Dynamic epistemic logic of belief change in legal judgments

Abstract

This study realizes belief/reliability change of a judge in a legal judgment by dynamic epistemic logic (DEL). A key feature of DEL is that possibilities in an agent’s belief can be represented by a Kripke model. This study addresses two difficulties in applying DEL to a legal case. First, since there are several methods for constructing a Kripke model, our question is how we can construct the model from a legal case. Second, since this study employs several dynamic operators, our question is how we can decide which operators are to be applied for belief/reliability change of a judge. In order to solve these difficulties, we have implemented a computer system which provides two functions. First, the system can generate a Kripke model from a legal case. Second, the system provides an inconsistency solving algorithm which can automatically perform several operations in order to reduce the effort needed to decide which operators are to be applied. By our implementation, the above questions can be adequately solved. With our analysis method, six legal cases are analyzed to demonstrate our implementation.

Appendices

Appendix 1: Details of target legal cases

The story and the judgment of six target legal cases can be summarized as follows:

• 1. Legal case from Thailand [in Jirakunkanok et al. (2014)] occurred on January 26, 2003 in Trang province, Thailand.¹¹ The story can be summarized as follows:

  One day, Choochart (v) had a drink with his friends including Saichol ($f_1$), Ekachai ($f_2$), and Sommai (d) at $f_2$’s house. After that, v was punched and stabbed with a hand scraper in the back by an offender, and as a result, v had bleeding in the lung. However, v was still alive.

  The details of judgment can be summarized as follows:

  In the inquiry stage, four witnesses v, $f_1$, $f_2$, and mo (mother of v) were interviewed by a police po, who is an official inquiry, as follows: v, $f_1$, and mo told that d was the offender, while $f_2$ told that d was not the offender. After the interview, d was charged with attempted murder. In the Civil Court, v and $f_1$ changed their statements, i.e., both of them told that d was not the offender. po was called to be a witness for testifying all statements in the inquiry stage. From these testimonies, the judge believed that the statements of v and $f_1$ in the Civil Court are less reliable than that in the inquiry stage. Thus, the judge believed that d was the offender and decided that d was guilty.

• 2. Legal case from Canada [in Jirakunkanok et al. (2015a)] occurred on April 24, 1988 in Ontario, Canada.¹² The story can be summarized as follows:

  One day, Joseph (v) and his brother, Steven (b), got off a bus at an intersection. At the same time, the respondent, K.G.B. (d), and three other men including P.L. ($f_1$), P.M. ($f_2$), and M.T. ($f_3$) were driving past the same intersection. An argument started among them and shortly thereafter a fight occurred. v and b were unarmed. In the course of the fight, one of the four men from the car pulled a knife and then stabbed v in the chest. Finally, v died.

  The details of judgment can be summarized as follows:

  In the inquiry stage, four witnesses b, $f_1$, $f_2$, and $f_3$ were interviewed as follows: b told that d was not the offender, while $f_1$, $f_2$, and $f_3$ told that d was the offender. After the interview, d was charged with murder. In the Youth Court, since all witnesses recanted their statements, the judge could not consider the prior statements of all witnesses as evidence. Thus, the judge acquitted d.

• 3. Legal case from British Columbia [in Jirakunkanok et al. (2015b)] occurred on August 31, 1996 in Surrey, British Columbia.$^{12}$ The story can be summarized as follows:

  One day, while Basant Singh (v) with new friends including Sher ($f_1$), Jarnail ($f_2$), and the others gathered for social purposes, a van consisting of two respondents Sukhminder ($d_1$), Ajmer ($d_2$), and the others slowly approached the group of v. A burst of gun fire swept the group of v, shooting on a low trajectory into the ground. As a result, v died and three others including $f_2$ were wounded.

  The details of judgment can be summarized as follows:

  In the inquiry stage, two witnesses $f_1$ and $f_2$ were interviewed as follows: $f_1$ told that $d_1$ was a driver of the van and $d_2$ was the shooter, while $f_2$ told that $d_1$ was a driver of the van and $d_2$ was not the shooter. After the interview, both $d_1$ and $d_2$ were charged with the first degree murder, the attempted murder of three other persons, and aggravated assault on the same three persons. In the Crown Court, $f_2$ changed his statement, i.e., he told that $d_2$ was the shooter. Since there is an inconsistency in the statements of $f_2$, the judge considered only $f_1$’s statement for identifying the shooter to be truthful. In addition, the judge believed that both $d_1$ and $d_2$ did not intend to kill v. For this reason, the judge acquitted both $d_1$ and $d_2$ of first degree murder but convicted them of manslaughter. The judge also convicted them of aggravated assault instead of attempted murder of the other three victims.

