1.1 Formal framework

We work with a sentential language $\mathcal {L}$ with a countable set $\mathcal {L}_{AT}$ of atomic formulas, $p, q, r \ (p_1, p_2, \dots )$ , negation $\neg $ , conjunction $\wedge $ , a normal (global) modal operator $\Box $ , and a two-place (intentional modal) operator X, and round parentheses as auxiliary symbols. We use $\varphi , \psi , \eta \ (\varphi _1, \varphi _2, ...)$ , as metavariables for formulas of $\mathcal {L}$ . The well-formed formulas of the language $\mathcal {L}$ are given by the following grammar in BNF form:

$$ \begin{align*}p \ | \ \neg \varphi \ | \ (\varphi\wedge \psi) \ | \ \Box \varphi \ | \ X^\varphi\psi\end{align*} $$

where $p\in \mathcal {L}_{AT}$ . We use $\vee $ for disjunction, $\supset $ for the material conditional, and $\equiv $ for material equivalence, defined in the usual manner as $\varphi \vee \psi :=\neg (\neg \varphi \wedge \neg \psi )$ , $\varphi \supset \psi :=\neg \varphi \vee \psi $ , and $\varphi \equiv \psi :=(\varphi \supset \psi ) \wedge (\psi \supset \varphi )$ . We follow the usual rules for the elimination of the parentheses.

The topic of a complex sentence $\varphi $ , defined from its primitive components in $\mathfrak {At}(\varphi )$ , makes all the logical connectives and modal operators in $\mathcal {L}$ topic-transparent:

• • $t(\neg \varphi ) {= t(\Box \varphi )}= t(\varphi )$ ;

• • $t(\varphi \wedge \psi ) =t(X^\varphi \psi )=t(\varphi )\oplus t(\psi )$ .

Observe that $(\mathcal {T}, \oplus )$ is a join-semilattice of topics, and $(\mathcal {T}, \leq )$ a poset. [Reference Berto3, p. 1877] commits firmly to a mereological reading of the structure of topics, interpreting $\oplus $ as a topic fusion operator, and showing how topic parthood $\leq $ behaves as a partial order on $\mathcal {T}$ . A minor matter worth highlighting: having a join-semilattice together with a partial order does not suffice for a ‘mereology’; notably missing is any kind of decomposition principle.Footnote ⁴ Still, treating bare join-semilattices as ‘quasi-mereological’ for the purposes of formal semantics is commonplace going back to Sharvy [Reference Sharvy24]; Link [Reference Link, Bäuerle, Christoph and Stechow20], and complicating this structure is beyond the scope of this paper.Footnote ⁵

Definition 2 (Semantics for $\mathcal {L}$ ( $\Vdash $ ))

Given a ts-model $\mathcal {M}=\langle W, \{R_\varphi \ | \ \varphi \in \mathcal {L}\}, \mathcal {T}, \oplus , t, V \rangle $ and a state $w\in W$ , the $\Vdash $ -semantics for the language $\mathcal {L}$ is defined recursively as:

$$\begin{align*}\begin{array}{llll} \mathcal{M}, w \Vdash p & \text{iff}& w\in V(p)\\ \mathcal{M}, w \Vdash \neg \varphi& \text{iff}& \text{not} \ \mathcal{M}, w\Vdash \varphi\\ \mathcal{M}, w \Vdash \varphi\wedge \psi & \text{iff}& \mathcal{M}, w\Vdash \varphi \ \text{and} \ \mathcal{M}, w\Vdash \psi\\ \mathcal{M}, w \Vdash \Box \varphi & \text{iff} & \text{for all } w'\in W (\mathcal{M}, w'\Vdash \varphi) \\ \mathcal{M}, w\Vdash X^\varphi\psi & \text{iff} & \text{for all } w'\in W (\text{if} \ wR_\varphi w' \text{ then } \mathcal{M}, w'\Vdash \psi) \ \text{and} \ t(\psi)\leq t(\varphi). \\ \end{array}\end{align*}$$

The intension of $\varphi $ with respect to $\mathcal {M}$ is $|\varphi |_{\mathcal {M}} := \{ w\in W : \mathcal {M}, w \Vdash \varphi \}$ . We omit the subscript $\mathcal {M}$ and write $|\varphi |$ when the model is contextually clear. To ease notation in proofs, we define $R_\varphi (w)=\{w'\in W : wR_\varphi w'\}$ . Note the clause for $X^\varphi \psi $ to be true in $\mathcal {M}$ at w. This clause requires both that (i) $\psi $ must be true at all $\varphi $ -accessible worlds $w'$ , and (ii) that the topic of $\psi $ be part of the topic of $\varphi $ . The latter corresponds to the aboutness preservation (AP) and is subject to concerns raised above.

Logical consequence is defined in the usual way, as truth preservation at all worlds of all models. With $\Sigma \subseteq \mathcal {L}$ ,

• Logical consequence $\Sigma \Vdash \varphi $ iff for all models $\mathcal {M}=\langle W, \{R_\varphi \ | \ \varphi \in \mathcal {L}\}, \mathcal {T}, \oplus , t, V \rangle $ and all $w\in W$ : if $\mathcal {M}, w\Vdash \psi $ for all $\psi \in \Sigma $ , then $\mathcal {M}, w\Vdash \varphi $ .Footnote ⁶

2.1 Mereotopology

Our approach supplements the quasi-mereological structure of $(\mathcal {T}, \oplus )$ with a topological closure operation $f: \mathcal {T} \rightarrow \mathcal {T}$ satisfying the following so-called ‘Kuratowski’ axioms.

• Inclusion $x \leq f(x)$

• Additivity $f(x \oplus y) = f(x) \oplus f(y)$

• Idempotence $f(f(x)) = f(x)$

These three constraints entail that the operation f is monotone increasing with respect to topic parthood.

• Monotone Increasingness if $x \leq y $ then $f(x) \leq f(y)$

This constraint will become important in §3.

Intuitively, f takes a topic of an imaginative input and maps it to (possibly) another topic that integrates the topics of the contextual, relevant background information that is taken on board in the imaginative episode initiated by the input, allowing for imaginative jumps to relevant topics. For example, given the input that ‘Gwenny is at her favourite lake’, f maps the topic of the input to a topic that includes topics of relevant background information such as ‘Gwenny likes swimming’, which, in turn, includes the topic of ‘Gwenny swims in her favourite lake’. Given the input that ‘we keep burning fossil fuels at this pace’, f maps the topic of this input to another one that includes also the topics of, e.g., ‘burning fossil fuels increases the emission of CO $_2$ ’, ‘more CO $_2$ in the air raises the global temperatures’, ‘ice melts in high temperatures’ etc., which will, in turn, include the topic of ‘the polar ice melts’.Footnote ⁷

Given this intuitive reading of f, the Kuratowski axioms are well-motivated from the philosophical standpoint of this paper. Inclusion ensures that the closure of a topic is only ever an expansion; it will never take us completely away from the topic of the input, but only broaden its scope. We allow for imaginative jumps from the topic of an imaginative input to a larger topic by integrating the topics of the contextual, relevant background information.

