A Stit Logic of Intentionality

Abstract

We extend epistemic stit theory with a modality $I_\alpha \varphi$, meant to express that at some moment agent $\alpha$ had a present-directed intention toward the realization of $\varphi$. The semantics is based on the extension of stit frames with special topologies associated to agents. The open sets of the associated topology are interpreted as present-directed intentions, that support whether an agent had an intention of realizing a specific state of affairs when it chose one of its available actions and executed it. As an important application, we use $I_\alpha \varphi$ to formalize intentional action and intentional responsibility. We present an axiom system for our logic of intentionality, and prove that it is sound and complete.

Appendix A Proofs of Soundness and Completeness

1.1 A.1 Soundness

Proposition 1

The system ${\varLambda _I}$ is sound with respect to the class of iebt-models.

Proof

The proof of soundness is routine: the validity of (SET) and (IA) is standard from BST; the validity of (OAC) and $(Unif-H)$ is shown exactly as [1]; the validity of (InN) follows straightforwardly from Definitions 4 and 5; and the validity of (KI) follows from frame condition $(\texttt{KI})$.

1.2 A.2 Completeness

Definition 7

(Alexandrov spaces). A topological space $(X,\tau )$ is said to be an Alexandrov space iff the intersection of any collection of open sets of X is an open set as well.

Notice that a space is Alexandrov iff every point $x\in X$ has a $\subseteq$-smallest open set including it, namely the intersection of all the open sets around x.

Definition 8

For a given frame (X, R) such that R is reflexive and transitive, a set $A\subseteq X$ is called upward-closed iff for $x\in A$, if $x\le y$ for some $y\in X$, then $y\in A$ as well. For $x\in X$, $x\uparrow _R$ denotes the set $\left\{ y\in X \ | \ x R y\right\}$, which is clearly upward closed.

Remark 2

For a frame (X, R) such that R is reflexive and transitive, the set of all R-upward-closed sets forms an Alexandrov topology on X, which will be denoted by $\tau _R$. For $x\in X$, the $\subseteq$-smallest open set including x is precisely $x\uparrow _R$. This implies that $\left\{ x\uparrow _\le \ | \ x\in X\right\}$ is a basis for the topology $\tau _R$.

Definition 9

(Kripke- ies -frames & models).

A tuple

$$\begin{aligned} \left\langle W, Ags, R_\square , Ags, \texttt{Choice}, \left\{ \mathtt {\approx }_\alpha \right\} _{\alpha \in Ags},\left\{ R_\alpha ^I\right\} _{\alpha \in Ags}\right\rangle \end{aligned}$$

is called a Kripke-ies-frame (where the acronym ‘ies’ stands for ‘epistemic intentional stit’) iff

• W is a set of possible worlds. $R_\square$ is an equivalence relation over W. For $w\in W$, the class of w under $R_\square$ is denoted by $\overline{w}$.

• $\texttt{Choice}$ is a function that assigns to each $\alpha \in Ags$ and each $\square$-class $\overline{w}$ a partition $\texttt{Choice}_\alpha ^{\overline{w}}$ of $\overline{w}$ given by an equivalence relation, denoted by $R_\alpha ^{\overline{w}}$. $\texttt{Choice}$ must satisfy the following constraint:

  • $\mathtt {(IA)_K}$ For $w\in W$, each function $s:Ags\rightarrow 2^{\overline{w}}$ that maps $\alpha$ to a member of $\texttt{Choice}^{\overline{w}}_\alpha$ is such that $\bigcap _{\alpha \in Ags} s(\alpha ) \ne \emptyset$.

  For $\alpha \in Ags$, $w\in W$, and $v\in \overline{w}$, the class of v in the partition $\texttt{Choice}^{\overline{w}}_\alpha$ is denoted by $\texttt{Choice}^{\overline{w}}_\alpha (v)$.

• For $\alpha \in Ags$, $\approx _\alpha$ is an (epistemic) equivalence relation on W that satisfies the following conditions:

  • $\mathtt {(OAC)_{\texttt{K}}}$ For $w\in W$ and $v\in \overline{w}$, $v\approx _\alpha u$ for every $u\in \texttt{Choice}^{\overline{w}}_\alpha (v)$.

  • $\mathtt {(Unif-H)_{\texttt{K}}}$ Let $\alpha \in Ags$ and $v, u\in W$ such that $v\approx _\alpha u$. For $v'\in \overline{v}$, there exists $u'\in \overline{u}$ such that $v'\approx _\alpha u'$.

  For $w, v\in W$ such that $w\approx _\alpha v$ and $L\subseteq \overline{w}$, L’s epistemic cluster at $\overline{v}$ is the set $[\![L]\!]_\alpha ^{\overline{v}}:=\left\{ u\in {\overline{v}} ; \text{ there } \text{ is } o\in L \ \text { such that }\ o \approx _\alpha u \right\} .$

  For $\alpha \in Ags$ and $w\in W$, $\alpha$’s ex ante information set at w is defined as $\pi _\alpha ^\square [w]:=\left\{ v; w\approx _\alpha \circ R_\square v \right\}$, which by frame condition $(\mathtt {Unif-H})_\texttt{K}$ coincides with the set $\left\{ v; w R_\square \circ \approx _\alpha v \right\}$. To clarify, $(\mathtt {Unif-H})_\texttt{K}$ implies that $R_\square \circ \approx _\alpha = \approx _\alpha \circ R_\square$. Thus $\approx _\alpha \circ R_\square$ is an equivalence relation such that $\pi _\alpha ^\square [w]=\pi _\alpha ^\square [v]$ for every $w,v \in W$ such that $w\approx _\alpha \circ R_\square v$.

• For $\alpha \in Ags$, $R_\alpha ^I$ is a serial, transitive, and euclidean relation on W such that $R_\alpha ^I\subseteq \approx _\alpha \circ R_\square$ and such that the following condition is satisfied:

  • $\mathtt {(Den)_K}$ For $v, u\in W$ such that $v \approx _\alpha \circ R_\square u$, there exists $u'\in W$ such that $vR_\alpha ^I u'$ and $uR_\alpha ^I u'$.

  For $\alpha \in Ags$, $R_\alpha ^{I+}$ denotes the reflexive closure of $R_\alpha ^I$.

A Kripke-ies-model $\mathcal {M}$ consists of the tuple that results from adding a valuation function $\mathcal {V}$ to a Kripke-ies-frame, where $\mathcal {V}: P\rightarrow 2^{W}$ assigns to each atomic proposition a set of worlds.

Kripke-ies-models allow us to evaluate the formulas of $\mathcal L_{\textsf {I}}$ with semantics that are analogous to the ones provided for iebt-frames. The semantics for the formulas of $\mathcal L_{\textsf {I}}$ are given in the definition below.

Definition 10

(Evaluation rules on Kripke models). Let $\mathcal {M}$ be a Kripke-ies-model. The semantics on $\mathcal {M}$ for the formulas of $\mathcal {L}_{\textsf {I}}$ are defined recursively by the following truth conditions, evaluated at a given world w:

$$ \begin{array}{lll} \mathcal {M}, w \models p &{} \text{ iff } &{} w \in \mathcal {V}(p) \\ \mathcal {M}, w \models \lnot \varphi &{} \text{ iff } &{} \mathcal {M}, w \not \models \varphi \\ \mathcal {M}, w \models \varphi \wedge \psi &{} \text{ iff } &{} \mathcal {M}, w \models \varphi \text{ and } \mathcal {M}, w \models \psi \\ \mathcal {M}, w \models \Box \varphi &{} \text{ iff } &{} \text{ for } v\in \overline{w},\mathcal {M}, v \models \varphi \\ \mathcal {M}, w \models [\alpha ] \varphi &{} \text{ iff } &{} \text{ for } v\in \texttt{Choice}^{\overline{w}}_\alpha (w), \mathcal {M}, v \models \varphi \\ \mathcal {M}, w \models K_{\alpha } \varphi &{} \text{ iff } &{} \text{ for } v \text{ s.t. } w \approx _{\alpha }v, \mathcal {M}, v \models \varphi \\ \mathcal {M}, w \models I_\alpha \varphi &{} \text{ iff } &{} \text{ there } \text{ exists } x \in \pi _\alpha ^\square [w] \text{ s.t. } x\uparrow _{R_\alpha ^{I+}} \subseteq |\varphi |. \end{array} $$

where we write $|\varphi |$ to refer to the set $\left\{ w\in W;\mathcal {M},w \models \varphi \right\}$. Satisfiability, validity on a frame, and general validity are defined as usual.

