Boolean connectives, and $\varphi :=\Box \psi$ are trivial. So assume inductively that the result holds for ψ and show that it holds also for φ := B_Pψ. Observe that the inductive hypothesis says that $[\![{\psi }]\!]^{O_a}_{{\mathscr{M}}}=[\![{\psi }]\!]^{U_{c}}_{{\mathscr{M}}}$. We then have

$$ \begin{array}{@{}rcl@{}} \mathcal{M}, (w, O_a)\Vdash B_{P}\psi & \text{ iff } O^{\cap}\subseteq [\![ \psi]\!]^{O_a}_{\mathcal{M}} \text{ and } t(\varphi)\sqsubseteq \mathfrak{b} ~~~~~~(\text{by the semantics of } B_{P}\psi)\\ & \text{ iff } O^{\cap}\subseteq [\![ \psi]\!]^{U_c}_{\mathcal{M}} \text{ and } t(\varphi)\sqsubseteq \mathfrak{b} ~~~(\text{by the induction hypothesis})\\ & \text{ iff } \mathcal{M}, (w, U_c)\Vdash B_{P}\psi ~~~~~~~~~~~~~~~~~~(\text{by the semantics of } B_{P}\psi) \end{array} $$

1.2 A.2 Proof of Theorem 2

1.2.1 A.2.1 Soundness of L

Soundness is a matter of routine validity check, so we spell out only the relatively tricky cases.

Proof

Let ${\mathscr{M}}=\langle W, \mathcal {O}, T, \oplus , t, \nu \rangle$ be a model and $(w, O_a)\in P({\mathscr{M}})$. Checking the soundness of the system S5 for $\Box$ is standard: recall that $\Box$ is interpreted as the global modality. Validity of $\mathsf {D}_{B_{A}}$ is guaranteed since O≠∅ by the definition of memory cells. Validity of $\mathsf {Ax}1_{{B_{\star }}}$ is an immediate consequence of the semantic clauses for B_⋆ and the definition of $\bar {\varphi }$. $\mathsf {Ax}3_{{B_{\star }}}$ is valid since truth of a belief sentence B_⋆φ is state independent: it is easy to see that either $[\![{{B_{\star }}\varphi }]\!]^{O_a}_{{\mathscr{M}}}=W$ or $[\![{{B_{\star }}\varphi }]\!]^{O_a}_{{\mathscr{M}}}=\emptyset$, for any $\varphi \in {\mathscr{L}}$. Here we spell out the details only for $\mathsf {C}_{B_{A}}$, $\mathsf {Ax}2_{{B_{P}}}$, and Inc.

$\mathsf {C}_{B_{A}}$:

$$ \begin{array}{@{}rcl@{}} &&\mathcal{M}, (w, O_a)\Vdash B_{A}(\varphi\wedge \psi)\\ && \text{ iff } O\subseteq [\![ \varphi\wedge \psi]\!]^{O_a} \text{ and } t(\varphi\wedge \psi)\sqsubseteq a\\ && \text{ iff } O\subseteq [\![ \varphi]\!]^{O_a} \cap [\![ \psi]\!]^{O_a} \text{ and } t(\varphi)\oplus t(\psi)\sqsubseteq a\\ && \text{ iff } (O\subseteq [\![ \varphi]\!]^{O_a} \text{ and } O\subseteq [\![ \psi]\!]^{O_a}) \text{ and } (t(\varphi) \sqsubseteq a \text{ and } t(\psi)\sqsubseteq a) \\ && \text{ iff } (O\subseteq [\![ \varphi]\!]^{O_a} \text{ and } t(\varphi) \sqsubseteq a) \text{ and } (O\subseteq [\![ \psi]\!]^{O_a} \text{ and } t(\psi)\sqsubseteq a) \\ && \text{ iff } \mathcal{M}, (w, O_a)\Vdash B_{A}\varphi \wedge B_{A}\psi \end{array} $$

Validity proof for $\mathsf {C}_{B_{P}}$ follows similarly: replace O by O^∩ and a by $\mathfrak {b}$.

$\mathsf {Ax}2_{B_{P}}$: Suppose that ${\mathscr{M}}, (w, O_a)\Vdash \Box (\varphi \rightarrow \psi )\wedge B_{P}\varphi \wedge B_{P}\bar {\psi }$, i.e., (1) ${\mathscr{M}}, (w, O_a)\Vdash \Box (\varphi \rightarrow \psi )$, (2) ${\mathscr{M}}, (w, O_a)\Vdash B_{P}\varphi$, and (3) ${\mathscr{M}}, (w, O_a)\Vdash B_{P}\bar {\psi }$. (1) means that $[\![{\varphi }]\!]^{O_a}\subseteq [\![{\psi }]\!]^{O_{a}}$, (2) implies that $O^{\cap }\subseteq [\![{\varphi }]\!]^{O}_{a}$. Therefore, (1) and (2) together implies that $O^{\cap }\subseteq [\![{\psi }]\!]^{O}_{a}$. Moreover, (3) is the case if and only if $t(\psi )\sqsubseteq \mathfrak {b}$. We therefore conclude that ${\mathscr{M}}, (w, O_a)\models {B_{P}}\psi$. Validity proof for $\mathsf {Ax}2_{B_{A}}$ follows similarly: replace O^∩ by O and $\mathfrak {b}$ by a.