Additivity ensures that the result of closing the topic of a whole sentence $\varphi $ is not different from the result of closing the topics of the atoms within $\varphi $ and then fusing those topics. Expansions of the whole never outstrip the expansions of its parts.

Finally, Idempotence ensures that the expansion by imagination can’t be repeated unless given different inputs. So for example, we might take an input, expand the topic by imagining, and then once in a new topic find new inputs and go again. But absent new inputs, the possible outputs of repeated imaginings cannot change.

A new proposal emerges for how to understand what happens to topics in imagination. The idea is that the topics of a given input can be expanded to other, distinct, but connected topics in the output of imaginings. Not just any connection will do, only the points of connection — those topics most closely related to the inputs — will be permitted. That is, the topic of outputs of imagination must be within the closure of the topic of the input. To formalize this idea, we endow the ts-models with an additional function f and then place constraints on it.

$X^\varphi \psi $ is then interpreted in ts-models with functions as

(F-Sem) $$ \begin{align} w \models X^\varphi\psi \ \ \text{iff} \ \ (\text{for all } w' \text{ if } wR_{\varphi}w' \text{ then } w' \models \psi) \text{ and } t(\psi) \leq f(t(\varphi)). \end{align} $$

These generic models place no constraints on f, and will become important later in §3 where we provide a sound and complete axiomatization for them. For the time being, we add the topological constraints on f as follows.

Similarly $X^\varphi \psi $ is then interpreted in topo-ts-models as

(C-Sem) $$ \begin{align} w \models X^\varphi\psi \ \ \text{iff} \ \ (\text{for all } w' \text{ if } wR_{\varphi}w' \text{ then } w' \models \psi) \text{ and } t(\psi) \leq c(t(\varphi)). \end{align} $$

Imaginative episodes can therefore lead to proper expansions of the subject matter — even to output topics that fail to mereologically overlap input topics. This can happen when connected output topics are contained within the closure of the input topic.

2.2 Resulting logic for the topological closure operator

What logic results from this new mereotopological approach? Let’s now compare the logic of ts-models wrt the semantics given in Definition 2 to the logic of topo-ts-models wrt the so-called closure semantics as in (C-Sem). For the sake of presentation, some formal definitions and notational conventions are given below.

The semantic clauses of the other components of $\mathcal {L}$ in topo-ts-models are as given in Definition 2 and we denote the intension of $\varphi $ with respect to topo-ts-model $\mathcal {X}$ by $[\![ \varphi ]\!]_{\mathcal {X}} := \{ w\in W : \mathcal {X}, w \models \varphi \}$ .

The notion of logical consequence wrt the C-Sem in topo-ts-models is defined standardly (as in §1.1) and we write $\Gamma \models \varphi $ when $\varphi $ is a logical consequence of $\Gamma $ wrt the semantics given in Definition 2 with C-Sem in topo-ts-models. Recall that $\Gamma \Vdash \varphi $ says $\varphi $ is a logical consequence of $\Gamma $ wrt the class of ts-models (without a closure operator). The notion of logical validity in topo-ts-models, $\models \varphi $ , is standardly defined as entailment by the empty set of premises ( $\emptyset \models \varphi $ ). To be precise, we take the logic of ts-models wrt the semantics given in Definition 2 to be the set of all logical consequence relations, namely the set $\{(\Gamma , \varphi )\in \mathcal {P}(\mathcal {L})\times \mathcal {L} \ | \ \Gamma \Vdash \varphi \}$ . Similarly, the logic of all topo-ts-models wrt the C-Sem is defined as $\{(\Gamma , \varphi )\in \mathcal {P}(\mathcal {L})\times \mathcal {L} \ | \ \Gamma \models \varphi \}$ . In the remainder of this section, we will show that these two sets are equivalent.

It is easy to see that every ts-model is a topo-model: take the closure operator c on $\mathcal {T}$ as the identity function, i.e., define $c: \mathcal {T}\rightarrow \mathcal {T}$ such that $c(x)=x$ for all $x\in \mathcal {T}$ . Such a c trivially satisfies inclusion, idempotence, and additivity. Moreover, given a topo-ts-model $\mathcal {X}=\langle W, \{R_\varphi \ | \ \varphi \in \mathcal {L}\}, \mathcal {T}, \oplus , c, t, V \rangle $ , we can construct a ts-model $\mathcal {M}_{\mathcal {X}}={\langle W, \{R_\varphi \ | \ \varphi \in \mathcal {L}\}, \mathcal {T}, \oplus , t_{\mathcal {X}}, V \rangle }$ where $t_{\mathcal {X}}(\varphi )=c(t(\varphi ))$ for all $\varphi \in \mathcal {L}$ .

Proof We first need to show that $\mathcal {M}_{\mathcal {X}}={\langle W, \{R_\varphi \ | \ \varphi \in \mathcal {L}\}, \mathcal {T}, \oplus , t_{\mathcal {X}}, V \rangle }$ is a ts-model:

• • $t_{\mathcal {X}}$ is well-defined: let $\varphi , \psi \in \mathcal {L}$ such that $t_{\mathcal {X}}(\varphi )\not =t_{\mathcal {X}}(\psi )$ . This means, by the definition of $t_{\mathcal {X}}$ , that $c(t(\varphi ))\not =c(t(\psi ))$ . Since c is well-defined, we obtain $t(\varphi )\not =t(\psi )$ . Similarly, since t is well-defined, we conclude that $\varphi \not =\psi $ .

• • for any $\varphi \in \mathcal {L}$ , $t_{\mathcal {X}}(\varphi )=\oplus \mathfrak {At}(\varphi )= t_{\mathcal {X}}(p_1)\oplus \dots \oplus t_{\mathcal {X}}(p_n)$ where $\mathfrak {At}(\varphi )=\{p_1, \dots , p_n\}$ :

  (by the defn. of t _𝒳) $$\begin{align} t_{\mathcal{X}}(\varphi) & =c(t(\varphi)) \qquad\qquad\qquad\qquad\qquad \end{align}$$
  (by the defn. of t) $$\begin{align} & = c(t(p_1)\oplus \dots \oplus t(p_n)) \qquad\end{align} $$
  (by additivity of c) $$\begin{align} & = c(t(p_1))\oplus \dots \oplus c(t(p_n)) \ \end{align}$$
  (by the defn. of t _𝒳) $$\begin{align} & = t_{\mathcal{X}}(p_1)\oplus \dots \oplus t_{\mathcal{X}}(p_n). \quad \end{align}$$

The proof of $[\![ \varphi ]\!]_{\mathcal {X}}=|\varphi |_{\mathcal {M}_{\mathcal {X}}}$ follows by induction on the structure of $\varphi $ , where cases for the atomic formulas, the Boolean connectives, and $\varphi :=\Box \psi $ are trivial. So assume inductively that the result holds for $\psi $ and $\eta $ , and show that it holds also for $\varphi :=X^\psi \eta $ . Let $w\in W$ :

(C-Sem) $$ \begin{align} \mathcal{X}, w\models X^\psi\eta & \text{ iff } R_{\psi}(w)\subseteq [\![ \eta]\!]_{\mathcal{X}} \text{ and } t(\eta) \leq c(t(\psi))\qquad\qquad \qquad\qquad\quad \end{align} $$

(induction hyp. and Lemma 1) $$ \begin{align} & \text{ iff } R_{\psi}(w)\subseteq |\eta|_{\mathcal{M}_{\mathcal{X}}} \text{ and } c(t(\eta)) \leq c(t(\psi)) \end{align} $$

(by the defn. of t _𝒳) $$\begin{align} & \text{ iff } R_{\psi}(w)\subseteq |\eta|_{\mathcal{M}_{\mathcal{X}}} \text{ and } t_{\mathcal{X}}(\eta) \leq t_{\mathcal{X}}(\psi)\qquad\qquad \qquad \end{align} $$

(Defn.2) $$ \begin{align} & \text{ iff } \mathcal{M}_{\mathcal{X}}, w\Vdash X^\psi\eta. \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \end{align} $$

We then obtain the following result.