• 4. Legal case from Nova Scotia occurred on December 31, 2009 in Halifax, Nova Scotia.$^{12}$ The story can be summarized as follows:

  One day, Welsh (v) went to a New Year’s Eve Party with his girlfriend Gautreau (f). While f was drinking in the party, v went outside the party to have a cigarette. Later on, f went outside and found v was punched then fell backward and struck his head on the pavement. Finally, v died.

  The details of judgment can be summarized as follows:

  In the inquiry stage, only one witness f was interviewed as follows: f told that Leeds (d) was the offender. After the interview, d was charged with manslaughter. In the Crown Court, the judge found that f’s recollection of the event was affected by her alcohol assumption, and there were many inconsistencies in f’s evidence such as the identification of d as the offender. Thus, the judge believed that f was not a reliable witness. Accordingly, the judge decided that d was not guilty.

• 5. Legal case from Nova Scotia occurred on August 6, 2011 in Halifax, Nova Scotia.$^{12}$ The story can be summarized as follows:

  One day, Barry (v) and his friends including Fisher ($f_1$), Marsh ($f_2$), and Slaunwhite ($f_3$) were drinking alcohol and smoking marijuana at v’s home. Then, v together with his friends $f_1$, $f_2$, and $f_3$ drove to the house of Neil (d). While v was driving the vehicle at d, d was scared and fired the shot that injured v. However, v was still alive.

  The details of judgment can be summarized as follows:

  In the inquiry stage, four witnesses v, $f_1$, $f_2$, and Beaupre (b) who was d’s neighbor were interviewed as follows: v and $f_1$ told that d intended to kill v, while $f_2$ and b told that d did not intend to kill v. After the interview, d was charged with the following offences: attempted murder, aggravated assault, using of a weapon in committing an assault, discharging a firearm with intent to endanger the life, intentionally discharging a firearm into a place, using of a firearm in a careless manner, and possessing a weapon for a purpose dangerous to the public peace. In the Crown Court, the judge considered v and $f_1$ to be unreliable because v could not recall the events because of a combination of his intoxication by both drugs and alcohol on the evening in the events, and $f_1$’s evidence was inconsistent within itself. Thus, the judge only accepted the evidence from $f_2$ and b that d did not intend to kill v; in fact, d just defended himself against v’s attack. That is, d’s actions were justified to be self-defense. Therefore, the judge decided that d was not guilty of all counts in the indictment.

• 6. Legal case from Nova Scotia occurred on July 17, 2004 in Bedford, Nova Scotia.$^{12}$ The story can be summarized as follows:

  One day, Bobby (v) was intoxicated at Busters Bar and having been denied further drinks from the bar. While Comer ($d_1$) and his friends including Warner ($f_1$), Maes ($f_2$), Southwell ($f_3$), and Morrison ($f_4$) were drinking, v approached $d_1$’s table and asked for some beer, but his request was refused. Then, v attempted to take $d_1$’s beer, but his attempt was prevented from $f_1$. After that, Smith ($d_2$) and his friend, Hodgson ($f_5$), arrived at the bar and joined the group at $d_1$’s table. v left the bar first, then $d_1$, $d_2$, and $f_5$ left the bar. When $f_1$, $f_2$, $f_3$, and $f_4$ exited the bar, they came upon a verbal exchange between v, $d_1$, $d_2$, and $f_5$. Then, v kicked $d_2$ first, then all three including v, $d_1$, and $d_2$ were punching each other. The fight was of short duration. After v fell to the ground, $d_1$, $d_2$, and $f_5$ ran off.

  The details of judgment can be summarized as follows:

  In the inquiry stage, five witnesses $f_1$, $f_2$, $f_3$, $f_4$, and $f_5$ were interviewed as follows: $f_1$ and $f_2$ told that they could not see what happened when v was on the ground, but $f_2$ stated that he saw $d_2$ kicked v once above the belt. $f_3$ told that he did not see anyone kick v while v was on the ground. $f_4$ told that v kicked $d_2$ first, then $d_2$ kicked v while v was on the ground. However, $f_4$ was not sure if $d_2$’s kick was to v’s head or not. $f_5$ told that he could not say where $d_1$’s kick landed on v. After the interview, both $d_1$ and $d_2$ were charged with manslaughter in the death of v. In the Crown Court, three witnesses $f_1$, $f_2$, and $f_5$ changed their statements as follows: $f_1$ testified that $d_1$ and $d_2$ kicked v while v was on the ground, and all the kicks he saw landed on v’s upper body between the belt and the head. $f_2$ told that he could not say if anyone kicked v while v was on the ground because people were in front of him and blocking his view. $f_5$ told that v kicked $d_2$ first, then $d_1$ kicked v in the head while v was on the ground. The judge found that the reliability of evidence of all witnesses was questionable because of the following reasons: $f_1$ and $f_5$ gave inconsistent statements, $f_2$’s view of the events was affected by the fact that he was not wearing his eyeglasses, and $f_3$ and $f_4$ turned away from the fight. Based on these reasons, the judge was not satisfied on the evidence that $d_2$ kicked v while v was on the ground. Thus, the judge believed that $d_2$’s act was in self-defense and was not excessive. Accordingly, $d_2$ was found not guilty. On the other hand, the judge believed that the kicking of $d_1$ was not in self-defense and was excessive because of the evidence that $d_1$ kicked v while v was on the ground. However, the judge cannot conclude that the kicking of $d_1$ was the cause of v’s death because there is no evidence to support a finding that $d_1$ kicked v in the head. Accordingly, $d_2$ was found not guilty.

Appendix 2: Omitted proofs

1.1 Appendix 2.1: Proof of Theorem 1

Proof

Since the soundness is straightforward, we will focus on the completeness proof. Let us write our axiomatization $\mathbf {HBSR}$ in Table 2 by $\varLambda$. We show that any unprovable formula $\varphi$ in $\varLambda$ is falsified in some model. Let $\varphi$ be an unprovable formula in $\varLambda$. We define the canonical model ${\mathfrak {M}}$ where $\varphi$ is falsified at some point of ${\mathfrak {M}}$. We say that a set $\varGamma$ of formulas is $\varLambda$-consistent (for short, consistent) if $\lnot (\bigwedge \varGamma ')$ is unprovable in $\varLambda$, for all finite subsets $\varGamma '$ of $\varGamma$, and that $\varGamma$ is maximally consistent, denoted $\varLambda$-MCS if $\varGamma$ is consistent and $\varphi \in \varGamma$ or $\lnot \varphi \in \varGamma$ for all formulas $\varphi$. Note that $\psi$ is unprovable in $\varLambda$ iff $\lnot \psi$ is $\varLambda$-consistent, for any formula $\psi$. Let $\varSigma$ be a $\varLambda$-MCS. We define the canonical model ${\mathfrak {M}}_{\varSigma }^\varLambda = ({W^\varLambda ,(R^\varLambda _{a})_{a \in G}, (S^\varLambda _{a})_{a \in G}, (\preccurlyeq ^\varLambda _{a})_{a \in G}, V^\varLambda }),$ for $\varLambda$ by:

• $W^\varLambda$ := $\{ \varGamma | \varGamma$ is a $\varLambda$-MCS and $\{ \varphi | {\mathsf {A}}\varphi \in \varSigma \} \subseteq \varGamma \}$;

• $\varGamma R^\varLambda _{a} \varDelta$ iff ($\mathsf {Bel}(a,\psi ) \in \varGamma$ implies $\psi \in \varDelta$) for all $\psi$;

• $\varGamma S^\varLambda _{a} \varDelta$ iff ($\mathsf {Sign}(a,\psi ) \in \varGamma$ implies $\psi \in \varDelta$) for all $\psi$;

• $b\, \preccurlyeq _{a}^{\varGamma } c$ iff $b \leqslant _a c \in \varGamma$;

• $\varGamma \in V^\varLambda (p)$ iff $p \in \varGamma$.

Then, we can show the following equivalence : ${\mathfrak {M}}_{\varSigma }^\varLambda ,\varGamma \models \psi$ iff $\psi \in \varGamma$ for all formulas $\psi$ and $\varGamma \in W$. Given any unprovable formula $\varphi$ in $\varLambda$, we can find a maximal consistent set $\varSigma$ such that $\lnot \varphi \in \varSigma$. Then, by the equivalence above, $\varphi$ is falsified at $\varSigma$ of the canonical model ${\mathfrak {M}}_{\varSigma }^\varLambda$ for $\varLambda$, where we can assure that ${\mathfrak {M}}_{\varSigma }^\varLambda$ is our intended model. $\square$