Inc: Suppose that ${\mathscr{M}}, (w, O_a)\Vdash B_{A}\varphi$, i.e, that $O\subseteq [\![ \varphi ]\!]^{O}_{a}$ and $t(\varphi )\sqsubseteq a$. By the definitions of O^∩ and $\mathfrak {b}$, we have that $O^{\cap } \subseteq O$ and $a\sqsubseteq \mathfrak {b}$. Therefore, $O^{\cap }\subseteq [\![{\varphi }]\!]^{O_a}$ and $t(\varphi )\sqsubseteq \mathfrak {b}$, i.e., ${\mathscr{M}}, (w, O_a)\Vdash {B_{P}}\varphi$. □

1.2.2 A.2.2 Completeness of L

We establish the completeness result via a canonical model construction. While the construction of memory cells uses methods presented by [26], the construction of canonical topics is inspired by the canonical model construction of awareness models (see, e.g., [27]).

1.2.3 Auxiliary Definitions and Lemmas:

The notion of derivation, denoted by ⊩, in L is defined as usual. Thus, ⊩ φ means φ is a theorem of L.

Lemma 3

The following are derivable in L:

$$ \textit{1. } (\Box\varphi \wedge B_{\star}\overline{\varphi})\rightarrow B_{\star}\varphi \textit{2. } B_{\star}\bar{\varphi}\rightarrow B_{\star}\bar{\psi}, if \text{Var}\psi\subseteq \text{Var}\varphi $$

Proof

• 1. $\vdash (\Box \varphi \wedge B_{\star }\bar \varphi )\supset B_{\star }\varphi$:

  1. $\vdash (\Box (\overline {\varphi } \rightarrow \varphi ) \wedge B_{\star }\overline {\varphi }) \rightarrow B_{\star }\varphi$ $\mathsf {Ax}2_{B_{\star }}$

  2. $\vdash (\Box (\overline {\varphi } \rightarrow \varphi ) \wedge B_{\star }\overline {\varphi }) \equiv (\Box \varphi \wedge B_{\star }\overline {\varphi })$ CPL, $\mathsf {S5}_{\Box }$

  3. $\vdash (\Box \varphi \wedge B_{\star }\bar {\varphi })\supset B_{\star }\varphi$ 1, 2, CPL

• 2. $\vdash B_{\star }\bar {\varphi }\rightarrow B_{\star }\bar {\psi }$, if $\text {Var}\psi \subseteq \text {Var}\varphi$

  Follows directly from $\mathsf {C}_{B_{\star }}$ and CPL.

□

For any set of formulas ${\Gamma }\subseteq {\mathscr{L}}$ and any $\varphi \in {\mathscr{L}}$, we write Γ ⊩ φ if there exists finitely many formulas $\varphi _1, \dots , \varphi _n\in {\Gamma }$ such that $\vdash (\varphi _1\wedge {\dots } \wedge \varphi _n){\rightarrow } \varphi$. We say that Γ is L-consistent if Γ⊯⊥, and L-inconsistent otherwise. A sentence φ is L-consistent with Γ if Γ ∪{φ} is L-consistent (or, equivalently, if Γ⊯¬φ). Finally, a set of formulas Γ is a maximallyL-consistent set (or, in short, mcs) if it is L-consistent and any set of formulas properly containing Γ is L-inconsistent [13]. We drop mention of the logic L when it is clear from the context.

Lemma 4

For every mcs Γ of L and $\varphi , \psi \in {\mathscr{L}}$, the following hold:

• 1. Γ ⊩ φ iff φ ∈Γ,

• 2. if φ ∈Γ and $\varphi \supset \psi \in {\Gamma }$ then ψ ∈Γ,

• 3. if ⊩ φ then φ ∈Γ,

• 4. φ ∈Γ and ψ ∈Γ iff φ ∧ ψ ∈Γ,

• 5. φ ∈Γ iff ¬φ∉Γ.

Proof

Standard. □

In the following proofs, we make repeated use of Lemma 4 in a standard way and often omit mention of it.

Lemma 5 (Lindenbaum’s Lemma)

Every L-consistent set can be extended to a maximally L-consistent one.