Theorem 3 shows that the addition of the closure operator does not change the resulting logic. From a purely formal perspective, the theorem shows that the aboutness preservation requirement (AP) in Berto’s logic is not essential to the system. The logic defined on mereotopological models, where AP clearly fails, results in the same logical system.

The result is also philosophically significant. An extremely natural and simple way of relaxing AP does not ultimately change the logic of imagination. So criticisms of AP (including Badura’s [Reference Berto and Özgün8]) do not ultimately refute Berto’s logic of the imagination operator.Footnote ⁸ However, these criticisms do display that certain sentences (e.g., ‘In an act of imagining that Gwenny is at her favourite lake, Helena imagines that Gwenny swims in the lake’) should be satisfiable, but simply aren’t on Berto’s semantics.Footnote ⁹ There is no model of them with topic assignments that allow for the required expansion. In our view, this simply refutes the semantic framework as an empirically adequate proposal for modelling the relevant truth conditions. Our semantic proposal arguably does a better job on this front, if one accepts the underlying philosophical contention that ‘swimming in the lake’ is a topic connected closely enough to the topic ‘Gwenny’s favourite lake’ as to be contained in its closure.Footnote ¹⁰

2.3 Expressivity

As noted, TSIMs are useful for more than just imagination; in other applications where the aboutness preservation constraint (AP) is more plausible, there is at least one potential benefit to Berto’s semantics over the closure semantics we have given above: Berto’s semantics permits topic parthood to be expressible in the language, while our mereotopological semantics does not. We show this expressive advantage below. On the flipside (and unsurprisingly) the mereotopological semantics can express topic-parthood and topic equivalence under the topic expansion operator, as shown in what follows.

The following abbreviation will be useful in the following: we will use ‘ $\overline {\varphi }$ ’ to denote the tautology $\bigwedge _{p\in \mathfrak {At}(\varphi )}(p\vee \neg p)$ Footnote ¹¹ , following a similar idea in [Reference Giordani15]. The reader should not identify $\overline {\varphi }$ with $\top $ : they will turn out to be logically equivalent for all $\varphi \in \mathcal {L}$ , but our bimodal operator X will discern them.

Notice that, since the topic component of the semantic clause for ‘ $X^\varphi \psi $ ’ uses the closure operator, $\mathcal {L}$ with respect to C-Sem is not expressive enough to speak of parthood relations. Instead it is expressive enough to speak of topic-parthood under closure. That is, $X^\psi {\overline {\varphi }}$ expresses, with respect to C-Sem, ‘The topic of $\varphi $ is included in the closure of the topic of $\psi $ ’ and also, via Lemma 1, ‘The closure of the topic of $\varphi $ is included in the closure of the topic of $\psi $ ’:

(t() = t(φ), since 𝔄𝔱() = 𝔄𝔱(φ)) $$\begin{align} \kern-10pt \mathcal{X}, w \models X^\psi{\overline{\varphi}}& \text{ iff } R_{\psi}(w)\subseteq [\![ \overline{\varphi}]\!] \text{ and } t(\overline{\varphi}) \leq c(t(\psi)) \notag\\ & \text{ iff }R_{\psi}(w)\subseteq W \text{ and } t(\varphi) \leq c(t(\psi)) \end{align}$$

(Lemma1) $$ \begin{align} & \kern-92pt \text{iff }t(\varphi) \leq c(t(\psi)) \notag \\& \kern-92pt \text{iff } c(t(\varphi)) \leq c(t(\psi)). \qquad\qquad\qquad \end{align} $$

On the other hand, in $\mathcal {L}$ we cannot say things like ‘The topic of $\varphi $ is included in the topic of $\psi $ ’, or ‘ $\varphi $ and $\psi $ have exactly the same topic’. The opposite is the case in Berto [Reference Berto3]: since the proposal in Berto [Reference Berto3] does not accommodate the closure operator, $X^\psi {\overline {\varphi }}$ there states precisely that the topic of $\varphi $ is included in that of $\psi $ (as also observed in [Reference Özgün and Berto21]):

(t() = t(φ), since 𝔄𝔱() = 𝔄𝔱(φ)) $$\begin{align} \mathcal{M}, w \Vdash X^\psi{\overline{\varphi}}& \text{ iff } R_{\psi}(w)\subseteq |\overline{\varphi}| \text{ and } t(\overline{\varphi}) \leq t(\psi) \notag\\ & \text{ iff }R_{\psi}(w)\subseteq W \text{ and } t(\varphi) \leq t(\psi) \\ & \text{ iff }t(\varphi) \leq t(\psi).\notag\end{align} $$

To see that $\mathcal {L}$ is not expressive enough to state ‘The topic of $\varphi $ is included in the topic of $\psi $ ’ with respect to C-Sem, consider the models $\mathcal {X}_1=\langle \{w\}, \{R_\varphi \ | \ \varphi \in \mathcal {L}\}, \{x_1, y_1, z_1\}, \oplus _1, c_1, t_1, V\rangle $ and $\mathcal {X}_2=\langle \{w\}, \{R_\varphi \ | \ \varphi \in \mathcal {L}\}, \{x_2, y_2, z_2\}, \oplus _2, c_2, t_2, V\rangle $ , where $R_\varphi =\{(w, w)\}$ for all $\varphi \in \mathcal {L}$ , $V(p)=V(q)=\{w\}$ , and $( \{x_1, y_1, z_1\} , \oplus _1, t_1)$ and $(\{x_2, y_2, z_2\}, \oplus _2, t_2)$ are as given in Figure 1, and both $c_1$ and $c_2$ are constant functions such that $c_i(x_i)=c_i(y_i)=c_i(z_i)=z_i$ for $i\in \{1, 2\}$ .Footnote ¹² We have ‘The topic of q is included in the topic of p’ true in $\mathcal {X}_1$ at w (since $t_1(q)=x_1\leq _1 y_1=t_1(p)$ ) and false in $\mathcal {X}_2$ at w (since $y_2=t_2(q)\not \leq _2 x_2 =t_2(p)$ ). However, as shown in Lemma 4, $\mathcal {X}_1, w$ and $\mathcal {X}_2, w$ are modally equivalent wrt C-Sem for the language $\mathcal {L}$ .