Definition 11

(Associated iebt -frame).

Let

$$\mathcal {F}=\left\langle W, Ags, R_\square , \texttt{Choice}, \left\{ \mathtt {\approx }_\alpha \right\} _{\alpha \in Ags}, \left\{ R_\alpha ^I\right\} _{\alpha \in Ags} \right\rangle $$

be a Kripke-ies-frame. Then $\mathcal {F}^T:=\left\langle M_W, \sqsubset , Ags, \textbf{Choice},\left\{ \sim _\alpha \right\} _{\alpha \in Ags}, \tau \right\rangle$ is called the iebt-frame associated to $\mathcal {F}$ iff

• $M_W:={W}\cup \left\{ \overline{w} ; w\in W \right\} \cup \left\{ W\right\}$, and $\sqsubset$ is a relation on $M_W$ such that $\sqsubset$ is defined as the transitive closure of the union $\left\{ (\overline{w},v) ; w\in W \text{ and } v\in \overline{w} \right\} \cup \left\{ (W,\overline{w}) ; w\in W\right\}$.

  It is clear that $\sqsubset$ is a strict partial order on $M_W$ that satisfies “no backward branching” straightforwardly. Since the tuple $\left\langle M_W, \sqsubset \right\rangle$ is thus a tree, let us refer to the maximal $\sqsubset$-chains in $M_W$ as histories, and let us denote by $H_W$ the set of all histories of $M_W$. Observe that the definition of $\sqsubset$ yields that there is a bijective correspondence W and $H_W$. For $v\in W$, let $h_v$ be the history $\left\{ W, \overline{v},v\right\}$. For $o\in W$, it is clear that $o\in h_v$ iff $o=v$. Therefore, each history in $H_W$ can be identified using the world at its terminal node. Consequently, for $w\in W$, if $H_{\overline{w}}$ denotes the set of histories passing through $\overline{w}$, then $H_{\overline{w}}=\left\{ h_v ; v\in \overline{w}\right\}$—since $\overline{w}\in h_v$ iff $v\in \overline{w}$. Observe, then, that $H_W=\left\{ h_v ; v\in W\right\}$.

• For $B\in 2^{W}$, let $B^T$ denote the set $\left\{ h_v ; \ v\in B\right\}$. With such a terminology, we define $\textbf{Choice}$ as a function on $Ags\times M_W$ given by the rules:

  • For $\alpha \in Ags$ and $v\in W$, $\textbf{Choice}(\alpha ,v)=\left\{ \left\{ h_v\right\} \right\}$.

  • For $\alpha \in Ags$ and $w\in W$, $\textbf{Choice}(\alpha , \overline{w})=\left\{ C_\alpha ^T ; C_\alpha \in \texttt{Choice}_\alpha ^{\overline{w}}\right\}$.

  • For $\alpha \in Ags$, $\textbf{Choice}(\alpha , W)=\left\{ H_W\right\}$.

  To keep notation consistent, the sets of the form $\textbf{Choice}(\alpha , \overline{w})$ will be denoted by $\textbf{Choice}_\alpha ^{\overline{w}}$. The choice-cell of a given $h_v$ in $\textbf{Choice}_\alpha ^{\overline{w}}$ is denoted by $\textbf{Choice}_\alpha ^{\overline{w}}(h_v)$.

• For $\alpha \in Ags$, $\thicksim _\alpha$ is a relation on $I\left( M_W\times H_W\right)$ defined as follows:

  $$\begin{array}{lll} \sim _\alpha &{}:= &{} \left\{ \left( \left\langle \overline{w},h_v\right\rangle , \left\langle \overline{w'},h_{v'}\right\rangle \right) ; w, w'\in W \text{ and } v\approx _\alpha v'\right\} \cup \\ &{}&{} \left\{ \left( \left\langle z,h_z\right\rangle ,\left\langle z,h_z\right\rangle \right) ; z\in W\right\} \cup \\ &{}&{}\left\{ \left( \left\langle W,h_v\right\rangle ,\left\langle W,h_{v'}\right\rangle \right) ; v,v'\in W\right\} . \end{array}$$

  It is clear that this definition entails that $\sim _\alpha$ is an equivalence relation for every $\alpha \in Ags$ and that, for $w\in W$ and $L\in \texttt{Choice}_\alpha ^{\overline{w}}$, $v\in [\![L]\!]_\alpha ^{\overline{w}}$ iff $h_v\in \left[ L^T\right] ^{\overline{w}}_\alpha$.

• $\tau$ is a function defined as follows:

  • For $\alpha \in Ags$ and $z\in W$, $\tau _\alpha ^{\left\langle z, h_z \right\rangle }= \left\{ \emptyset , \pi _\alpha ^\square \left[ \left\langle z, h_z \right\rangle \right] \right\}$

  • For $\alpha \in Ags$, we first define a relation $R_\alpha ^{IT}$ on $\left\{ \left\langle \overline{w},h_v\right\rangle ; w\in W \text{ and } v\in \overline{w}\right\}$ by the rule: $\left\langle \overline{w},h_v\right\rangle R_\alpha ^{IT} \left\langle \overline{w'},h_{v'}\right\rangle$ iff $vR_\alpha ^I v'$. For $\alpha \in Ags$, $w\in W$, and $v\in \overline{w}$, then, we define $\tau _\alpha ^{\left\langle \overline{w}, h_v \right\rangle }$ as the subspace topology of $\tau _{R_\alpha ^{IT+}}$ on $\pi _\alpha ^\square \left[ \left\langle \overline{w}, h_v \right\rangle \right]$.¹¹ Observe that, for $\alpha \in Ags$, $w\in W$, and $v\in \overline{w}$, $\pi _\alpha ^\square \left[ \left\langle \overline{w}, h_v\right\rangle \right] =\left\{ \left\langle \overline{v'}, h_{v'}\right\rangle ; v'\in \pi _\alpha ^\square [v]\right\}$. Thus, the fact that $R_\alpha ^I\subseteq \approx _\alpha \circ R_\square$ implies that, for $\left\langle \overline{x}, h_{x}\right\rangle \in \pi _\alpha ^\square \left[ \left\langle \overline{w}, h_v \right\rangle \right]$, $\left\langle \overline{x}, h_{x}\right\rangle \uparrow _{R_\alpha ^{IT+}}\subseteq \pi _\alpha ^\square \left[ \left\langle \overline{w}, h_v \right\rangle \right]$, so that $\pi _\alpha ^\square \left[ \left\langle \overline{w}, h_v \right\rangle \right]$ is open in $\tau _{R_\alpha ^{IT+}}$.

  • For $\alpha \in Ags$ and $v\in W$, $\tau _\alpha ^{\left\langle W, h_v \right\rangle }= \left\{ \emptyset , \pi _\alpha ^\square \left[ \left\langle W, h_v \right\rangle \right] \right\}$.

Proposition 2

Let $\mathcal {F}$ be a Kripke-ies-frame. Then $\mathcal {F}^T$ is an iebt-frame.

Proof

It amounts to showing that $\sqsubset$ is a strict partial order that satisfies no backward branching, that $\textbf{Choice}$ is a function that satisfies frame conditions $(\texttt{NC})$ and $(\texttt{IA})$, that $\left\{ \sim _\alpha \right\} _{\alpha \in Ags}$ is such that $\approx _\alpha$ is an equivalence relation for every $\alpha \in Ags$ and frame conditions $(\texttt{OAC})$ and $(\mathtt {Unif-H})$ are met, and that $\tau$ is a function that meets the requirements of Definition 4.