Proof

Standard. □

1.2.4 Canonical Model of ${\mathscr{L}}$ for a mcs Γ₀:

Let $\mathcal {X}^{c}$ be the set of all maximally L-consistent sets. For each ${\Gamma }\in \mathcal {X}^{c}$, define

$$ \begin{array}{@{}rcl@{}} {\Gamma}[\Box] \ &:=&\{\varphi\in \mathcal{L}: \Box\varphi\in {\Gamma}\}, \\ {\Gamma}[B_{\star}] \ &:=& \ \{\varphi \in \mathcal{L} : B_{\star}\psi\wedge \Box (\psi\supset \varphi) \in {\Gamma} \text{ for some } \psi\in \mathcal{L}\}, \text{ and } \\ {\Gamma}[B_{\star},\Box] \ &: = & \ \{\varphi \in \mathcal{L} : B_{\star}\psi\wedge \Box (\psi\supset \varphi) \in {\Gamma} \text{ for some } \psi\in \mathcal{L}\}\cup\\&&\quad\{\varphi\in \mathcal{L}: \Box\varphi\in {\Gamma}\}. \end{array} $$

In short, ${\Gamma }[B_{\star },\Box ] ={\Gamma }[B_{\star }]\cup {\Gamma }[\Box ]$. By axiom Inc, we have ${\Gamma }[B_{A}]\subseteq {\Gamma } [B_{P}]$, therefore, also have ${\Gamma }[B_{A},\Box ]\subseteq {\Gamma } [{B_{P}},\Box ]$. Moreover, we define $\sim _{\Box }$ on $\mathcal {X}^c$ as

$$ {\Gamma} \sim_{\Box} {\Delta} \ \text{ iff } \ {\Gamma}[\Box] \subseteq {\Delta}. $$

Since $\Box$ is an S5 modality, it is easy to see that $\sim _{\Box }$ is an equivalence relation. For any maximally L-consistent set Γ, we denote by $[{\Gamma }]_{\Box }$ the equivalence class of Γ induced by $\sim _{\Box }$, i.e., $[{\Gamma }]_{\Box }=\{\Delta \in \mathcal {X}^c : {\Gamma }\sim _{\Box } {\Delta }\}$. It is easy to see that if ${\Gamma }[{B_{\star }}, \Box ]\subseteq {\Delta }$, then ${\Delta }\in [{\Gamma }]_{\Box }$.

Lemma 6

For any two maximally consistent sets Γ and Δ such that ${\Gamma }\sim _{\Box }{\Delta }$, ${\Gamma }[\Box ]={\Delta }[\Box ]$ and Γ[B_⋆] = Δ[B_⋆]. Therefore, also ${\Gamma }[B_{\star }, \Box ]={\Delta }[{B_{\star }}, \Box ]$.

Proof

${\Gamma }[\Box ]={\Delta }[\Box ]$ follows from the axioms and rules of $\mathsf {S5}_{\Box }$. For ${\Gamma }[B_{\star }]\subseteq {\Delta }[B_{\star }]$, let φ ∈Γ[B_⋆]. This means that there is $\psi \in {\mathscr{L}}$ such that ${B_{\star }}\psi \in \Box (\psi \rightarrow \varphi )\in {\Gamma }$. Then, by $\mathsf {Ax}3_{{B_{\star }}}$ and $\mathsf {S5}_{\Box }$, we have $\Box {B_{\star }}\psi \wedge \Box \Box (\psi \rightarrow \varphi )\in {\Gamma }$. As ${\Gamma }\sim _{\Box } {\Delta }$, we obtain that ${B_{\star }}\psi \wedge \Box (\psi \rightarrow \varphi )\in {\Delta }$, thus, φ ∈Δ[B_⋆]. The other direction follows similarly since $\sim _{\Box }$ is symmetric. We then also have ${\Gamma }[{B_{\star }}, \Box ]={\Gamma }[\Box ]\cup {\Gamma }[{B_{\star }}]={\Delta }[\Box ]\cup {\Delta }[{B_{\star }}]={\Delta }[{B_{\star }}, \Box ]$. □

Given a mcs Γ₀ of L, the canonical model for Γ₀ is a tuple ${\mathscr{M}}^c=\langle [{\Gamma }_0]_{\Box }, \mathcal {O}^{c}, T^{c}, \oplus ^{c}, t^{c}, \nu ^{c} \rangle$ where

• $\mathcal {O}^{c}=\{\{\Delta \in \mathcal {X}^{c}: {\Gamma }_{0}[B_{A},\Box ]\subseteq {\Delta }\}, \{\Delta \in \mathcal {X}^{c} : {\Gamma }_0[B_{P},\Box ]\subseteq {\Delta }\} \}$. To simplify the notation, we denote

  $$ \begin{array}{@{}rcl@{}} O &:= \{\Delta \in \mathcal{X}^{c}: {\Gamma}_0[B_{A},\Box]\subseteq {\Delta}\}\\ O^{\cap} & :=\{\Delta \in \mathcal{X}^{c} : {\Gamma}_0[B_{P},\Box]\subseteq {\Delta}\}. \end{array} $$

  Since ${\Gamma }_0[\Box ]\subseteq {\Gamma }_0[B_{\star },\Box ]$, we guarantee that $O, O^{\cap }\subseteq [{\Gamma }_{0}]_{\Box }$. Moreover, since ${\Gamma }[B_{A},\Box ]\subseteq {\Gamma } [B_{P},\Box ]$, we have $O^{\cap } \subseteq O$. Therefore, $\bigcap \mathcal {O}^{c}={O}^{\cap }$.