To see that $\mathcal {L}$ is not expressive enough to state ‘ $\varphi $ and $\chi $ have exactly the same topic’ with respect to C-Sem, compare the model $\mathcal {X}_1$ given in Figure 1a with $\mathcal {X}_3=\langle \{w\}, \{R_\varphi \ | \ \varphi \in \mathcal {L}\}, \{x_3, y_3\} , \oplus _3, c_3, t_3, V\rangle $ , where $(\{x_3, y_3\}, \oplus _3, t_3)$ is as given in Figure 2 and $c_3(x_3)=c_3(y_3)=y_3$ . It is then easy to see that ‘p and q have exactly the same topic’ is true in $\mathcal {X}_3$ at w (since $t_3(p)=x_3=t_3(q)$ ), whereas it is false in $\mathcal {X}_1$ at w (since $t_1(q)=x_1\not = y_1=t_1(p)$ ). However, $\mathcal {X}_1, w$ and $\mathcal {X}_3, w$ are modally equivalent with respect to the language $\mathcal {L}$ , that is, for all $\varphi \in \mathcal {L}$ , $\mathcal {X}_1, w\models \varphi $ iff $\mathcal {X}_3, w\models \varphi $ (the proof follows similarly to the proof of Lemma 4).Footnote ¹³

Figure 1

Models $\mathcal {X}_1$ and $\mathcal {X}_2$ . (In figures of models, circles represent possible worlds, diamonds represent possible topics. Lines between topics represent the parthood relation going upwards (e.g., $x\leq z$ ). Valuation and topic assignment are given by labelling each node with atomic formulas. We omit labelling when a node is assigned every element in $\mathcal {L}_{AT}$ . The same conventions apply to all our diagrams.)

Figure 2

Model $\mathcal {X}_3$ .

These results suggest one possible reason to prefer Berto’s semantics for TSIMs over ours in applications where the aboutness preservation constraint (AP) is not in question and topic-parthood is desired to be expressible in the modal language $\mathcal {L}$ .

3.1 Definability of the properties of f

In this section, we prove that inclusion and monotone increasingness are both definable in $\mathcal {L}$ , while additivity is not. To do so, we need to define a few auxiliary notions and a notion of frame definability for ts-frames with functions, adapting the corresponding standard definitions of modal logic (see, e.g., [Reference Blackburn, de Rijke and Venema9, Chapter 3] and [Reference Pacuit23, Chapter 2.5]).

Given a ts-frame with functions $\mathcal {E}=\langle W, \{R_\varphi \ | \ \varphi \in \mathcal {L}\}, \mathcal {T}, \oplus , f \rangle $ , a topic assignment function $t: \mathcal {T}\times \mathcal {T}\rightarrow \mathcal {T}$ , and a valuation function $V:\mathcal {L}_{AT} \rightarrow \mathcal {P}(W)$ , the tuple $\mathcal {X}^{\mathcal {E}}=\langle W, \{R_\varphi \ | \ \varphi \in \mathcal {L}\}, \mathcal {T}, \oplus , f, t, V \rangle $ is called a ts-model with functions based on $\mathcal {E}$ . We say a formula $\varphi \in \mathcal {L}$ is valid in $\mathcal {E}$ , denoted by $\mathcal {E} \models \varphi $ if $\mathcal {X}^{\mathcal {E}}, w\models \varphi $ for all models $\mathcal {X}^{\mathcal {E}}$ based on $\mathcal {E}$ and $w\in W$ .

Proof Suppose, toward contradiction, that additivity is definable in $\mathcal {L}$ , that is, there is a formula $\varphi $ such that for all frames $\mathcal {E}$ , $\mathcal {E}\models \varphi $ iff f in $\mathcal {E}$ satisfies additivity. Now consider the following two frames $\mathcal {E}_1{\kern-1pt}={\kern-1pt}\langle \{w\}, \{R_\psi | \psi {\kern-1pt}\in{\kern-1pt} \mathcal {L}\}, \mathcal {T}_1, \oplus _1, f_1 \rangle $ and $\mathcal {E}_2{\kern-1pt}={\kern-1pt}\langle \{w\}, \{R_\psi | \psi {\kern-1pt}\in{\kern-1pt} \mathcal {L}\}, \mathcal {T}_2, \oplus _2, f_2 \rangle $ where $(\mathcal {T}_1, \oplus _1, f_1)$ and $(\mathcal {T}_2, \oplus _2, f_2)$ are as given in Figure 3a and 3b, respectively, such that $f_1(x_1)=x_1, f_1(y_1)=y_1, f_1(z_1)=f_1(u_1)=u_1$ (represented by the arrows in Figure 3a), $f_2$ is the identity function (represented by the arrows in Figure 3b).Footnote ¹⁸ $R_\psi =\{(w, w)\}$ for all $\psi \in \mathcal {L}$ .

It is easy to verify that $f_2$ is additive whereas $f_1$ is not ( $f_1(x_1\oplus y_1)=f(z_1)=u_1\not = z_1 = f(x_1)\oplus f(y_1)$ .) Therefore, it should be that $\mathcal {E}_2 \models \varphi $ but $\mathcal {E}_1\not \models \varphi $ , as $\varphi $ is assumed to define additivity. The latter means that there is a model $\mathcal {X}^{\mathcal {E}_1}=\langle \{w\}, \{R_\psi | \psi \in \mathcal {L}\}, \mathcal {T}_1, \oplus _1, f_1, t_1, V_1 \rangle $ based on $\mathcal {E}_1$ such that $\mathcal {X}^{\mathcal {E}_1}, w\not \models \varphi $ . However, as shown below, for all $\beta \in \mathcal {L}$ , $\mathcal {X}^{\mathcal {E}_1}, w\models \beta $ iff $\mathcal {X}^{\mathcal {E}_2}, w\models \beta $ , where $\mathcal {X}^{\mathcal {E}_2}=\langle \{w\}, \{R_\psi | \psi \in \mathcal {L}\}, \mathcal {T}_2, \oplus _2, f_2, t_2, V_2 \rangle $ such that $V_1=V_2$ and for $\varphi \in \mathcal {L}$ :

$$ \begin{align*}t_2(\varphi)= \begin{cases} x_2, \text{ if }t_1(\varphi)=x_1\\ y_2, \text{ if }t_1(\varphi)=y_1\\ z_2, \text{ if }t_1(\varphi)=z_1\text{ or }t_1(\varphi)=u_1.\\ \end{cases} \end{align*} $$

(It is easy to verify that $t_2$ is a well-defined topic assignment function.) Therefore, $\mathcal {X}^{\mathcal {E}_2}, w\not \models \varphi $ , contradicting $\mathcal {E}_2\models \varphi $ (as $\mathcal {X}^{\mathcal {E}_2}$ is a model based on $\mathcal {E}_2$ ). All that is left to show is the above-mentioned equivalence claim of $\mathcal {X}^{\mathcal {E}_1}, w$ and $\mathcal {X}^{\mathcal {E}_2}, w$ . We prove this by induction on the complexity of $\beta $ . The cases for the propositional variables, Booleans, and $\varphi :=\Box \psi $ are standard. We here prove the case $\beta :=X^\psi \eta $ . Moreover, since $R_\psi (w)\subseteq |\eta |_{\mathcal {X}^{\mathcal {E}_1}}$ iff $R_\psi (w)\subseteq |\eta |_{\mathcal {X}^{\mathcal {E}_2}}$ follows by the induction hypothesis and the fact that the possible-worlds components of both models are the same, we only need to show that $t_1(\eta )\leq f_1(t_1(\psi ))$ iff $t_2(\eta )\leq f_2(t_2(\psi ))$ .