• As mentioned in Definition 11, it is straightforward to show that $\sqsubset$ is a strict partial order that satisfies no backward branching. It is also clear from Definition 11 that $\sim _\alpha$ is an equivalence relation for every $\alpha \in Ags$.

• $(\texttt{NC})$ is vacuously validated at moment W. It is validated in moments of the form $\overline{w}$ ($w\in W$), since two different histories never intersect in a moment later than $\overline{w}$. Finally, it is also validated in moments of the form v such that $v\in W$ (since there are no moments above v).

• For $(\texttt{IA})$, we reason by cases:

  • (a) At moment W, $(\texttt{IA})$ is validated straightforwardly, since $\textbf{Choice}(\alpha , W)=\left\{ H\right\}$ for each $\alpha \in Ags$.

  • (b) For a moment of the form $\overline{w}$ (with $w\in W$), let s be a function that assigns to each agent $\alpha$ a member of $\textbf{Choice}^{\overline{w}}_\alpha =\left\{ (C_\alpha )^T; C_\alpha \in \texttt{Choice}^{\overline{w}}_\alpha \right\}$. Let $s_k:Ags \rightarrow \bigcup _{\alpha \in Ags}\texttt{Choice}^{\overline{w}}_\alpha$ be a function such that $s_k(\alpha )=C_\alpha$ iff $s(\alpha )=(C_\alpha )^T$. Since $\mathcal {M}$ satisfies condition $(\texttt{IA})_{\texttt{K}}$, then $\bigcap _{\alpha \in Ags} s_k(\alpha ) \ne \emptyset$. Take $v\in \bigcap _{\alpha \in Ags} s_k(\alpha )$. Then $v\in C_\alpha$ for every $\alpha \in Ags$. This implies that $h_v\in (C_\alpha )^T$ for every $\alpha \in Ags$, so $\bigcap _{\alpha \in Ags} s(\alpha ) \ne \emptyset$.

  • (c) At moments of the form v such that $v\in W$, if s is a function that assigns to each agent $\alpha$ a member of $\textbf{Choice}(v,\alpha )$, s must be constant and $\bigcap _{\alpha \in Ags} s(\alpha ) = \left\{ h_v\right\}$.

• For $(\texttt{OAC})$, again we reason by cases:

  • (a) Assume that $\left\langle \overline{w}, h_v \right\rangle \sim _\alpha \left\langle \overline{w'}, h_{v'} \right\rangle$ (for $w, w' \in W$). This means that $v\approx _\alpha v'$. We want to show that, for every $h_u\in \textbf{Choice}^{\overline{w}}_\alpha$ such that $h_u\in \textbf{Choice}^{\overline{w}}_\alpha (h_v)$, $\left\langle \overline{w}, h_u \right\rangle \sim _\alpha \left\langle \overline{w'}, h_{v'} \right\rangle$. Therefore, let $h_u\in \textbf{Choice}^{\overline{w}}_\alpha (h_v)$. By definition, this means that $u\in \texttt{Choice}^{\overline{w}}_\alpha (v)$. Since $\mathcal {M}$ satisfies condition $(\texttt{OAC})_{\texttt{K}}$, this last fact implies, with $v\approx _\alpha v'$, that $u\approx _\alpha v'$, which in turn yields that $\left\langle \overline{w}, h_u \right\rangle \sim _\alpha \left\langle \overline{w'}, h_v \right\rangle$.

  • (b) For indices based on moments of the form v such that $v\in W$, $(\texttt{OAC})$ is met straightforwardly, since for $h_v$ the choice-cell in $\textbf{Choice}(\alpha ,v)$ to which $h_v$ belongs is just $\left\{ h_v\right\}$.

  • (c) For indices based on moment W, $(\texttt{OAC})$ is also met straightforwardly, since for every $\alpha \in Ags$ $\sim _\alpha$ is defined such that $\left\langle W,h_v\right\rangle \sim _\alpha \left\langle W, h_{v'}\right\rangle$ for every pair of histories $h_v, h_{v'}$ in H.

• For $(\mathtt {Unif-H})$, again we reason by cases:

  • (a) Assume that $\left\langle \overline{w}, h_v \right\rangle \sim _\alpha \left\langle \overline{w'}, h_{v'} \right\rangle$ (for $w, w' \in W$). This means that $v\in \overline{w}$, $v'\in \overline{w'}$, and $v\approx _\alpha v'$. Let $h_{z}\in H_{\overline{w}}$ (which means that $z\in \overline{w}$). We want to show that there exists $h\in H_{\overline{w'}}$ such that $\left\langle \overline{w}, h_z \right\rangle \sim _\alpha \left\langle \overline{w'}, h \right\rangle$. Condition $(Unif-H)_{\texttt{K}}$ gives us that there exists $z'\in \overline{w'}$ such that $z\approx _\alpha z'$, which by definition of $\sim _\alpha$ means that $\left\langle \overline{w}, h_z \right\rangle \sim _\alpha \left\langle \overline{w'}, h_{z'} \right\rangle$.

  • (b) For indices based on moments of the form v such that $v\in W$, $\left\langle v, h_v \right\rangle$ is $\sim _\alpha$-related only to itself, so $(\mathtt {Unif-H})$ is met straightforwardly.

  • (c) For indices based on moment W, $(\mathtt {Unif-H})$ is also met straightforwardly, since $\left\langle W,h_v\right\rangle \sim _\alpha \left\langle W,h_{v'}\right\rangle$ for every $v,v'\in W$.

• As for $\tau$, it is clear that, for $\alpha \in Ags$ and index $\langle m, h\rangle$ either of the form $\left\langle z, h_z \right\rangle (z\in W)$ or of the form $\left\langle W, h_v \right\rangle (v\in W)$, $\tau _\alpha ^{\langle m, h \rangle }$ is a topology on $\pi _\alpha ^\square [\left\langle m, h \right\rangle ]$ that satisfies frame conditions $(\texttt{CI})$ and $(\texttt{KI})$.

  Assume, then, that $\langle m, h\rangle$ is of the form $\left\langle \overline{w}, h_v \right\rangle$ such that $v\in \overline{w}$. Let $\alpha \in Ags$. By Definition 11, $\tau _\alpha ^{\left\langle \overline{w}, h_v \right\rangle }$ is the subspace topology of $\tau _{R_\alpha ^{IT+}}$ on $\pi _\alpha ^\square [\left\langle \overline{w}, h_v \right\rangle ]$. Thus, it is clear that $\tau _\alpha ^{\left\langle \overline{w}, h_v \right\rangle }$ is a topology on $\pi _\alpha ^\square [\left\langle \overline{w}, h_v \right\rangle ]$, so that $\tau$ straightforwardly satisfies $(\texttt{KI})$. Let us show that condition $(\texttt{CI})$ is also satisfied: let $U,V\in \tau _\alpha ^{\left\langle \overline{w}, h_v \right\rangle }$ such that U and V are non-empty. Let $\left\langle \overline{u}, h_u \right\rangle \in U$ and $\left\langle \overline{x}, h_x \right\rangle \in V$. Definition 11 implies that $u\approx _\alpha \circ R_\square x$. $\mathcal {F}$’s condition $\mathtt {(Den)_K}$ implies that there exists $u'\in W$ such that $uR_\alpha ^I u'$ and $xR_\alpha ^I u'$, which implies that $\left\langle \overline{u}, h_u\right\rangle R_\alpha ^{IT} \left\langle \overline{u'}, h_{u'}\right\rangle$ and $\left\langle \overline{x}, h_x\right\rangle R_\alpha ^{IT} \left\langle \overline{u'}, h_{u'}\right\rangle$, by definition of $R_\alpha ^{IT}$. This means that $\left\langle \overline{u'}, h_{u'}\right\rangle \in \left\langle \overline{u}, h_{u}\right\rangle \uparrow _{R_\alpha ^{IT+}}$ and $\left\langle \overline{u'}, h_{u'}\right\rangle \in \left\langle \overline{x}, h_{x}\right\rangle \uparrow _{R_\alpha ^{IT+}}$. Since $\pi _\alpha ^\square \left[ \left\langle \overline{w}, h_v \right\rangle \right]$ is open in $\tau _{R_\alpha ^{IT+}}$, we know that $\left\langle \overline{u}, h_{u}\right\rangle \uparrow _{R_\alpha ^{IT+}}\subseteq U$ and that $\left\langle \overline{x}, h_{x}\right\rangle \uparrow _{R_\alpha ^{IT+}}\subseteq V$. Thus, $\left\langle \overline{u'}, h\right\rangle \in U\cap V$, so that U and V are $\tau _\alpha ^{\left\langle \overline{w}, h_v \right\rangle }$-dense.