• $T^{c}=\{\mathfrak {a}, {\mathfrak {b}_{P}}, {\mathfrak {b}_{A}}\}$, where $\mathfrak {a}=\{q\in \mathsf {Prop} : \neg B_{P}\overline {q}\in {\Gamma }_{0}\}$ and ${\mathfrak {b}_{P}}=\{q\in \mathsf {Prop} : B_{P}\overline {q}\wedge \neg B_{A}\overline {q}\in {\Gamma }_0\}$, and $\mathfrak {b}_{A}=\{q\in \mathsf {Prop} : B_{A}\overline {q}\in {\Gamma }_0\}$.

• $\oplus ^{c}: T^{c}\times T^{c}\rightarrow T^{c}$ such that the corresponding strict partial order $\sqsubset ^c$ is ${\mathfrak {b}_{A}}\sqsubset ^{c} {\mathfrak {b}_{P}} \sqsubset ^{c} \mathfrak {a}$ (see Fig. 2).

  Fig. 2

  

  (T^c,⊕^c) for Γ₀, where ${\mathfrak {b}_{A}}\sqsubset ^{c} {\mathfrak {b}_{P}} \sqsubset ^{c} \mathfrak {a}$

• $t^{c}: {\mathscr{L}}\cup \mathcal {O}^{c} \rightarrow T^{c}\cup \mathcal {P}(T^{c})$ such that, for every a ∈ T^c and q ∈Prop, t^c(q) = a iff q ∈ a, and t^c extends to ${\mathscr{L}}$ by t^c(φ) = ⊕^cVarφ. Moreover, $t^{c}(O^{\cap }) = \{\mathfrak {b}_{P}\}$ and $t^{c}(O)=\{\mathfrak {b}_{A}\}$.

• $\nu ^{c}: \mathsf {Prop} \rightarrow \mathcal {P}([{\Gamma }_0]_{\Box })$ such that $\nu ^c(p)=\{\Gamma \in [{\Gamma }_0]_{\Box } : p\in {\Gamma }\}$.

In order to show that the canonical model ${\mathscr{M}}^{c}$ for Γ₀ is a topic-sensitive subset space model, we need the following auxiliary lemmas.

Lemma 7

Given a mcs Γ, for all finite ${\Phi } \subseteq {\Gamma }[B_{\star }]$ and ${\Phi }^{\prime }\subseteq {\Gamma }[\Box ]$, we have $\bigwedge {\Phi }\in {\Gamma }[B_{\star }]$ and $\bigwedge {\Phi }^{\prime }\in {\Gamma }[\Box ]$.

Proof

Given a finite ${\Phi }^{\prime }\subseteq {\Gamma }[\Box ]$, $\bigwedge {\Phi }^{\prime }\in {\Gamma }[\Box ]$ follows via a standard argument since $\Box$ is a normal modal operator. We only show the case for Γ[B_⋆]. Let ${\Phi }=\{\varphi _1, \dots , \varphi _n\} \subseteq {\Gamma }[{B_{\star }}]$. This means that, for each φ_j with 1 ≤ j ≤ n, there is a $\psi _{j}\in {\mathscr{L}}$ such that ${B_{\star }}\psi _{j}\wedge \Box (\psi _{j}{\rightarrow } \varphi _j)\in {\Gamma }$. Thus, $\bigwedge _{1\leq j\leq n} {B_{\star }}\psi _{j} \wedge \bigwedge _{1\leq j\leq n} \Box (\psi _{j}{\rightarrow } \varphi _j)\in {\Gamma }$. Then, by $\mathsf {C}_{{B_{\star }}}$, we obtain that ${B_{\star }}(\bigwedge _{j\leq n}\psi _{j}) \in {\Gamma }$. By $\mathsf {S5}_{\Box }$, we also have $\Box (\bigwedge _{j\leq n}\psi _{j} {\rightarrow } \bigwedge _{j\leq n} \varphi _j)$. Therefore, $\bigwedge {\Phi }\in {\Gamma }[{B_{\star }}]$. □

Lemma 8

Given a mcs Γ, both ${\Gamma }[\Box ]$ and Γ[B_A] are consistent. Moreover, ${\Gamma }[B_{A},\Box ]$ is consistent.