( $\Rightarrow $ ) Suppose that $t_1(\eta )\leq f_1(t_1(\psi ))$ . We then have three cases:

(Case 1) $f_1(t_1(\psi ))=x_1$ : This means that $t_1(\psi )=x_1$ and $t_1(\eta )=x_1$ . Then, by the defn. $t_2$ and $f_2$ , we have that $t_2(\eta )=x_2\leq x_2=f_2(x_2)=f_2(t_2(\psi ))$ .

(Case 2) $f_1(t_1(\psi ))=y_1$ : Similar to the above case.

(Case 3) $f_1(t_1(\psi ))=u_1$ : This means that $t_1(\psi )=z_1$ or $t_1(\psi )=u_1$ . No matter which is the case, by the defn. of $t_2$ and $f_2$ , $t_2(\psi )=z_2=f_2(t_2(\psi ))$ . Therefore, by the structure of $(\mathcal {T}_2, \oplus _2, f_2)$ , we obtain that $t_2(\eta )\leq f_2(t_2(\psi ))$ .

( $\Leftarrow $ ) Suppose that $t_2(\eta )\leq f_2(t_2(\psi ))$ . We again have three cases:

(Case 1) $f_2(t_2(\psi ))=x_2$ : This means that $t_2(\psi )=x_2$ and $t_2(\eta )=x_2$ . Then, by the defn. of $t_2$ and $f_2$ , we have that $t_1(\eta )=x_1\leq x_1=f_1(x_1)=f_1(t_1(\psi ))$ .

(Case 2) $f_2(t_2(\psi ))=y_2$ : Similar to the above case.

(Case 3) $f_2(t_2(\psi ))=z_2$ : This means that $t_2(\psi )=z_2$ . Then, by the defn. of $t_2$ and $f_2$ , we have that $t_1(\psi )=z_1$ or $t_1(\psi )=u_1$ . No matter which one is the case, we have $t_1(\eta )\leq u_1=f_1(t_1(\psi ))$ .

3.2 Logics for weaker expansion operators

3.2.1 Inclusion and monotone increasing

In this section we provide strongly sound and complete axiomatizations for the logics of inclusive ts-models with functions, and inclusive and monotone increasing ts-models with functions. Soundness is a matter of routine validity check so we skip its proof. We prove the completeness results via a canonical model construction and present the proofs in full detail. Note that our canonical model is in fact more general and provides a completeness proof with respect to all ts-models with functions.

A sound and complete axiomatization $\mathsf {Log}_{incl}$ of the logic of inclusive ts-models with functions is presented in Table 1.

Table 1

Axiomatization $\mathsf {Log}_{incl}$ for the logic of inclusive ts-models with functions.

The notion of derivation, denoted by $\vdash _{incl}$ , in $\mathsf {Log}_{incl}$ is defined as usual [Reference Blackburn, de Rijke and Venema9, p. 192]. For any set of formulas $\Gamma \subseteq \mathcal {L}$ and any $\varphi \in \mathcal {L}$ , we write $\Gamma \vdash _{incl} \varphi $ if there exists finitely many formulas $\varphi _1, \dots , \varphi _n\in \Gamma $ such that $\vdash _{incl}(\varphi _1\wedge \dots \wedge \varphi _n)\supset \varphi $ . We say that $\Gamma $ is $\mathsf {Log}_{incl}$ -consistent if $\Gamma \not \vdash _{incl} \bot $ , and $\mathsf {Log}_{incl}$ -inconsistent otherwise. A sentence $\varphi $ is $\mathsf {Log}_{incl}$ -consistent with $\Gamma $ if $\Gamma \cup \{\varphi \}$ is $\mathsf {Log}_{incl}$ -consistent (or, equivalently, if $\Gamma \not \vdash _{incl} \neg \varphi $ ). Finally, a set of formulas $\Gamma $ is a maximally $\mathsf {Log}_{incl}$ -consistent set (or, in short, mcs) if it is $\mathsf {Log}_{incl}$ -consistent and any set of formulas properly containing $\Gamma $ is $\mathsf {Log}_{incl}$ -inconsistent [Reference Blackburn, de Rijke and Venema9]. We drop mention of the logic $\mathsf {Log}_{incl}$ when it is clear from the context and also simply write $\vdash $ for $\vdash _{incl}$ . Similar definitions also hold for the logic $\mathsf {Log}_{incl + mon}$ of inclusive and monotone increasing ts-models with functions obtained by replacing $\mathsf {Ax3}$ in Table 1 by the following stronger principle:

$$ \begin{align*}\mathsf{sAx3}: \ X^\psi\overline{\varphi} \supset X^\eta\overline{\varphi} \text{ if } \mathfrak{At}(\psi) \subseteq \mathfrak{At}(\eta).\end{align*} $$

Lemma 7 The following is derivable in $\mathsf {Log}_{incl}$ : $X^\psi \overline {\varphi }\supset X^\psi \overline {\eta }$ , if $\mathfrak {At}(\eta )\subseteq \mathfrak {At}(\varphi )$ .

Proof Follows from $\mathsf {Ax2}$ and $\mathsf {CPL}$ .

Lemma 8 For every mcs $\Gamma $ of $\mathsf {Log}_{incl}$ and $\varphi , \psi \in \mathcal {L}$ , the following hold:

1. 1. $\Gamma \vdash \varphi $ iff $\varphi \in \Gamma $ ,

2. 2. if $\varphi \in \Gamma $ and $\varphi \supset \psi \in \Gamma $ then $\psi \in \Gamma $ ,

3. 3. if $\vdash \varphi $ then $\varphi \in \Gamma $ ,

4. 4. $\varphi \in \Gamma $ and $\psi \in \Gamma $ iff $\varphi \wedge \psi \in \Gamma $ ,

5. 5. $\varphi \in \Gamma $ iff $\neg \varphi \not \in \Gamma $ .

Proof Standard.

Lemma 9 (Lindenbaum’s Lemma)

Every $\mathsf {Log}_{incl}$ -consistent set can be extended to a maximally $\mathsf {Log}_{incl}$ -consistent one.

Proof Standard.

Canonical Model Construction: Our canonical model is similar to the one presented in Giordani [Reference Giordani15] except for the construction of the topic relevant components. We follow Siemers [Reference Siemers25] for the construction of $\mathcal {T}^c$ and $\oplus ^c$ and, to the best of our knowledge, component $f^c$ is novel.

