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 mattersensitivity 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.
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Acknowledgements
This research is published within the project ‘The Logic of Conceivability’, funded by the European Research Council (ERC CoG), grant number 681404. We thank the anonymous reviewers of the Journal of Philosophical Logic for their valuable comments. Versions of this paper were presented at the Cognitive Lunch, Indiana University, April 7, 2021; the workshop ‘Beyond the Impossible’, University of Padua, October 22, 2021; the London Group for Formal Philosophy (UCL/KCL/BBK), November 18, 2021; the Department of Philosophy, University of Frankfurt, November 30, 2021; the Moral Science Club, University of Cambridge, January 25, 2022; the Department of Philosophy, Kansas State University, March 25, 2022. Thanks to the audiences for helpful discussion.
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Appendix A: Proofs
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_{P}ψ. Observe that the inductive hypothesis says that \([\![{\psi }]\!]^{O_a}_{{\mathscr{M}}}=[\![{\psi }]\!]^{U_{c}}_{{\mathscr{M}}}\). We then have
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.
\(\mathsf {C}_{B_{A}}\):
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:
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 Lconsistent if Γ⊯⊥, and Linconsistent otherwise. A sentence φ is Lconsistent with Γ if Γ ∪{φ} is Lconsistent (or, equivalently, if Γ⊯¬φ). Finally, a set of formulas Γ is a maximallyLconsistent set (or, in short, mcs) if it is Lconsistent and any set of formulas properly containing Γ is Linconsistent [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 Lconsistent set can be extended to a maximally Lconsistent one.
Proof
Standard. □
1.2.4 Canonical Model of \({\mathscr{L}}\) for a mcs Γ_{0}:
Let \(\mathcal {X}^{c}\) be the set of all maximally Lconsistent sets. For each \({\Gamma }\in \mathcal {X}^{c}\), define
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
Since \(\Box \) is an S5 modality, it is easy to see that \(\sim _{\Box }\) is an equivalence relation. For any maximally Lconsistent 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 Γ_{0} of L, the canonical model for Γ_{0} 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).

\(t^{c}: {\mathscr{L}}\cup \mathcal {O}^{c} \rightarrow T^{c}\cup \mathcal {P}(T^{c})\) 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) = ⊕^{c}Varφ. 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 Γ_{0} is a topicsensitive 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_{A}ψ ∈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_{A}ψ ∈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_{A}χ ∈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_{A}χ ∈Γ implies ¬B_{A}¬χ ∈Γ, by \(\mathsf {D}_{B_{A}}\). Therefore, \({\Gamma }[B_{A}, \Box ]\) is consistent. □
Lemma 9
Given a mcs Γ_{0}, the canonical model \({\mathscr{M}}^c=\langle [{\Gamma }_{0}]_{\Box }, \mathcal {O}^{c}, T^{c}, \oplus ^{c}, t^{c}, \nu ^{c} \rangle \) for Γ_{0} is a topicsensitive subset space model.
Proof
Observe that, since Γ_{0} 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 welldefined. 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 lefttoright 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 Γ_{0}, 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.
□
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 Γ_{0} 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 Γ_{0}. 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 welldefined worldmemory 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 welldefined worldmemory 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
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_{P}ψ
(⇐) Suppose that B_{P}ψ ∈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_{P}ψ. 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_{P}ψ ∈Γ, by \(\mathsf {Ax}3_{{B_{P}}}\) and the fact that \({\Gamma } \sim _{\Box } {\Gamma }_0\), we also have B_{P}ψ ∈Γ_{0}. Moreover, \(\Box (\psi \rightarrow \psi )\in {\Gamma }_0\) (by \(\mathsf {S5}_{\Box }\)). We then have that ψ ∈Γ_{0}[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_{P}ψ ∈Γ.
Case φ := B_{A}ψ: Follows similarly to the proof of case φ := B_{P}ψ, using Corollary 11.1. □
Corollary 12
L is complete with respect to the class of topicsensitive 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 Γ_{0} such that φ∉Γ_{0}. 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 Γ_{0}. □
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Berto, F., Özgün, A. The Logic of Framing Effects. J Philos Logic (2023). https://doi.org/10.1007/s10992022096940
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DOI: https://doi.org/10.1007/s10992022096940
Keywords
 Aboutness
 Subject matter
 Framing
 Hyperintensionality
 Logical omniscience
 Knowledge representation
 Working memory
 Long term memory