Proof

Consistency of ${\Gamma }[\Box ]$ follows via a standard argument since $\Box$ is an S5 operator, in particular, since $\neg \Box \bot$ is a theorem of L.To show that Γ[B_A] is consistent, assume, toward contradiction, that Γ[B_A] is not consistent, i.e., Γ[B_A] ⊩⊥. This means that there is a finite subset ${\Phi }=\{\varphi _{1}, \dots , \varphi _{n}\}\subseteq {\Gamma }[B_{A}]$ such that $\vdash \bigwedge {\Phi }\supset \neg \varphi _{j}$ for some j ≤ n. By Lemma 7, we have that $\bigwedge {\Phi }\in {\Gamma }[B_{A}]$, thus, there is a $\psi \in {\mathscr{L}}$ such that B_Aψ ∈Γ and $\Box (\psi \supset \bigwedge {\Phi })\in {\Gamma }$. Since $\vdash \bigwedge {\Phi }\supset \neg \varphi _j$, by $\mathsf {S5}_{\Box }$, we also have $\Box (\psi \supset \neg \varphi _{j})\in {\Gamma }$. Hence, ¬φ_j ∈Γ[B_A] too. As φ_j ∈Γ[B_A], we also have a $\psi ^{\prime }\in {\mathscr{L}}$ with $B_{A}\psi ^{\prime }\in {\Gamma }$ and $\Box (\psi ^{\prime }\supset \varphi _{j})\in {\Gamma }$. From $\Box (\psi \supset \neg \varphi _{j})\in {\Gamma }$ and $\Box (\psi ^{\prime }\supset \varphi _{j})\in {\Gamma }$, by $\mathsf {S5}_{\Box }$, we obtain that $\Box (\psi \supset \neg \psi ^{\prime })\in {\Gamma }$. As $B_{A}\psi ^{\prime }\in {\Gamma }$, by $\mathsf {Ax}1_{B_{A}}$ and Lemma 3.2, $B_{A}\overline {\neg \psi ^{\prime }}\in {\Gamma }$. Therefore, $B_{A}\overline {\neg \psi ^{\prime }}\in {\Gamma }$, $\Box (\psi \supset \neg \psi ^{\prime })\in {\Gamma }$, B_Aψ ∈Γ, by $\mathsf {Ax}2_{B_{A}}$, imply that $B_{A}\neg \psi ^{\prime }\in {\Gamma }$, contradicting the consistency of Γ: $B_{A}\psi ^{\prime }\in {\Gamma }$ implies $\neg B_{A}\neg \psi ^{\prime }\in {\Gamma }$, by D_B. Therefore, Γ[B_A] is consistent. Suppose now, toward contradiction, that ${\Gamma }[B_{A},\Box ]$ is inconsistent. Recall that ${\Gamma }[B_{A},\Box ]={\Gamma }[B_{A}]\cup {\Gamma }[\Box ]$ and both ${\Gamma }[\Box ]$ and Γ[B_A] are consistent. Therefore, ${\Gamma }[B_{A},\Box ]$ being inconsistent implies (by Lemma 7) that there is ψ ∈Γ[B_A] and $\phi \in {\Gamma }[\Box ]$ such that $\vdash (\varphi \wedge \psi ) \rightarrow \bot$, i.e., $\vdash \varphi \rightarrow \neg \psi$, while both φ and ψ are consistent. Then, by $\mathsf {S5}_{\Box }$ and since $\Box \varphi \in {\Gamma }$, we obtain that $\Box \neg \psi \in {\Gamma }$. Moreover, ψ ∈Γ[B_A] implies that there is a $\chi \in {\mathscr{L}}$ such that $B_{A}\chi \wedge \Box (\chi \rightarrow \psi )\in {\Gamma }$. This implies that $\Box (\neg \psi \rightarrow \neg \chi )\in {\Gamma }$. Then, by $\mathsf {K}_{\Box }$, we have that $\Box \neg \psi \rightarrow \Box \neg \chi \in {\Gamma }$. Thus, as $\Box \neg \psi \in {\Gamma }$, we obtain that $\Box \neg \chi \in {\Gamma }$. Moreover, by B_Aχ ∈Γ, $\mathsf {Ax}1_{B_{A}}$, and Lemma 3.2, we have $B_{A}\overline {\neg \chi }\in {\Gamma }$. Then, as $\Box \neg \chi \wedge B_{A}\overline {\neg \chi }\in {\Gamma }$, by Lemma 3.1, we have B_A¬χ ∈Γ, contradicting the consistency of Γ: B_Aχ ∈Γ implies ¬B_A¬χ ∈Γ, by $\mathsf {D}_{B_{A}}$. Therefore, ${\Gamma }[B_{A}, \Box ]$ is consistent. □

Lemma 9

Given a mcs Γ₀, the canonical model ${\mathscr{M}}^c=\langle [{\Gamma }_{0}]_{\Box }, \mathcal {O}^{c}, T^{c}, \oplus ^{c}, t^{c}, \nu ^{c} \rangle$ for Γ₀ is a topic-sensitive subset space model.