1.2 Appendix 2.2: Proof of Proposition 1

Proof

Our goal is to show that all axioms are valid with respect to the semantics of ${[}\mathbb {E}_{!^a_{\varphi }},e{]}$ that is straightforward. We will show only the most important axiom, i.e.,

$$\begin{aligned} {[}\mathbb {E}_{!^a_{\varphi }},e{]} \mathsf {Bel}(a,\psi ) \leftrightarrow \bigwedge _{f \in D_{a}(e)} \mathsf {Bel}\big (a, \mathrm {pre}(f) \rightarrow [\mathbb {E}_{!^a_{\varphi }},f]\psi \big ). \end{aligned}$$

Let us fix any model ${\mathfrak {M}}$ and any state w of ${\mathfrak {M}}$. We suffice to show:

$$\begin{aligned} {\mathfrak {M}},w \models {[}\mathbb {E}_{!^a_{\varphi }},e{]} \mathsf {Bel}(a,\psi ) \,\,{\mathrm {iff}}\,\,{\mathfrak {M}},w \models \bigwedge _{f \in D_{a}(e)} \mathsf {Bel}\big (a, \mathrm {pre}(f) \rightarrow [\mathbb {E}_{!^a_{\varphi }},f]\psi \big ). \end{aligned}$$

The left-hand-side is equivalent to:

$$\begin{aligned}&{\mathfrak {M}}^{\otimes \mathbb {E}_{!^a_{\varphi }}}, (w,e) \models \mathsf {Bel}(a,\psi ) \\&\,\,{\mathrm {iff}}\,\,\forall (v,f)\left( (w,e) R_{a}' (v,f) \Rightarrow {\mathfrak {M}}^{\otimes \mathbb {E}_{!^a_{\varphi }}}, (v,f) \models \psi \right) \\&\,\,{\mathrm {iff}}\,\,\forall (v,f)\left( (wR_a v\,\text { and }\,(e,f) \in D_a \,\text { and } \,{\mathfrak {M}},v \models \mathrm {pre}(f)) \Rightarrow {\mathfrak {M}},v \models [\mathbb {E}_{!^a_{\varphi }},f]\psi \right) \\&\,\,{\mathrm {iff}}\,\,\forall f \left( (e,f) \in D_{a} \Rightarrow \forall v \left( (wR_{a}v\, \text { and }\, {\mathfrak {M}},v \models \mathrm {pre}(f)) \Rightarrow {\mathfrak {M}},v \models [\mathbb {E}_{!^a_{\varphi }},f]\psi \right) \right) \\&\,\,{\mathrm {iff}}\,\,\forall f \left( (e,f) \in D_{a} \Rightarrow \forall v \left( wR_{a}v \Rightarrow ({\mathfrak {M}},v \models \mathrm {pre}(f) \Rightarrow {\mathfrak {M}},v \models [\mathbb {E}_{!^a_{\varphi }},f]\psi )\right) \right) \\&\,\,{\mathrm {iff}}\,\,\forall f \left( (e,f) \in D_{a} \Rightarrow \forall v \left( wR_{a}v \Rightarrow ({\mathfrak {M}},v \models \mathrm {pre}(f) \rightarrow [\mathbb {E}_{!^a_{\varphi }},f]\psi )\right) \right) \\&\,\,{\mathrm {iff}}\,\,\forall f \left( (e,f) \in D_{a} \Rightarrow {\mathfrak {M}},w \models \mathsf {Bel}(a,\mathrm {pre}(f) \rightarrow [\mathbb {E}_{!^a_{\varphi }},f]\psi ) \right) \\&\,\,{\mathrm {iff}}\,\,{\mathfrak {M}},w \models \bigwedge _{f \in D_{a}(e)} \mathsf {Bel}(a,\mathrm {pre}(f) \rightarrow [\mathbb {E}_{!^a_{\varphi }},f]\psi ). \end{aligned}$$

1.3 Appendix 2.3: Proof of Proposition 2

Proof

Our goal is to show that all axioms are valid with respect to the semantics of ${[}\mathbb {E}_{{{\textexclamdown }}^a_{\varphi }},e{]}$ that is straightforward. We will show only the most important axiom, i.e.,

$$\begin{aligned} {[}\mathbb {E}_{{{\textexclamdown }}^a_{\varphi }},e{]} \mathsf {Bel}(a,\psi ) \leftrightarrow \bigwedge _{f \in D_{a}(e)} \big ( \mathsf {Bel}(a, [\mathbb {E}_{{{\textexclamdown }}^a_{\varphi }},f]\psi ) \wedge {\mathsf {A}}(\mathrm {pre}(f) \rightarrow [\mathbb {E}_{{{\textexclamdown }}^a_{\varphi }},f]\psi ) \big ). \end{aligned}$$