Let $W^c$ be the set of all maximally $\mathsf {Log}_{incl}$ -consistent sets. For each $\Gamma \in W^c$ , define

$$ \begin{align*} \Gamma[\Box] \ & :=\{\varphi\in \mathcal{L}: \Box\varphi\in \Gamma\}, \notag\\ \Gamma[X^\psi] \ & := \ \{\varphi \in \mathcal{L} : X^\psi\eta \wedge \Box (\eta\supset \varphi) \in \Gamma \text{ for some } \eta\in \mathcal{L}\}, \text{ and } \notag\\ \Gamma/\psi \ & := \ \{\varphi \in \mathcal{L} : X^\psi\varphi\in \Gamma \}. \notag \end{align*} $$

Moreover, we define $\sim _\Box $ and $\rightarrow ^c_\varphi $ on $W^c$ , respectively, as

$$ \begin{align*} \Gamma \sim_\Box \Delta \ & \text{ iff } \ \Gamma[\Box] \subseteq \Delta,\notag\\ \Gamma \rightarrow^c_\psi \Delta \ & \text{ iff } \ \Gamma[X^\psi] \subseteq \Delta. \notag \end{align*} $$

Since $\Box $ is an $\mathsf {S5}$ modality, it is easy to see that $\sim _\Box $ is an equivalence relation [Reference Blackburn, de Rijke and Venema9]. For any maximally $\mathsf {Log}_{incl}$ -consistent set $\Gamma $ , we denote by $[\Gamma ]_\Box $ the equivalence class of $\Gamma $ induced by $\sim _\Box $ , i.e., $[\Gamma ]_\Box =\{\Delta \in W^c : \Gamma \sim _\Box \Delta \}$ . Moreover, as shown by the lemma below, $\rightarrow ^c_\psi \subseteq \sim _\Box $ .

Lemma 10 For all $\psi \in \mathcal {L}$ and $\Gamma , \Delta \in W^c$ , if $\Gamma \rightarrow ^c_\psi \Delta $ , then $\Gamma \sim _\Box \Delta $ , i.e., $\rightarrow ^c_\psi \subseteq \sim _\Box $

Proof Let $\psi \in \mathcal {L}$ and $\Gamma , \Delta \in W^c$ such that $\Gamma \rightarrow ^c_\psi \Delta $ , i.e., that $\Gamma [X^\psi ]\subseteq \Delta $ . Let $\varphi \in \Gamma [\Box ]$ . This means that $\Box \varphi \in \Gamma $ . Then, by $\mathsf {S5}_\Box $ (since $\vdash \Box \varphi \equiv \Box (\overline {\psi } \rightarrow \varphi )$ ) and Lemma 8, we have that $\Box (\overline {\psi } \supset \varphi )\in \Gamma $ . Moreover, by $\mathsf {Ax1}$ , we have that $X^\psi \overline {\psi }\in \Gamma $ . Then, by the definition of $\Gamma [X^\psi ]$ , we conclude that $\varphi \in \Gamma [X^\psi ]$ . Then, by the first assumption that $\Gamma [X^\psi ]\subseteq \Delta $ , we obtain $\varphi \in \Delta $ . Therefore, $\Gamma [\Box ]\subseteq \Delta $ , i.e., $\Gamma \sim _\Box \Delta $ .

Given a mcs $\Gamma _0$ of $\mathsf {Log}_{incl}$ , the canonical model for $\Gamma _0$ is a tuple $\mathcal {X}^c=\langle [\Gamma _0]_\Box , \{R^c_\psi \ | \ \psi \in \mathcal {L}\}, \mathcal {T}^c, \oplus ^c, f^c, t^c, V^c \rangle $ where

• • $[\Gamma _0]_\Box $ is as described above.

• • $R^c_\psi = \rightarrow ^c_\psi \cap ([\Gamma _0]_\Box \times [\Gamma _0]_\Box ).$

• • $\mathcal {T}^c:= \mathcal {P}(\mathcal {L}_{AT}).$

• • $\oplus ^c:\mathcal {T}^c \times \mathcal {T}^c \rightarrow \mathcal {T}^c$ such that for all $A, B\in \mathcal {P}(\mathcal {L}_{AT})$ , $A\oplus ^c B=A\cup B$ .

• • $t^c: \mathcal {L} \rightarrow \mathcal {T}^c$ such that, for all $\varphi \in \mathcal {L}$ , $t^c(\varphi )=\mathfrak {At}(\varphi )$ .

• • $f^c: \mathcal {T}^c \rightarrow \mathcal {T}^c$ such that

  $$ \begin{align*}f^c(A)= \begin{cases} \bigcup \{t(\varphi) : X^{(\bigwedge A)}\overline{\varphi} \in \Gamma_0\}, \text{ if }A\text{ is finite}\\ \mathcal{L}_{AT}, \text{ otherwise.}\\ \end{cases} \end{align*} $$

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

The canonical topic-parthood relation is the subset relation on $\mathcal {P}(\mathcal {L}_{AT})$ : $A\oplus ^c B= B$ iff $A\cup B=B$ iff $A\subseteq B$ .

Lemma 11 Given a mcs $\Gamma _0$ , the canonical model $\mathcal {X}^c=\langle [\Gamma _0]_\Box , \{R^c_\psi \ | \ \psi \in \mathcal {L}\}, \mathcal {T}^c, \oplus ^c, f^c, t^c, V^c \rangle $ for $\Gamma _0$ is an inclusive ts-model with functions.

Proof Observe that $\oplus ^c$ and $t^c$ are well-defined. Moreover, for all $\varphi \in \mathcal {L}$ , $t^c(\varphi )=\mathfrak {At}(\varphi ) = \{t^c(p) : p\in \mathfrak {At}(\varphi )\} = \oplus ^c\mathfrak {At}(\varphi )$ . Finally, $f^c$ satisfies inclusion, that is, for all $A\in \mathcal {T}^c$ , $A\subseteq f^c(A)$ : if A is infinite, by definition, $A\subseteq f^c(A)= \mathcal {L}_{AT}$ . Let A be finite and $p\in A$ . We know, by $\mathsf {Ax1}$ , that $\vdash X^{(\bigwedge A)}\overline {p}$ , thus, $X^{(\bigwedge A)}\overline {p}\in \Gamma _0$ . This means, by the definition of $f^c$ and $t^c$ , that $t^c(p)=\{p\}\subseteq f^c(A)$ . We then conclude that $A\subseteq f^c(A)$ .

Inclusivity of $f^c$ is guaranteed by $\mathsf {Ax1}$ (it is indeed guaranteed by the weaker theorem $\vdash X^\varphi \overline {\varphi }$ ). Therefore, the axiomatization given in Table 1 without $\mathsf {Ax1}$ is a strongly sound and complete axiomatization for all ts-models with functions (corresponding completeness proof follows the same steps as the proof presented in this section). Since we are interested in topic expansion operators, we focus on the logics of at least inclusive ts-models.

Lemma 12 For any mcs $\Gamma $ and $\psi , \varphi \in \mathcal {L}$ , $X^\psi \overline \varphi \in \Gamma \text { iff } X^\psi \overline {p}\in \Gamma \text { for all } p\in \mathfrak {At}(\varphi ).$

Proof The direction from left-to-right follows from Lemma 7. For the opposite direction, let $\mathfrak {At}(\varphi )=\{p_1, \dots , p_n\}$ and observe that $\overline {\varphi }:= \overline {p_1}\wedge \dots \wedge \overline {p_n}$ . If $X^\psi \overline {p_i}\in \Gamma $ for all $p_i\in \{p_1, \dots , p_n\}$ , then $\bigwedge _{i\leq n}X^\psi \overline {p_i}\in \Gamma $ (by Lemma 8.4). Then, by $\mathsf {Ax2}$ , we obtain that $X^\psi (\bigwedge _{i\leq n}\overline {p_i})\in \Gamma $ , i.e., $X^\psi \overline {\varphi }\in \Gamma $ .