Proof

Observe that, since Γ₀ is consistent, by axiom Inc, topics $\mathfrak {a}, {\mathfrak {b}_{P}}$, and ${\mathfrak {b}_{A}}$ are mutually disjoint. This implies that the canonical topic assignment function t^c is well-defined. Moreover, both $\mathcal {O}^{c}$ and, for each $O\in \mathcal {O}^{c}$, t^c(O) are finite. Finally, we show that $\mathcal {O}^{c} \not =\{\emptyset \}$: by Lemma 8, we know that ${\Gamma }_{0}[B_{A}, \Box ]$ is consistent. Therefore, by Lemma 5, there is a mcs Δ such that ${\Gamma }_{0}[B_{A}, \Box ]\subseteq {\Delta }.$ Therefore, Δ ∈ O≠∅. □

Lemma 10

For any mcs Γ and $\varphi \in {\mathscr{L}}$, $B_{\star }\overline \varphi \in {\Gamma } \text { iff } B_{\star }\overline {p}\in {\Gamma } \text { for all } p\in \text {Var}\varphi .$

Proof

The direction from left-to-right follows from Lemma 3.2. For the opposite direction, let $\text {Var}\varphi =\{p_1, \dots , p_n\}$ and observe that $\bar {\varphi }:= \bar {p_1}\wedge {\dots } \wedge \bar {p_n}$. If ${B_{\star }}\overline {p_i}\in {\Gamma }$ for all $p_i\in \{p_1, \dots , p_n\}$, then $\bigwedge _{i\leq n}{B_{\star }}\overline {p_i}\in {\Gamma }$ (by Lemma 4.4). Then, by $\mathsf {C}_{{B_{\star }}}$, we obtain that ${B_{\star }}(\bigwedge _{i\leq n}\overline {p_i})\in {\Gamma }$, i.e., ${B_{\star }}\bar {\varphi }\in {\Gamma }$. □

Corollary 9

Given the canonical model ${\mathscr{M}}^c=\langle [{\Gamma }_0]_{\Box }, \mathcal {O}^c, T^c, \oplus ^c, t^c, \nu ^c \rangle$ for Γ₀, for any mcs ${\Gamma }\in [{\Gamma }_0]_{\Box }$ and $\varphi \in {\mathscr{L}}$,

• 1. $B_{A}\overline \varphi \in {\Gamma } \text { iff } t^c(\varphi )\sqsubseteq {\mathfrak {b}_{A}}$, and

• 2. $B_{P}\overline \varphi \in {\Gamma } \text { iff } t^c(\varphi )\sqsubseteq {\mathfrak {b}_{P}}$.

Proof

We prove item (2) only, item (1) follows similarly.

$$ \begin{array}{@{}rcl@{}} B_{P}\overline \varphi\in {\Gamma}\!\!\! && \text { iff } B_{P}\overline{q}\in {\Gamma} \text{ for all } q\in \text{Var} \varphi \qquad\qquad \qquad \qquad \qquad \textup{({Lemma 10})}\\ && \text { iff } B_{P}\overline{q}\in {\Gamma}_{0} \text{ for all } q\in \text{Var}\varphi \qquad\qquad\qquad ({\mathsf{Ax}3_{B_{P}} \textup{and}~ {\Gamma}\in[{\Gamma}_0]_{\Box}})\\ && \text { iff } t^{c}(q)={\mathfrak{b}_{A}} \text{ or } t^{c}(q)={\mathfrak{b}_{P}} \text{ for all } q\in \text{Var}\varphi \\ && \qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad ({\text{by the definitions of}~ {\mathfrak{b}_{P}}, {\mathfrak{b}_{A}}, \textup{and}~ t^{c}})\\ && \text { iff } t^{c}(\varphi)\sqsubseteq {\mathfrak{b}_{P}} ~~~~~({\text{by the definition of}~ (T^{c}, \oplus^{c}) \textup{ and}~ t^{c}(\varphi), \textup{and}~ {\mathfrak{b}_{A}}\sqsubseteq^{c} {\mathfrak{b}_{P}}}) \end{array} $$

□

Lemma 12

For every mcs Γ and $\varphi \in {\mathscr{L}}$, if ${\Gamma }[B_{\star }, \Box ]\vdash \varphi$ and $B_{\star }\overline {\varphi }\in {\Gamma }$, then B_⋆φ ∈Γ.

Proof

Suppose that ${\Gamma }[B_{\star }, \Box ]\vdash \varphi$ and $B_{\star }\overline {\varphi }\in {\Gamma }$. Recall that ${\Gamma }[B_{\star }, \Box ]={\Gamma }[B_{\star }]\cup {\Gamma }[\Box ]$. Then, the first assumption means that there are finite sets ${\Phi }\subseteq {\Gamma }[\Box ]$ and ${\Phi }^{\prime }\subseteq {\Gamma }[{B_{\star }}]$ such that $\vdash (\bigwedge {\Phi }\wedge \bigwedge {\Phi }^{\prime }){\rightarrow } \varphi$. By Lemma 7, we know that $\bigwedge {\Phi }^{\prime } \in {\Gamma }[{B_{\star }}]$. This means that there is a ψ such that ${B_{\star }}\psi \wedge \Box (\psi {\rightarrow } \bigwedge {\Phi }^{\prime })\in {\Gamma }$. Again by Lemma 7, we have $\bigwedge {\Phi }\in {\Gamma }[\Box ]$, i.e., $\Box (\bigwedge {\Phi })\in {\Gamma }$. Then, by $\mathsf {S5}_{\Box }$, we obtain that $\Box (\psi {\rightarrow } \bigwedge {\Phi })\in {\Gamma }$. Therefore, as $\Box (\psi {\rightarrow } \bigwedge {\Phi }^{\prime })\in {\Gamma }$ as well, we have $\Box (\psi {\rightarrow } (\bigwedge {\Phi } \wedge \bigwedge {\Phi }^{\prime }))\in {\Gamma }$. Then, since $\vdash (\bigwedge {\Phi }\wedge \bigwedge {\Phi }^{\prime }){\rightarrow } \varphi$, we have $\Box (\psi {\rightarrow } \varphi )\in {\Gamma }$. Hence, by $\mathsf {Ax}2_{{B_{\star }}}$ together with B_⋆ψ ∈Γ and ${B_{\star }}\overline {\varphi }\in {\Gamma }$, we obtain that B_⋆φ ∈Γ. □