Let us fix any model ${\mathfrak {M}}$ and any state w of ${\mathfrak {M}}$. We suffice to show:

$$\begin{aligned} {\mathfrak {M}},w \models {[}\mathbb {E}_{{{\textexclamdown }}^a_{\varphi }},e{]} \mathsf {Bel}(a,\psi ) \,\,{\mathrm {iff}}\,\,{\mathfrak {M}},w \models \bigwedge _{f \in D_{a}(e)} \big ( \mathsf {Bel}(a, [\mathbb {E}_{{{\textexclamdown }}^a_{\varphi }},f]\psi ) \wedge {\mathsf {A}}(\mathrm {pre}(f) \rightarrow [\mathbb {E}_{{{\textexclamdown }}^a_{\varphi }},f]\psi ) \big ). \end{aligned}$$

The left-hand-side is equivalent to:

$$\begin{aligned}&{\mathfrak {M}}^{\otimes \mathbb {E}_{{{\textexclamdown }}^a_{\varphi }}}, (w,e) \models \mathsf {Bel}(a,\psi ) \\&\,\,{\mathrm {iff}}\,\,\forall (v,f)\left( (w,e) R_{a}' (v,f) \Rightarrow {\mathfrak {M}}^{\otimes \mathbb {E}_{{{\textexclamdown }}^a_{\varphi }}}, (v,f) \models \psi \right) \\&\,\,{\mathrm {iff}}\,\,\forall (v,f)\left( \big ((wR_a v\,\text { and }\,(e,f) \in D_a) \text { or } ({\mathfrak {M}},v \models \mathrm {pre}(f)\,\text { and }\,(e,f) \in D_a) \big ) \Rightarrow {\mathfrak {M}},v \models [\mathbb {E}_{{{\textexclamdown }}^a_{\varphi }},f]\psi \right) \\&\,\,{\mathrm {iff}}\,\,\forall (v,f) \begin{pmatrix} \left( (wR_a v\,\text { and }\,(e,f) \in D_a) \Rightarrow {\mathfrak {M}},v \models [\mathbb {E}_{{{\textexclamdown }}^a_{\varphi }},f]\psi \right)\,\text { and }\,\\ \left( ({\mathfrak {M}},v \models \mathrm {pre}(f)\,\text { and }\,(e,f) \in D_a) \Rightarrow {\mathfrak {M}},v \models [\mathbb {E}_{{{\textexclamdown }}^a_{\varphi }},f]\psi \right) \end{pmatrix}\\&\,\,{\mathrm {iff}}\,\,\forall (v,f) \begin{pmatrix} \left( (e,f) \in D_a \Rightarrow (wR_a v \Rightarrow {\mathfrak {M}},v \models [\mathbb {E}_{{{\textexclamdown }}^a_{\varphi }},f]\psi )\right)\,\text { and }\,\\ \left( (e,f) \in D_a \Rightarrow ({\mathfrak {M}},v \models \mathrm {pre}(f) \Rightarrow {\mathfrak {M}},v \models [\mathbb {E}_{{{\textexclamdown }}^a_{\varphi }},f]\psi ) \right) \end{pmatrix}\\&\,\,{\mathrm {iff}}\,\,\forall (v,f) \begin{pmatrix} \left( (e,f) \in D_a \Rightarrow (wR_a v \Rightarrow {\mathfrak {M}},v \models [\mathbb {E}_{{{\textexclamdown }}^a_{\varphi }},f]\psi )\right)\,\text { and }\,\\ \left( (e,f) \in D_a \Rightarrow ({\mathfrak {M}},v \models \mathrm {pre}(f) \rightarrow [\mathbb {E}_{{{\textexclamdown }}^a_{\varphi }},f]\psi ) \right) \end{pmatrix}\\&\,\,{\mathrm {iff}}\,\,\forall (v,f) \left( (e,f) \in D_a \Rightarrow \left( (wR_a v \Rightarrow {\mathfrak {M}},v \models [\mathbb {E}_{{{\textexclamdown }}^a_{\varphi }},f]\psi )\,\text { and }\,({\mathfrak {M}},v \models \mathrm {pre}(f) \rightarrow [\mathbb {E}_{{{\textexclamdown }}^a_{\varphi }},f]\psi ) \right) \right) \\&\,\,{\mathrm {iff}}\,\,\forall f \left( (e,f) \in D_a \Rightarrow \left( \forall v (wR_a v \Rightarrow {\mathfrak {M}},v \models [\mathbb {E}_{{{\textexclamdown }}^a_{\varphi }},f]\psi )\,\text { and }\,\forall v ({\mathfrak {M}},v \models \mathrm {pre}(f) \rightarrow [\mathbb {E}_{{{\textexclamdown }}^a_{\varphi }},f]\psi ) \right) \right) \\&\,\,{\mathrm {iff}}\,\,\forall f \left( (e,f) \in D_a \Rightarrow \left( {\mathfrak {M}},w \models \mathsf {Bel}(a,[\mathbb {E}_{{{\textexclamdown }}^a_{\varphi }},f]\psi )\,\text { and }\,{\mathfrak {M}},w \models {\mathsf {A}}(\mathrm {pre}(f) \rightarrow [\mathbb {E}_{{{\textexclamdown }}^a_{\varphi }},f]\psi ) \right) \right) \\&\,\,{\mathrm {iff}}\,\,\forall f \left( (e,f) \in D_a \Rightarrow \left( {\mathfrak {M}},w \models \mathsf {Bel}(a,[\mathbb {E}_{{{\textexclamdown }}^a_{\varphi }},f]\psi ) \wedge {\mathsf {A}}(\mathrm {pre}(f) \rightarrow [\mathbb {E}_{{{\textexclamdown }}^a_{\varphi }},f]\psi ) \right) \right) \\&\,\,{\mathrm {iff}}\,\,{\mathfrak {M}},w \models \bigwedge _{f \in D_{a}(e)} \left( \mathsf {Bel}(a, [\mathbb {E}_{{{\textexclamdown }}^a_{\varphi }},f]\psi ) \wedge {\mathsf {A}}(\mathrm {pre}(f) \rightarrow [\mathbb {E}_{{{\textexclamdown }}^a_{\varphi }},f]\psi ) \right) . \end{aligned}$$