Let $\mathcal {M}$ be a Kripke-ies-model with valuation function $\mathcal {V}$. The frame upon which $\mathcal {M}$ is based has an associated iebt-frame. If to the tuple of this iebt-frame one adds a valuation function $\mathcal {V}^t$ such that $\mathcal {V}^t(p)=\left\{ \left\langle \overline{w},h_w\right\rangle ; w\in \mathcal {V}(p)\right\}$, the resulting model is called the iebt-model associated to $\mathcal {M}$.

Proposition 3

Let $\mathcal {M}$ be a Kripke-ies-model, and let $\mathcal {M}^T$ denote its associated iebt-model. For $\varphi$ of $\mathcal {L}_{\textsf {I}}$ and $w\in W$, $\mathcal {M} ,w\models \varphi$ iff $\mathcal {M} ^T,\left\langle \overline{w},h_w\right\rangle \models \varphi$.

Proof

We proceed by induction on $\varphi$. For the base case, take a propositional letter p and an arbitrary $w\in W$. Then $\mathcal {M} ,w\models p$ iff $w\in \mathcal {V}(p)$ iff $\left\langle \overline{w},h_w\right\rangle \in \mathcal {V}^T(p)$ iff $\mathcal {M} ^T,\left\langle \overline{w},h_w\right\rangle \models p$. The cases of Boolean connectives are standard, so let us deal with the modal operators. Let $w\in W$ and $\alpha \in Ags$.

• ($\square$) $\mathcal {M} ,w\models \square \varphi$ iff for every $v\in \overline{w}$, $\mathcal {M},v\models \varphi$, which by induction hypothesis happens iff $\mathcal {M} ^T,\left\langle \overline{v},h_v\right\rangle \models \varphi$ for every $v\in \overline{w}$, which happens iff $\mathcal {M} ^T,\left\langle \overline{w},h_w\right\rangle \models \square \varphi$, since it is the case that $h_v\in H_{\overline{w}}$ iff $v\in \overline{w}$.

• ($[\alpha ]$) $\mathcal {M} ,w\models [\alpha ]\varphi$ iff for every $v\in W$ such that $wR_\alpha ^{\overline{w}} v$, $\mathcal {M} ,v\models \varphi$, which by induction hypothesis happens iff $\mathcal {M} ^T,\left\langle \overline{w},h_v\right\rangle \models \varphi$ for every $h_v\in \textbf{Choice}^{\overline{w}}_\alpha (h_w)$, which in turn happens iff $\mathcal {M} ^T,\left\langle \overline{w},h_w\right\rangle \models [\alpha ]\varphi$.

• ($K_\alpha$) $\mathcal {M} ,w\models K_\alpha \varphi$ iff for every $v\in W$ such that $w\approx _\alpha v$, $\mathcal {M} ,v\models \varphi$, which by induction hypothesis occurs iff $\mathcal {M} ^T,\left\langle \overline{v},h_v\right\rangle \models \varphi$ for every $h_v\in H$ such that $\left\langle \overline{w},h_w\right\rangle \sim _\alpha \left\langle \overline{v},h_v\right\rangle$, which happens iff $\mathcal {M} ^T,\left\langle \overline{w},h_w\right\rangle \models K_\alpha \varphi$.

• ($I_\alpha$) First, observe that the induction hypothesis implies that $\Vert \varphi \Vert =\left\{ \left\langle \overline{w}, h_w \right\rangle ; w\in |\varphi |\right\}$. Therefore, $\mathcal {M} ,w\models I_\alpha \varphi$ iff there exists $x\in \pi _\alpha ^\square [w]$ such that $x\uparrow _{R_\alpha ^{I+}}\subseteq |\varphi |$ iff $\left\langle \overline{x}, h_{x}\right\rangle \uparrow _{R_\alpha ^{IT+}}\subseteq \Vert \varphi \Vert$ iff there exists $U\in \tau ^{\left\langle \overline{w}, h_w \right\rangle }_\alpha$ such that $U\subseteq \Vert \varphi \Vert$ iff $\mathcal {M} ^T,\left\langle \overline{w},h_w\right\rangle \models I_\alpha \varphi$.

1.3 A.3 Canonical Kripke-ies-structure

We show that the proof system ${\varLambda _I}$ is complete with respect to the class of Kripke-ies-models. For each ${\varLambda _I}$-consistent formula $\varphi$, we build a canonical structure from the syntax that satisfies $\varphi$.

Definition 12

(Canonical Structure). The tuple

$$\mathcal {M}=\left\langle W^{\varLambda _I}, R_\square , \texttt{Choice}, \left\{ \approx _\alpha \right\} _{\alpha \in Ags}, \left\{ R_\alpha ^I\right\} _{\alpha \in Ags} ,\mathcal {V} \right\rangle $$

is called a canonical structure for ${\varLambda _I}$ iff

• $W^{\varLambda _I}=\left\{ w ;w \text{ is } \text{ a } {\varLambda _I}\text {-MCS}\right\}$. $R_\square$ is a relation on $W^{\varLambda _I}$ defined by ther rule: for $w,v\in W^{\varLambda _I}$, $wR_{\square }v$ iff $\square \varphi \in w\Rightarrow \varphi \in v$ for every $\varphi$ of $\mathcal {L}_{\textsf {I}}$. For $w\in W^{\varLambda _I}$, the set $\left\{ v\in W^{\varLambda _I} ;wR_\square v \right\}$ is denoted by $\overline{w}$.

• $\texttt{Choice}$ is a function that assigns to each $\alpha$ and $\overline{w}$ a subset of $2^{\overline{w}}$, denoted by $\texttt{Choice}_\alpha ^{\overline{w}}$, and defined as follows: let $R_\alpha ^{\overline{w}}$ be a relation on ${\overline{w}}$ such that, for $w,v\in W^{\varLambda _I}$, $wR_{\alpha }^{\overline{w}}v$ iff $[\alpha ]\varphi \in w\Rightarrow \varphi \in v$ for every $\varphi$ of $\mathcal {L}_{\textsf {I}}$; if $\texttt{Choice}^{\overline{w}}_\alpha (v):=\left\{ u\in \overline{w} ; vR_\alpha ^{\overline{w}} u\right\}$, then $\texttt{Choice}^{\overline{w}}_\alpha :=\left\{ \texttt{Choice}^{\overline{w}}_\alpha (v); v\in \overline{w}\right\}$.

• For $\alpha \in Ags$, $\approx _\alpha$ is an epistemic relation on $W^{\varLambda _I}$ given by the rule: for $w,v\in W^{\varLambda _I}$, $w\approx _{\alpha }v$ iff $K_\alpha \varphi \in w\Rightarrow \varphi \in v$ for every $\varphi$ of $\mathcal {L}_{\textsf {I}}$.

• For $\alpha \in Ags$, $R_\alpha ^I$ is a relation on $W^{\varLambda _I}$ given by the rule: for $w,v\in W^{\varLambda _I}$, $wR_{\alpha }^Iv$ iff $I_\alpha \varphi \in w\Rightarrow \varphi \in v$ for every $\varphi$ of $\mathcal {L}_{\textsf {I}}$.