Lemma 13 (Truth Lemma)

Let Γ₀ be a mcs of L and ${\mathscr{M}}^{c}=\langle [{\Gamma }_{0}]_{\Box }, \mathcal {O}^{c}, T^{c}, \oplus ^{c}, t^{c}, \nu ^{c} \rangle$ be the canonical model for Γ₀. Then, for all $\varphi \in {\mathscr{L}}$ and ${\Gamma }\in [{\Gamma }_{0}]_{\Box }$, we have ${\mathscr{M}}^{c}, ({\Gamma }, O_{\mathfrak {b}_{A}})\Vdash \varphi \text { iff } \varphi \in {\Gamma }$.

Proof

First observer that $({\Gamma }, O_{\mathfrak {b}_{A}})$ is a well-defined world-memory pair in ${\mathscr{M}}^{c}$: ${\Gamma }\in [{\Gamma }_{0}]_{\Box }$, O≠∅ (see the proof of Lemma 9), and $\mathfrak {b}_{A}\in t^{c}(O)=\{\mathfrak {b}_{A}\}$ (by the definition of t^c). Due to latter two, we in fact have that $({\Delta }, O_{\mathfrak {b}_{A}})$ is a well-defined world-memory pair of ${\mathscr{M}}^{c}$, for all ${\Delta }\in [{\Gamma }_0]_{\Box }$.

The proof follows by induction on the structure of φ.

Base case: φ := p

$$ \begin{array}{@{}rcl@{}} \mathcal{M}^{c}, ({\Gamma}, O_{\mathfrak{b}_{A}})\Vdash p && \ \text{iff} \ {\Gamma}\in \nu^c(p) \qquad({\textup{by the semantics}})\\ && \ \text{iff} \ p\in{\Gamma} \qquad~~ ({\textup{by the definition of $\nu^c$}}) \end{array} $$

The cases for the Booleans are standard. We here prove the cases $\varphi :=\Box \psi$ and φ := B_⋆ψ.

Case $\varphi :=\Box \psi$

(⇐) Suppose that $\Box \psi \in {\Gamma }$ and let ${\Delta }\in [{\Gamma }_0]_{\Box }$. The latter implies that ${\Delta }\sim _{\Box } {\Gamma }$. Therefore, ψ ∈Δ. Then, by the induction hypothesis (IH), we have ${\mathscr{M}}^{c}, ({\Delta }, O_{\mathfrak {b}_{A}})\Vdash \psi$. As Δ has been chosen arbitrarily from $[{\Gamma }_0]_{\Box }$, we conclude that ${\mathscr{M}}^{c}, ({\Gamma }, O_{\mathfrak {b}_{A}})\Vdash \Box \psi$.

(⇒) Suppose that ${\mathscr{M}}^{c}, ({\Gamma }, O_{\mathfrak {b}_{A}})\Vdash \Box \psi$ and, toward contradiction, that $\Box \psi \not \in {\Gamma }$. The latter implies that ${\Gamma }[\Box ]\cup \{\neg \psi \}$ is consistent. Therefore, by Lemma 5, there is a mcs Δ such that ${\Gamma }[\Box ]\cup \{\neg \psi \}\subseteq {\Delta }$. Observe, by Lemma 6, that ${\Delta }\in [{\Gamma }_0]_{\Box }$ (since ${\Gamma }_{0}[\Box ]={\Gamma }[\Box ]$). And, as ¬ψ ∈Δ, we also obtain that ψ∉Δ. Then, by IH, we have that ${\mathscr{M}}^{c}, ({\Delta }, O_{\mathfrak {b}_{A}})\not \Vdash \psi$, contradicting the initial assumption ${\mathscr{M}}^{c}, ({\Gamma }, O_{\mathfrak {b}_{A}})\Vdash \Box \psi$. Therefore, $\Box \psi \in {\Gamma }$.