Corollary 13 Given the canonical model $\mathcal {M}^c=\langle [\Gamma _0]_\Box , \{R^c_\psi \ | \ \psi \in \mathcal {L}\}, \mathcal {T}^c, \oplus ^c, f^c, t^c, V^c \rangle $ for $\Gamma _0$ , $\varphi ,\psi \in \mathcal {L}$ , and $\Gamma \in [\Gamma _0]_\Box $ , $X^\psi \overline {\varphi }\in \Gamma $ iff $t^c(\varphi )\subseteq f^c(t^c(\psi ))$ .

Proof

(Lemma 12) $$ \begin{align} X^\psi\overline \varphi\in \Gamma & \text { iff } X^\psi\overline{p}\in \Gamma \text{ for all } p\in \mathfrak{At}(\varphi) \ \qquad\qquad\qquad\qquad\qquad\end{align} $$

(Ax5, S5 _□, and Γ ∈ [Γ₀]_□) $$\begin{align} & \text { iff } X^\psi \overline{p}\in \Gamma_0 \text{ for all } p\in \mathfrak{At}(\varphi) \end{align} $$

(Lemma 12) $$ \begin{align} & \text { iff } X^\psi \overline{\varphi}\in \Gamma_0\qquad\qquad\qquad\qquad\qquad\qquad\quad \ \ \end{align} $$

(Ax3) $$\begin{align} & \text { iff } X^{\bigwedge \mathfrak{At}(\psi)} \overline{\varphi}\in \Gamma_0 \qquad\quad\qquad\qquad\qquad\qquad\qquad\qquad\quad \ \ \end{align}$$

(defns. of f ^c and t ^c ) $$\begin{align} & \text { iff } t^c(\varphi)\subseteq f^c(t^c(\psi)).\qquad\qquad\qquad\quad \end{align} $$

Lemma 14 Given a mcs $\Gamma $ , for all finite $\Phi \subseteq \Gamma [X^\psi ]$ , we have $\bigwedge \Phi \in \Gamma [X^\psi ]$ .

Proof Let $\Phi =\{\varphi _1, \dots , \varphi _n\} \subseteq \Gamma [X^\psi ]$ . This means that, for each $\varphi _j$ with $1\leq j\leq n$ , there is a $\eta _j\in \mathcal {L}$ such that $X^\psi \eta _j\wedge \Box (\eta _j\supset \varphi _j)\in \Gamma $ . Thus, $\bigwedge _{1\leq j\leq n} X^\psi \eta _j \wedge \bigwedge _{1\leq j\leq n} \Box (\eta _j\supset \varphi _j)\in \Gamma $ . Then, by $\mathsf {Ax2}$ , we obtain that $X^\psi (\bigwedge _{j\leq n}\eta _j) \in \Gamma $ . By $\mathsf {S5}_\Box $ , we also have $\Box (\bigwedge _{j\leq n}\eta _j \supset \bigwedge _{j\leq n} \varphi _j) \in \Gamma $ . Therefore, $\bigwedge \Phi \in \Gamma [X^\psi ]$ .

Lemma 15 For every mcs $\Gamma $ and $\varphi \in \mathcal {L}$ , if $\Gamma [X^\psi ]\vdash \varphi $ and $X^\psi \overline {\varphi }\in \Gamma $ , then $X^\psi \varphi \in \Gamma $ .

Proof Suppose that $\Gamma [X^\psi ]\vdash \varphi $ and $X^\psi \overline {\varphi }\in \Gamma $ . The first assumption means that there is a finite $\Phi \subseteq \Gamma [X^\psi ]$ such that $\vdash \bigwedge \Phi \supset \varphi $ . By Lemma 14, we know that $\bigwedge \Phi \in \Gamma [X^\psi ]$ . This means that there is a $\eta $ such that $X^\psi \eta \wedge \Box (\eta \supset \bigwedge \Phi )\in \Gamma $ . Moroever, $\vdash \bigwedge \Phi \supset \varphi $ entails, by $\mathsf {S5}_\Box $ , that $\vdash \Box (\bigwedge \Phi \supset \varphi )$ , thus, $\Box (\bigwedge \Phi \supset \varphi )\in \Gamma $ . Therefore, by $\mathsf {CPL}$ , $\mathsf {S5}_\Box $ , and our first assumption, we obtain $X^\psi \eta \wedge \Box (\eta \supset \varphi ) \wedge X^\psi \overline {\varphi } \in \Gamma $ . Hence, by $\mathsf {Ax6}$ , $X^\psi \varphi \in \Gamma $ .

Lemma 16 (Truth Lemma)

Let $\Gamma _0$ be a mcs of $\mathsf {Log}_{incl}$ and $\mathcal {X}^c=\langle [\Gamma _0]_\Box , \{R^c_\psi \ | \ \psi \in \mathcal {L}\}, \mathcal {T}^c, \oplus ^c, f^c, t^c, V^c \rangle $ be the canonical model for $\Gamma _0$ . Then, for all $\varphi \in \mathcal {L}$ and $\Gamma \in [\Gamma _0]_\Box $ , we have $\mathcal {X}^c, \Gamma \models \varphi \text { iff } \varphi \in \Gamma $ .

Proof The proof follows by induction on the structure of $\varphi $ . The cases for the propositional variables, Booleans, and $\varphi :=\Box \psi $ are standard. We here prove the case $\varphi :=X^\psi \eta $ .

( $\Leftarrow $ ) Suppose $X^\psi \eta \in \Gamma $ . Since $X^\psi \eta \in \Gamma $ , by $\mathsf {Ax4}$ , $X^\psi \overline {\eta }\in \Gamma $ . Thus, by Corollary 13, $t^c(\eta )\subseteq f^c(t^c(\psi ))$ . Now let $\Delta \in [\Gamma _0]_\Box $ such that $\Gamma R^c_\psi \Delta $ . As $X^\psi \eta \in \Gamma $ and $\Box (\eta \supset \eta )\in \Gamma $ (the latter is by $\mathsf {S5}_\Box $ ), we have that $\eta \in \Gamma [X^\psi ]$ . Therefore, since $\Gamma R^c_\psi \Delta $ , we have $\eta \in \Delta $ . Then, by the induction hypothesis, we have $\mathcal {X}^c, \Delta \models \eta $ . As $\Delta $ has been chosen arbitrarily, we obtain that $\mathcal {X}^c, \Gamma \models X^\psi \eta $ .