• $\mathcal {V}$ is the canonical valuation, defined such that $w\in \mathcal {V}(p)$ iff $p\in w$.

Proposition 4

The canonical structure $\mathcal {M}$ for ${\varLambda _I}$ is a Kripke-ies-model.

Proof

We want to show that the tuple

$\left\langle W^{\varLambda _I}, R_\square , \texttt{Choice}, \left\{ \mathtt {\approx }_\alpha \right\} _{\alpha \in Ags}, \left\{ \texttt{R}_\alpha ^I\right\} _{\alpha \in Ags}\right\rangle$ is a Kripke-ies-frame, which amounts to showing that the tuple satisfies the items in the definition of Kripke-ies-frames (Definition 9).

• It is clear that $R_\square$ is an equivalence relation, since ${\varLambda _I}$ includes the $\textbf{S5}$ axioms for $\square$.

• Since ${\varLambda _I}$ includes the $\textbf{S5}$ schemata for $[\alpha ]$ ($\alpha \in Ags$), $R_\alpha ^{\overline{w}}$ is an equivalence relation for $\alpha \in Ags$ and $w\in W^{\varLambda _I}$. Moreover, since ${\varLambda _I}$ includes schema (SET), $R_\alpha ^{\overline{w}}\subseteq \overline{w}\times \overline{w}$ for every $w\in W^{\varLambda _I}$. Thus, $\texttt{Choice}$ indeed assigns to each $\alpha$ and $\overline{w}$ a partition of $\overline{w}$.

  To show that frame condition $\mathtt {(IA)_K}$ is satisfied, we first prove two intermediate results:

  • (a) For $w_*\in W^{\varLambda _I}$, $w\in \overline{w_*}$ iff $\left\{ \square \psi ; \square \psi \in w_*\right\} \subseteq w$. $(\Rightarrow )$ Let $w\in \overline{w_*}$ (which means that $w_*R_\square w$). Take $\varphi$ of $\mathcal {L}_{\textsf {Kx}}$ such that $\square \varphi \in w_*$. Since $w_*$ is closed under Modus Ponens, axiom (4) for $\square$ implies that $\square \square \varphi \in w_*$. By definition of $R_\square$, $\square \varphi \in w$. $(\Leftarrow )$ Assume that $\left\{ \square \psi ; \square \psi \in w_*\right\} \subseteq w$. Take $\varphi$ of $\mathcal {L}_{\textsf {KX}}$ such that $\square \varphi \in w_*$. By assumption, $\square \varphi \in w$. Since w is closed under Modus Ponens, axiom (T) for $\square$ implies that $\varphi \in w$. Thus, $w_*R_\square w$ and $w\in \overline{w_*}$.

  • (b) If $w_*\in W^{\varLambda _I}$ and $s:Ags\rightarrow 2^{\overline{w_*}}$ maps $\alpha$ to a member of $\texttt{Choice}^{\overline{w_*}}_\alpha$ such that $v_{s(\alpha )}\in s(\alpha )$, then $w\in s(\alpha )$ iff $\varDelta _{s(\alpha )}=\left\{ [\alpha ] \psi ;[\alpha ]\psi \in v_{s(\alpha )}\right\} \subseteq w$. $(\Rightarrow )$ Let $w\in s(\alpha )$ (which means that $v_{s(\alpha )}R_\alpha w$). Take $\varphi$ of $\mathcal {L}_{\textsf {KX}}$ such that $[\alpha ]\varphi \in v_{s(\alpha )}$. Since $v_{s(\alpha )}$ is closed under Modus Ponens, schema (4) for $[\alpha ]$ implies that $[\alpha ][\alpha ]\varphi \in v_{s(\alpha )}$. Therefore, by definition of $R_\alpha$, $[\alpha ]\varphi \in w$. $(\Leftarrow )$ Assume that $\varDelta _{s(\alpha )}=\left\{ [\alpha ] \psi ; [\alpha ]\psi \in v_{s(\alpha )}\right\} \subseteq w$. Take $\varphi$ of $\mathcal {L}_{\textsf {KX}}$ such that $[\alpha ]\varphi \in v_{s(\alpha )}$. By assumption, $[\alpha ]\varphi \in w$. Since w is closed under Modus Ponens, axiom (T) for $[\alpha ]$ implies that $\varphi \in w$. Thus, $v_{s(\alpha )}R_\alpha ^{\overline{w_*}} w$ and $w\in s(\alpha )$.

  Next we show that, for $w_*\in W^{\varLambda _I}$ and $s:Ags\rightarrow 2^{\overline{w_*}}$ just as in item b above, $\bigcup _{\alpha \in Ags}\varDelta _{s(\alpha )}\cup \left\{ \square \psi ; \square \psi \in w_*\right\}$ is $\varLambda _I$-consistent: first we show that $\bigcup _{\alpha \in Ags} \varDelta _{s(\alpha )}$ is consistent. Suppose that this is not the case. Then there exists a set $\left\{ \varphi _1,\dots ,\varphi _n\right\}$ of formulas of $\mathcal {L}_{\textsf {KX}}$ such that $[\alpha _{i} ]\varphi _i\in v_{s(\alpha _i)}$ for every $1\le i \le n$ and

  $$\begin{aligned} \vdash _{\varLambda _I}[\alpha _{1} ]\varphi _1\wedge \dots \wedge [\alpha _{n} ]\varphi _n\rightarrow \bot . \end{aligned}$$

  (1)

  Without loss of generality, assume that $\alpha _i\ne \alpha _j$ for all $j\ne i$ such that $j,i\in \left\{ 1,\dots ,n\right\}$—this assumption hinges on the fact that any stit operator distributes over conjunction. Notice that the fact that $[\alpha _{i} ]\varphi _i\in v_{s(\alpha _i)}$ for every $1\le i \le n$ implies that $\Diamond [\alpha _{i} ]\varphi _i \in w_*$ for every $1\le i \le n$. Since $w_*$ is closed under conjunction, $\Diamond [\alpha _{1} ]\varphi _1\wedge \dots \wedge \Diamond [\alpha _{n} ]\varphi _n\in w_*$.

  Axiom (IA) then implies that

  $$\begin{aligned} \vdash _{\varLambda _I}\Diamond [\alpha _{1} ]\varphi _1\wedge \dots \wedge \Diamond [\alpha _{n} ]\varphi _n\rightarrow \Diamond \left( [\alpha _{1} ]\varphi _1\wedge \dots \wedge [\alpha _{n} ]\varphi _n\right) . \end{aligned}$$

  (2)

  Therefore, Eqs. (2) and (1), imply that

  $$\begin{aligned} \vdash _{\varLambda _I}\Diamond [\alpha _{1} ]\varphi _1\wedge \dots \wedge \Diamond [\alpha _{n} ]\varphi _n\rightarrow \Diamond \bot . \end{aligned}$$

  (3)

  But this is a contradiction, since $\Diamond [\alpha _{1} ]\varphi _1\wedge \dots \wedge \Diamond [\alpha _{n} ]\varphi _n\in w_*$, and $w_*$ is a $\varLambda _I$-MCS. Therefore, $\bigcup _{\alpha \in Ags} \varDelta _{s(\alpha )}$ is consistent. Secondly, we show that the union $\bigcup _{\alpha \in Ags} \varDelta _{s(\alpha )}\cup \left\{ \square \psi ; \square \psi \in w_*\right\}$ is also consistent. Suppose that this is not the case. Since $\bigcup _{\alpha \in Ags} \varDelta _{s(\alpha )}$ is consistent, there must exist sets $\left\{ \varphi _1,\dots ,\varphi _n\right\}$ and $\left\{ \theta _1,\dots ,\theta _m\right\}$ of formulas of $\mathcal {L}_{\textsf {KX}}$ such that $[\alpha _i ]\varphi _i\in v_{s(\alpha _i)}$ for every $1\le i \le n$, $\square \theta _i\in w_*$ for every $1\le i\le m$, and

  $$\begin{aligned} \vdash _{\varLambda _I}[\alpha _1 ]\varphi _1\wedge \dots \wedge [\alpha _n]\varphi _n\wedge \square \theta _1\wedge \dots \wedge \square \theta _m\rightarrow \bot . \end{aligned}$$