Case φ := B_Pψ

(⇐) Suppose that B_Pψ ∈Γ. Then, by $\mathsf {Ax}1_{B_{P}}$, we also have that $B_{P}\overline {\psi }\in {\Gamma }$. This implies, by Corollary 11.2, that $t^{c}(\psi )\sqsubseteq ^{c}\mathfrak {b}_{P}$. Observe that $\oplus ^c(\bigcup _{O\in \mathcal {O}^{c}} t^c(O))=\oplus ^{c}\{\mathfrak {b}_{P}, \mathfrak {b}_{A}\}= \mathfrak {b}_{P}\oplus ^{c}\mathfrak {b}_{A}=\mathfrak {b}_{P}$, so we satisfy the topicality component of the semantic clause for B_Pψ. In order to complete the proof, we need to show that $\bigcap \mathcal {O}^{c} \subseteq [\![{\psi }]\!]^{O}_{\mathfrak {b}_{A}}$. As stated before, $\bigcap \mathcal {O}^c=O^{\cap }$. So, we need to show that $\{\Delta \in \mathcal {X}^{c} : {\Gamma }_0[{B_{P}},\Box ]\subseteq {\Delta } \}\subseteq [\![{\psi }]\!]^{O_{\mathfrak {b}_{A}}}$. Let Σ ∈ O^∩. This means that ${\Gamma }_0[{B_{P}},\Box ]\subseteq {\Sigma }$. As B_Pψ ∈Γ, by $\mathsf {Ax}3_{{B_{P}}}$ and the fact that ${\Gamma } \sim _{\Box } {\Gamma }_0$, we also have B_Pψ ∈Γ₀. Moreover, $\Box (\psi \rightarrow \psi )\in {\Gamma }_0$ (by $\mathsf {S5}_{\Box }$). We then have that ψ ∈Γ₀[B_P], thus, $\psi \in {\Gamma }_0[{B_{P}},\Box ]$. Hence, ψ ∈Σ. Then, by IH, we have ${\mathscr{M}}^{c}, ({\Sigma }, O_{\mathfrak {b}_{A}})\Vdash \psi$, i.e., that ${\Sigma }\in [\![{\psi }]\!]^{O_{\mathfrak {b}_{A}}}$. As Σ has been chosen arbitrarily from O^∩, we conclude that $O^{\cap } \subseteq [\![{\psi }]\!]^{O_{\mathfrak {b}_{A}}}$. Since $t^c(\psi )\sqsubseteq ^c\mathfrak {b}_{P}$ as well, we obtain that ${\mathscr{M}}^{c}, ({\Gamma }, O_{\mathfrak {b}_{A}})\Vdash {B_{P}}\psi$.

(⇒) Suppose that ${\mathscr{M}}^{c}, ({\Gamma }, O_{{\mathfrak {b}_{A}}})\Vdash B_{P}\psi$, i.e., that $\bigcap \mathcal {O}^c=O^{\cap }=\{\Delta \in \mathcal {X}^c : {\Gamma }_0[B_{P},\Box ]\subseteq {\Delta }\}\subseteq [\![ \psi ]\!]^{O_{{\mathfrak {b}_{A}}}}$ and $t^c(\psi )\sqsubseteq ^c\oplus ^c(\bigcup _{O\in \mathcal {O}^c} t^c(O))={\mathfrak {b}_{P}}$. By Corollary 11.2, the latter means that ${B_{P}}\overline {\psi }\in {\Gamma }$. Then, by the former and the IH, we have that whenever ${\Gamma }_0[{B_{P}},\Box ]\subseteq {\Delta }$, then ψ ∈Δ. This implies that ${\Gamma }_0[{B_{P}},\Box ]\vdash \psi$. Otherwise, ${\Gamma }_0[{B_{P}},\Box ]\cup \{\neg \psi \}$ would be consistent, thus, by Lemma 5, there would exist a mcs ${\Delta }^{\prime }$ such that ${\Gamma }_0[{B_{P}},\Box ]\cup \{\neg \psi \}\subseteq {\Delta }^{\prime }$, contradicting the fact that if ${\Gamma }_0[{B_{P}},\Box ]\subseteq {\Delta }$ then ψ ∈Δ. By Lemma 6, ${\Gamma }_0[{B_{P}},\Box ]\vdash \psi$ means ${\Gamma }[{B_{P}},\Box ]\vdash \psi$. Since ${B_{P}}\overline {\psi }\in {\Gamma }$, by Lemma 12, we obtain that B_Pψ ∈Γ.

Case φ := B_Aψ: Follows similarly to the proof of case φ := B_Pψ, using Corollary 11.1. □

Corollary 12

L is complete with respect to the class of topic-sensitive subset space models.

Proof

Let $\varphi \in {\mathscr{L}}$ such that ⊯φ. This mean that {¬φ} is consistent. Then, by Lindenbaum’s Lemma (Lemma 5), there exists a mcs Γ₀ such that φ∉Γ₀. Therefore, by Lemma 13, we conclude that ${\mathscr{M}}^c, ({{\Gamma }_0}, O_{\mathfrak {b}_{A}}) \not \Vdash \varphi$, where ${\mathscr{M}}^c$ is the canonical model for Γ₀. □