1.4 Appendix 2.4: Proof of Theorem 2

Proof

By $\vdash \psi$ (or $\vdash ^{+} \psi$), we mean that $\psi$ is a theorem of the axiomatization for $\mathcal {L}_{BSR}$ (or, $\mathcal {L}_{BSR}^{+}$, respectively.) The soundness part is mainly due to Propositions 1, 2, 3, 4 and 5. One can also check that the necessitation rules for ${[}\mathbb {E}_{!^a_{\varphi }},e{]}$, ${[}\mathbb {E}_{{{\textexclamdown }}^a_{\varphi }},e{]}$, $[\mathop {H\Downarrow ^a_{\varphi }}]$, $[\mathop {H\Uparrow ^a_{\varphi }}]$ and $[ \mathop {H\,\downdownarrows ^a} ]$ preserve the validity on the class of all models. As for the completeness part, we can reduce the completeness of our dynamic extension to the static counterpart (i.e., Theorem 1) as follows. With the help of the reduction axioms of Propositions 1, 2, 3, 4 and 5, we can define a mapping t sending a formula $\psi$ of $\mathcal {L}_{BSR}^{+}$ to a formula $t(\psi )$ of $\mathcal {L}_{BSR}$, where we start rewriting the innermost occurrences of ${[}\mathbb {E}_{!^a_{\varphi }},e{]}$, ${[}\mathbb {E}_{{{\textexclamdown }}^a_{\varphi }},e{]}$, $[\mathop {H\Downarrow ^a_{\varphi }}]$, $[\mathop {H\Uparrow ^a_{\varphi }}]$ and $[ \mathop {H\,\downdownarrows ^a} ]$. We can define this mapping t such that $\psi \leftrightarrow t(\psi )$ is valid on all models and $\vdash ^{+} \psi \leftrightarrow t(\psi )$. Then, we can proceed as follows. Fix any formula $\psi$ of $\mathcal {L}_{BSR}^{+}$ such that $\psi$ is valid on all models. By the validity of $\psi \leftrightarrow t(\psi )$ on all models, we obtain $t(\psi )$ is valid on all models. By Theorem 1, $\vdash t(\psi )$, which implies $\vdash ^{+} t(\psi )$. Finally, it follows from $\vdash ^{+} \psi \leftrightarrow t(\psi )$ that $\vdash ^{+} \psi$, as desired. $\square$