  (4)

  Let $\theta =\theta _1\wedge \dots \wedge \theta _m$. Since $\square$ distributes over conjunction, $\vdash _{\varLambda _I}\square \theta \leftrightarrow \square \theta _1\wedge \dots \wedge \square \theta _m$, where it is important to mention that, since $w_*$ is a $\varLambda _I$-MCS closed under logical equivalence, $\square \theta \in w_*$. Thus, (4) implies that

  $$\begin{aligned} \vdash _{\varLambda _I}([\alpha _1 ]\varphi _1\wedge \dots \wedge [\alpha _n ]\varphi _n)\rightarrow \lnot \square \theta .\end{aligned}$$

  (5)

  Once again, assume without loss of generality that $\alpha _i\ne \alpha _j$ for all $j\ne i$ such that $j,i\in \left\{ 1,\dots ,n\right\}$. By an argument analogous to the one used to show that $\bigcup _{\alpha \in Ags} \varDelta _{s(\alpha )}$ is consistent, (5) implies that

  $$\begin{aligned} \vdash _{\varLambda _I}\Diamond [\alpha _{1} ]\varphi _1\wedge \dots \wedge \Diamond [\alpha _{n} ]\varphi _n\rightarrow \Diamond \lnot \square \theta . \end{aligned}$$

  (6)

  This entails that $\Diamond \lnot \square \theta \in w_*$, but this is a contradiction, since the fact that $\square \theta \in w_*$ implies with axiom (4) for $\square$ that $\square \square \theta \in w_*$. Now, let $u_*$ be the $\varLambda _I$-MCS that includes $\bigcup _{\alpha \in Ags} \varDelta _{s(\alpha )}\cup \left\{ \square \psi ; \square \psi \in w_*\right\}$. By intermediate result a, it is clear that $u_*\in \overline{w_*}$. By intermediate result b, it is clear that $u_*\in s(\alpha )$ for every $\alpha \in Ags$. Therefore, we have shown that, for $w_*\in W$, each function $s:Ags\rightarrow 2^{\overline{w_*}}$ that maps $\alpha$ to a member of $\texttt{Choice}^{\overline{w_*}}_\alpha$ is such that $\bigcap _{\alpha \in Ags} s(\alpha ) \ne \emptyset$, which means that $\mathcal {M}$ satisfies $\mathtt {(IA)_K}$.

• Since the proof system ${\varLambda _I}$ includes the $\textbf{S5}$ axioms for $K_\alpha$ ($\alpha \in Ags$), $\approx _\alpha$ is an equivalence relation for $\alpha \in Ags$. We verify that $\mathcal {M}$ satisfies conditions $(\texttt{OAC})_{\texttt{K}}$ and $(\mathtt {Unif-H})_{\texttt{K}}$.

  For $(\texttt{OAC})_{\texttt{K}}$, let $w_*\in W^{\varLambda _I}$ and $\alpha \in Ags$. Assume that $v,u\in \overline{w_*}$ are such that $v\approx _\alpha u$. Let $v'\in \texttt{Choice}^{\overline{w_*}}_\alpha (v)$. This means that $vR_\alpha v'$. We want to show that $v'\approx _\alpha u$, so let $\varphi$ be a formula of $\mathcal {L}_{\textsf {KO}}$ such that $K_\alpha \varphi \in v'$. By schema (4) for $K_\alpha$, $K_\alpha K_\alpha \varphi \in v'$. Similarly, since all substitutions of axiom (OAC) lie within $v'$ and it is closed under Modus Ponens, $[\alpha ]K_\alpha \varphi$ also lies in $v'$. Since $v'R_\alpha v$, this implies that $K_\alpha \varphi \in v$. Therefore, our assumption that $v\approx _\alpha u$ entails that $\varphi \in u$. With this, we have shown that the fact that $K_\alpha \varphi \in v'$ implies that $\varphi \in u$, which means that $v'\approx _\alpha u$.

  For $\mathtt {(Unif-H)_K}$, let $v, u\in W^{\varLambda _I}$ such that $v\approx _\alpha u$. Take $v'\in \overline{v}$. We want to show that there exists $u'\in \overline{u}$ such that $v'\approx _\alpha u'$. We show that $u''=\left\{ \psi ; K_\alpha \psi \in v'\right\} \cup \left\{ \square \psi ; \square \psi \in u \right\}$ is consistent. To do so, we first show that $\left\{ \psi ; K_\alpha \psi \in v'\right\}$ is consistent. Suppose for a contradiction that it is not consistent. Then there exists a set $\left\{ \psi _1,\dots ,\psi _n\right\}$ of formulas of $\mathcal {L}_{\textsf {KX}}$ such that $K_\alpha \psi _i\in v'$ for every $1\le i \le n$ and (a) $\vdash _{\varLambda _I}\psi _1\wedge \dots \wedge \psi _n\rightarrow \bot$. By Necessitation for $K_\alpha$ and its distributivity over conjunction, (a) implies that $\vdash _{\varLambda _I}K_\alpha \psi _1\wedge \dots \wedge K_\alpha \psi _n\rightarrow K_\alpha \bot$, but this is a contradiction, since $v'$ is a $\varLambda _I$-MCS and it includes $K_\alpha \psi _1\wedge \dots \wedge K_\alpha \psi _n$. Next we show that $u''=\left\{ \psi ; K_\alpha \psi \in v'\right\} \cup \left\{ \square \psi ; \square \psi \in u \right\}$ is also consistent. Suppose for a contradiction that it is not consistent. Since $\left\{ \psi ; K_\alpha \psi \in v'\right\}$ and $\left\{ \square \psi ; \square \psi \in u \right\}$ are consistent, there must exist sets $\left\{ \varphi _1,\dots ,\varphi _n\right\}$ and $\left\{ \theta _1,\dots ,\theta _m\right\}$ of formulas of $\mathcal {L}_{\textsf {KX}}$ such that $K_\alpha \varphi _i\in v'$ for every $1\le i \le n$, $\square \theta _i\in w_2$ for every $1\le i\le m$, and (b) $\vdash _{\varLambda _I}\varphi _1\wedge \dots \wedge \varphi _n\wedge \square \theta _1\wedge \dots \wedge \square \theta _m\rightarrow \bot$. Let $\theta =\theta _1\wedge \dots \wedge \theta _m$ and $\varphi =\varphi _1\wedge \dots \wedge \varphi _n$. Since $\square$ distributes over conjunction, $\vdash _{\varLambda _I}\square \theta \leftrightarrow \square \theta _1\wedge \dots \wedge \square \theta _m$, where it is important to mention that, since u is a $\varLambda _I$-MCS, then $\square \theta \in u$ and ($\star$) $\square \square \theta \in u$ as well. In this way, (b) implies that $\vdash _{\varLambda _I}\varphi \rightarrow \lnot \square \theta$ and thus that (c) $\vdash _{\varLambda _I}\Diamond \varphi \rightarrow \Diamond \lnot \square \theta$. Notice that the facts that $K_\alpha \varphi _i\in v'$ for every $1\le i\le n$, that $K_\alpha$ distributes over conjunction, and that $v'$ is a $\varLambda _I$-MCS imply that $K_\alpha \varphi \in v'$. The fact that $v'\in \overline{v}$ implies that $\Diamond K_\alpha \varphi \in v$, so that $(Unif-H)$ entails that $K_\alpha \Diamond \varphi \in v$. Now, this last inclusion implies, with our assumption that $v\approx _\alpha u$, that $\Diamond \varphi \in u$, which by (c) in turn yields that $\Diamond \lnot \square \theta \in u$, contradicting $(\star )$. Therefore, $u''$ is consistent. Let $u'$ be the $\varLambda _I$-MCS that includes $u''$. It is clear from its construction that $u'\in \overline{u}$ and that $v'\approx _\alpha u'$. With this, we have shown that $\mathcal {M}$ satisfies condition $\mathtt {(Unif-H)_K}$.

• Since ${\varLambda _I}$ includes the $\textbf{KD45}$ schemata for $I_\alpha$ ($\alpha \in Ags$), then $R_\alpha ^I$ is a serial, transitive, and euclidean relation on W, for $\alpha \in Ags$. Since ${\varLambda _I}$ includes schema (InN), then $R_\alpha ^I \subseteq \approx _\alpha \circ R_\square$ for $\alpha \in Ags$.

  We now verify that frame condition $\mathtt {(Den)_K}$ is satisfied. Let $v, u\in W^{\varLambda _{I}}$ such that $v \approx _\alpha \circ R_\square u$. This means that there exists w such that $v\in \overline{w}$ and $w\approx _\alpha u$. We want to show that there exists $uR_\alpha ^I u'$ such that $vR_\alpha ^I u'$. We show that $u''=\left\{ \psi ; I_\alpha \psi \in v\right\} \cup \left\{ \psi ; I_\alpha \psi \in u \right\}$ is consistent. To do so, we first show that $\left\{ \psi ; I_\alpha \psi \in v\right\}$ is consistent. Suppose for a contradiction that it is not consistent. Then there exists a set $\left\{ \psi _1,\dots ,\psi _n\right\}$ of formulas of $\mathcal {L}_{\textsf {KX}}$ such that $I_\alpha \psi _i\in v$ for every $1\le i \le n$ and (a) $\vdash _{\varLambda _{I}}\psi _1\wedge \dots \wedge \psi _n\rightarrow \bot$. By Necessitation for $I_\alpha$ and its distributivity over conjunction, (a) implies that $\vdash _{\varLambda _{I}}I_\alpha \psi _1\wedge \dots \wedge I_\alpha \psi _n\rightarrow I_\alpha \bot$, but this is a contradiction, since v is a $\varLambda _{I}$-MCS and it includes $I_\alpha \psi _1\wedge \dots \wedge I_\alpha \psi _n$. Next we show that $u''=\left\{ \psi ; I_\alpha \psi \in v\right\} \cup \left\{ \psi ; I_\alpha \psi \in u \right\}$ is also consistent. Suppose for a contradiction that it is not consistent. Since $\left\{ \psi ; I_\alpha \psi \in v'\right\}$ and $\left\{ \psi ; I_\alpha \psi \in u \right\}$ are consistent, there must exist sets $\left\{ \varphi _1,\dots ,\varphi _n\right\}$ and $\left\{ \theta _1,\dots ,\theta _m\right\}$ of formulas of $\mathcal {L}_{\textsf {KX}}$ such that $I_\alpha \varphi _i\in v$ for every $1\le i \le n$, $I_\alpha \theta _i\in w_2$ for every $1\le i\le m$, and (b) $\vdash _{\varLambda _{I}}\varphi _1\wedge \dots \wedge \varphi _n\wedge \theta _1\wedge \dots \wedge \square \theta _m\rightarrow \bot$. Let $\theta =\theta _1\wedge \dots \wedge \theta _m$ and $\varphi =\varphi _1\wedge \dots \wedge \varphi _n$. Thus, (b) implies that $\vdash _{\varLambda _{I}}\varphi \rightarrow \lnot \theta$ and thus that (c) $\vdash _{\varLambda _{I}}\langle I_\alpha \rangle \varphi \rightarrow \langle I_\alpha \rangle \lnot \theta$. Notice that the facts that $I_\alpha \varphi _i\in v$ for every $1\le i\le n$, that $I_\alpha$ distributes over conjunction, and that v is a $\varLambda _{I}$-MCS imply that $I_\alpha \varphi \in v$. Analogously, one has that ($\star$) $I_\alpha \theta \in u$. The fact that $v\in \overline{w}$ implies that $\Diamond I_\alpha \varphi \in w$, so that (Den) entails that $K_\alpha \langle I_\alpha \rangle \varphi \in w$. Now, this last inclusion implies, with the fact that $w\approx _\alpha u$, that $\langle I_\alpha \rangle \varphi \in u$, which by (c) in turn yields that $\langle I_\alpha \rangle \lnot \theta \in u$, contradicting $(\star )$. Therefore, $u''$ is consistent. Let $u'$ be the $\varLambda _{I}$-MCS that includes $u''$. It is clear from its construction that $uR_\alpha ^I u'$ and that $vR_\alpha ^I u'$. With this, we have shown that $\mathcal {M}$ satisfies $\mathtt {(Den)_K}$.

Lemma 1

(Existence for non-intentional operators). Let $\mathcal {M}$ be the canonical Kripke-ies-model for ${\varLambda _I}$. Let $w\in W^{\varLambda _I}$. For $\varphi$ of $\mathcal L_{\textsf {I}}$, the following items hold:

• 1. $\square \varphi \in w$ iff $\varphi \in v$ for every $v\in \overline{w}$.

• 2. $[\alpha ]\varphi \in w$ iff $\varphi \in v$ for every $v\in \overline{w}$ such that $wR_\alpha ^{\overline{w}} v$.

• 3. $K_\alpha \varphi \in w$ iff $\varphi \in v$ for every $v\in W^{\varLambda _I}$ such that $w\approx _\alpha v$.

Proof

Let $w\in W^{\varLambda _I}$, and take $\varphi$ of $\mathcal L_{\textsf {I}}$. All items are shown in the same way. Let $\triangle \in \left\{ \square , [\alpha ], K_\alpha \right\}$, and let $R_\triangle$ stand for the relation upon which the semantics of $\triangle \varphi$ is defined. We show that $\triangle \varphi \in w$ iff $\varphi \in v$ for every $v\in W^{\varLambda _I}$ such that $wR_\triangle v$.

$(\Rightarrow )$ Assume that $\triangle \varphi \in w$. Let $v\in W^{\varLambda _I}$ such that $wR_\triangle v$. The definition of $R_\triangle$ straightforwardly gives that $\varphi \in v$.

$(\Leftarrow )$ We work by contraposition. Assume that $\triangle \varphi \notin w$. We show that there is a world v in $W^{\varLambda _I}$ such that $wR_\triangle v$ and such that $\varphi$ does not lie within it. For this, let $v'=\left\{ \psi ;\triangle \psi \in w\right\}$, which is shown to be consistent as follows: suppose for a contradiction that $v'$ is not consistent; then there exists a set $\left\{ \psi _1,\dots ,\psi _n\right\}$ of formulas of $\mathcal {L}_{\textsf {KX}}$ such that $\left\{ \psi _1,\dots ,\psi _n\right\} \subseteq v'$ and (a) $\vdash _{\varLambda _I}\psi _1\wedge \dots \wedge \psi _n\rightarrow \bot$; now, the fact that $\left\{ \psi _1,\dots ,\psi _n\right\} \subseteq v'$ means that $\triangle \psi _i\in w$ for every $1\le i \le n$; Necessitation for $\triangle$ and its distributivity over conjunction yield that (a) implies that $\vdash _{\varLambda _I}\triangle \psi _1\wedge \dots \wedge \triangle \psi _n\rightarrow \triangle \bot$, but this is a contradiction, since w is a $\varLambda _I$-MCS which includes $\triangle \psi _1\wedge \dots \wedge \triangle \psi _n$. Now, we define $v':'=v'\cup \left\{ \lnot \varphi \right\}$, and we show that it is also consistent: suppose for a contradiction that it is not consistent; since $v'$ is consistent, there exists a set $\left\{ \psi _1,\dots ,\psi _n\right\}$ of formulas of $\mathcal {L}_{\textsf {KX}}$ such that $\left\{ \psi _1,\dots ,\psi _n\right\} \subseteq v'$ and $\vdash _{\varLambda _I}\psi _1\wedge \dots \wedge \psi _n \wedge \lnot \varphi \rightarrow \bot$, which then implies that (b) $\vdash _{\varLambda _I}\psi _1\wedge \dots \wedge \psi _n \rightarrow \varphi$; By Necessitation for $\triangle$ and its distributivity over conjunction, (b) implies that $\vdash _{\varLambda _I}\triangle \psi _1\wedge \dots \wedge \triangle \psi _n\rightarrow \triangle \varphi$; but, since w is a $\varLambda _I$-MCS, then $\triangle \psi _1\wedge \dots \wedge \triangle \psi _n\in w$, so that (b) and the fact that w is closed under Modus Ponens entail that $\triangle \varphi \in w$, contradicting the initial assumption that $\triangle \varphi \notin w$. Let v be the $\varLambda _I$-MCS that includes $v''$. It is clear from its construction that $\varphi \notin v$ and that $wR_\triangle v$, by definition of $R_\triangle$.

Lemma 2

(Truth Lemma). Let $\mathcal {M}$ be the canonical Kripke-ies-model for ${\varLambda _I}$. For $\varphi$ of $\mathcal {L}_{\textsf {I}}$ and $w\in W^{\varLambda _I}$, $\mathcal {M},w\models \varphi \ \text {iff} \ \varphi \in w.$

Proof

We proceed by induction on $\varphi$. The cases of Boolean connectives are standard. For formulas involving $\square , [\alpha ]$, and $K_\alpha$, both directions follow straightforwardly from Lemma 1 (items 1, 2, and 3, respectively). As for $I_\alpha$, we have the following arguments:

• (“$I_\alpha$”)

  ($\Rightarrow$) We work by contraposition. Suppose that $I_\alpha \varphi \notin w$. Take $x\in \pi _\alpha ^\square [w]$. The assumption that $\lnot I_\alpha \varphi \in w$ implies, by schema (KI) and closure of w under Modus Ponens, that $\square K_\alpha \lnot I_\alpha \varphi \in w$. Since $x\in \pi _\alpha ^\square [w]$, this implies that $\lnot I_\alpha \varphi \in x$. By an argument analogous to the one used in Proposition 4 to show that the canonical model satisfies $(\texttt{Den})_{\texttt{K} }$, the set $\left\{ \psi ; I_\alpha \psi \in x\right\}$ is consistent. Next, observe that $\left\{ \psi ; I_\alpha \psi \in w\right\} \cup \left\{ \lnot \varphi \right\}$ is consistent. Suppose it is not consistent. Since $\left\{ \psi ; I_\alpha \psi \in w\right\}$ is consistent, there must exist a set $\left\{ \varphi _1,\dots ,\varphi _n\right\}$ such that $I_\alpha \varphi _i\in w$ for every $1\le i \le n$ and $\vdash _{\varLambda _I}(\varphi _1\wedge \dots \wedge \varphi _n)\wedge \lnot \varphi \rightarrow \bot$ Now, this ${\varLambda _I}$-theorem implies that $\vdash _{\varLambda _I}(\varphi _1\wedge \dots \wedge \varphi _n)\rightarrow \varphi$. By Necessitation of $I_\alpha$, its schema (K), and its distributivity over conjunction, one then has that ($\star$) $\vdash _{\varLambda _I}(I_\alpha \varphi _1\wedge \dots \wedge I_\alpha \varphi _n)\rightarrow I_\alpha \varphi$. Now, closure of w under conjunction then implies that $\left( \bigwedge _{1\le i\le n}I_\alpha \varphi _i\right) \in x$, so that the antecedent in ${\varLambda _I}$-theorem $(\star )$ lies in x. Closure of x under Modus Ponens then implies that $I_\alpha \varphi \in x$, but this contradicts the previously shown fact that $I_\alpha \varphi \notin x$. Therefore, $\left\{ \psi ; I_\alpha \psi \in w\right\} \cup \left\{ \lnot \varphi \right\}$ is in fact consistent. Let u be the ${\varLambda _I}$-MCS that includes $\left\{ \psi ; I_\alpha \psi \in w\right\} \cup \left\{ \lnot \varphi \right\}$. It is clear from construction that $x R_\alpha ^I u$, so that $u\in x\uparrow _{R_\alpha ^{+}}$. It also clear from construction that $\lnot \varphi \in x$, so that the induction hypothesis yields that $\mathcal {M}, x\models \lnot \varphi$. Thus, x is such that $x\in \pi _\alpha ^\square [w]$ and such that $x\uparrow _{R_\alpha ^{+}}\not \subseteq |\varphi |$, which implies that $\mathcal {M}, w\not \models \varphi$.

  ($\Leftarrow$) Assume that $I_\alpha \varphi \in w$. Suppose for a contradiction that $\mathcal {M}, w\not \models I_\alpha \varphi$. This means that, for $x\in \pi _\alpha ^\square [w]$, there exists y such that $xR_\alpha ^{I+} y$ and $\mathcal {M}, y\not \models \varphi$. Now, we have two cases. If for every $x\in \pi _\alpha ^\square [w]$ the y such that $xR_\alpha ^{I+} y$ and $\mathcal {M}, y\not \models \varphi$ is actually x itself, then $\mathcal {M}, x\not \models \varphi$ for every $x\in \pi _\alpha ^\square [w]$. By induction hypothesis, this implies that $\lnot \varphi \in x$ for every $x\in \pi _\alpha ^\square [w]$, which, by items 1 and 3 of Lemma 1, implies that $\square K_\alpha \lnot \varphi \in x$ for every $x\in \pi _\alpha ^\square [w]$. In particular, $\square K_\alpha \lnot \varphi \in w$. Schema (InN) and closure of w under Modus Ponens then imply that $I\lnot \varphi \in w$, but this is a contradiction, since the fact that $I_\alpha \varphi \in w$, with schema (D) for $I_\alpha$ and closure of w under Modus Ponens, implies that $\lnot I\lnot \varphi \in w$. The other case is that there exist $x, y\in \pi _\alpha ^\square [w]$ such that $xR_\alpha ^{I+} y$, $\mathcal {M}, y\not \models \varphi$, and $y\ne x$. By induction hypothesis, $\varphi \notin y$. Since $xR_\alpha ^{I+} y$ and $y\ne x$, then $xR_\alpha ^{I} y$, so the definition of $R_\alpha ^{I}$ implies that $I_\alpha \varphi \not \in x$. As such, $\lnot I_\alpha \varphi \in x$, which, by schema (KI) and closure of x under Modus Ponens, implies that $\square K_\alpha \lnot I_\alpha \varphi \in x$. Since $x\in \pi _\alpha ^\square [w]$, this implies that $\lnot I_\alpha \varphi \in w$, but this is a contradiction to the initial assumption.

Theorem 2

(Completeness w.r.t. Kripke-ies-models). The proof system ${\varLambda _I}$ is complete with respect to the class of Kripke-ies-models.

Proof

Let $\varphi$ be a ${\varLambda _I}$-consistent formula of $\mathcal {L}_{\textsf {I}}$. Let w be the ${\varLambda _I}$-MCS including $\varphi$. The canonical Kripke-ies-model $\mathcal {M}$ for $\varLambda _I$ is such that $\mathcal {M}, w\models \varphi$.

Back to Branching-Time Models

Theorem 3

(Completeness w.r.t. iebt-models). The proof system ${\varLambda _I}$ is complete with respect to the class of iebt-models.

Proof

Let $\varphi$ be a ${\varLambda _I}$-consistent formula of $\mathcal {L}_{\textsf {I}}$. Theorem 2 implies that there exists a Kripke-ies-model $\mathcal {M}$ and a world w in its domain such that $\mathcal {M} , w \models \varphi$. Proposition 3 then ensures that the iebt-model $\mathcal {M}^T$ associated to $\mathcal {M}$ is such that $\mathcal {M}^T, \left\langle \overline{w},h_w \right\rangle \models \varphi$.

Therefore, the following result, appearing in the main body of the paper, has been shown:

Theorem 1

The proof system ${\varLambda _I}$ is sound and complete with respect to the class of iebt-models.

   $\square$