( $\Rightarrow $ ) Suppose $\mathcal {X}^c, \Gamma \models X^\psi \eta $ , i.e., for all $\Delta \in [\Gamma _0]_\Box $ such that $\Gamma R^c_\psi \Delta $ , $\mathcal {X}^c, \Delta \models \eta $ and $t^c(\eta )\subseteq f^c(t^c(\psi ))$ . By Corollary 13, the latter means that $X^\psi \overline {\eta }\in \Gamma $ . Moreover, the former, by the induction hypothesis, implies that $\eta \in \Delta $ for all $\Delta \in [\Gamma _0]_\Box $ with $\Gamma R^c_\psi \Delta $ . This implies that $\Gamma [X^\psi ]\vdash \eta $ . Otherwise, $\Gamma [X^\psi ]\cup \{\neg \eta \}$ would be consistent, thus, by Lemma 9, there exists a mcs $\Delta '$ such that $\Gamma [X^\psi ]\cup \{\neg \eta \}\subseteq \Delta '$ . As $\Gamma [X^\psi ]\subseteq \Delta '$ , we have $\Gamma \rightarrow ^c_\psi \Delta '$ . Since $\rightarrow ^c_\psi \subseteq \sim _\Box $ (Lemma 10) and $\Gamma \in [\Gamma _0]_\Box $ , we obtain that $\Delta '\in [\Gamma _0]_\Box $ , therefore, $\Gamma R^c_\psi \Delta '$ . Hence, that $\neg \eta \in \Delta '$ contradicts with the assumption that $\eta \in \Delta $ for all $\Delta $ with $\Gamma R^c_\psi \Delta $ . Since $X^\psi \overline {\eta }\in \Gamma $ , by Lemma 15, we obtain that $X^\psi \eta \in \Gamma $ .

Corollary 17 $\mathsf {Log}_{incl}$ is strongly complete with respect to the class of inclusive ts-models with functions.

Proof Let $\Phi _0\subseteq \mathcal {L}$ be a $\mathsf {Log}_{incl}$ -consistent set of formulas. Then, by Lindenbaum’s Lemma (Lemma 9), there exists a mcs $\Gamma _0$ such that $\Phi _0\subseteq \Gamma _0$ . We can then construct a canonical model $\mathcal {X}^c=\langle [\Gamma _0]_\Box , \{R^c_\psi \ | \ \psi \in \mathcal {L}\}, \mathcal {T}^c, \oplus ^c, f^c, t^c, V^c \rangle $ for $\Gamma _0$ as described above. Then, by Lemma 16, we obtain that $\mathcal {X}^c, {\Gamma _0} \models \varphi $ for all $\varphi \in \Phi _0$ .

Corollary 18 $\mathsf {Log}_{incl + mon}=\mathsf {Log}_{incl} + \mathsf {sAx3}$ is strongly complete with respect to the class of inclusive and monotone increasing ts-models with functions.

Proof The proof follows similarly to the proof for $\mathsf {Log}_{incl}$ , by constructing the corresponding canonical model in the same way. The only additional step we need to show is that $f^c$ in the canonical model for $\mathsf {Log}_{incl + mon}$ is monotone increasing and this follows from Lemma 5.2.

3.2.2 Preclosure

As shown by Lemma 6, additivity is not definable in $\mathcal {L}$ . This suggests that obtaining a complete axiomatization for a class of models satisfying additivity requires either more involved techniques or a more expressive language. We leave the investigations of a complete axiomatization for ts-models with functions satisfying additivity to future work. Nevertheless, we here show that the logic of preclosure ts-models with functions is strictly weaker than the logic of topo-ts-models and, in turn, Berto’s logic of imagination in [Reference Berto3].

In the following theorem, ${\models _{pre}} \varphi $ denotes that $\varphi $ is valid in preclosure ts-models with functions, that is, it is true in all worlds of all preclosure ts-models.

Theorem 19 For all $\varphi \in \mathcal {L}$ , if ${\models _{pre}}\varphi $ then $\Vdash \varphi $ . However, there is a $\psi \in \mathcal {L}$ such that $\Vdash \psi $ but $\not \models _{pre}\psi $ .

Proof The first part immediately follows from Theorem 3 and the fact that ${\models _{pre}} \subseteq \models $ . To show the latter, consider the following sentence:

$$\begin{align*} \psi:=(X^p \overline{q} \wedge X^{p\wedge q}\overline{r}) \supset X^p \overline{r}. \end{align*}$$

It is then easy to see that we have $\Vdash \psi $ . However, $\psi $ is falsified by the preclosure ts-model $\mathcal {X}=\langle \{w\}, \{R_\varphi \ | \ \varphi \in \mathcal {L}\}, \{x, y, z, z'\}, \oplus , f, t, V\rangle $ where $(T, \oplus , t)$ is as given in Figure 4 and $f(x)=f(z)=f(z')=z'$ and $f(y)=z$ , $R_\varphi =\{(w, w)\}$ for all $\varphi \in \mathcal {L}$ , $V(p)=V(q)= V(r)=\{w\}$ : we have that $\mathcal {X}, w\models X^p \overline {q}$ (since $t(q)=x\leq z=f(t(p))=f(y)$ ) and $\mathcal {X}, w\models X^{p\wedge q} \overline {r}$ (since $t(r)=z' =f(z)=f(t(p)\oplus t(q))$ ). However, $\mathcal {X}, w\not \models X^p \overline {r}$ (since $t(r)=z' \not \leq z =f(t(p)$ ). It is easy to verify that $\mathcal {X}$ is indeed a preclosure ts-model, in particular, f satisfies inclusion and additivity.Footnote ¹⁹

Figure 4

Counterexample for the invalidity of $\psi $ .

A few notes about the invalidated formula in the proof of Theorem 19 seem to be warranted. First of all, it is closely related to the so-called Cautious Transitivity principle ( $(X^\varphi \psi \wedge X^{\varphi \wedge \psi }\eta )\supset X^\varphi \eta $ ); in fact, by replacing q in ‘ $X^{p\wedge q}\overline {r}$ ’ by $\overline {q}$ , we obtain an instance of Cautious Transitivity that is invalidated by the same counterexample. Cautious Transitivity (also called ‘Special Transitivity’ in [Reference Berto3]) has come to be controversial in the literature. On the one hand, it is sometimes seen as part of minimal conditional logics for non-monotonic inference. [Reference Berto3, p. 1883] provides a constraint on $R_\varphi $ that validates Cautious Transitivity but leaves open the possibility that there might be intuitive counterexamples against the principle. In [Reference Berto and Hawke7], Cautious Transitivity is invalidated due to the (lack of) constraints imposed on $R_\varphi $ , whereas our models violate Cautious Transitivity due to the topicality component as shown in Theorem 19.

We think that there is an intuitive case against Cautious Transitivity at least where TSIMs have application to imagination. Here is a version of the Helena example from Section 3 which seems to violate Cautious Transitivity. In an act of imagining that Helena is on her way to meet John, she imagines that they go to the movies when they meet. In an act of imagining that Helena is on her way to meet John and they go to the movies together, she imagines buying some popcorn at the movie theatre. However, in an act of imagining that Helena is on her way to meet John, she might not imagine that she buys some popcorn at the movie theatre. The imaginative input ‘they go to the movies together’ in addition to ‘Helena is on her way to meet John’ might lead to further expansion of the topic of the latter even when the former is imagined in an act of imagining the latter.

Such failures of Cautious Transitivity can be connected to empirical theories of imagination. The core thought is that non-inferential embellishment has topical limits. Where $X^\varphi \psi $ one imagines that $\psi $ via embellishing the input $\varphi $ . The topic of input $\varphi $ has a limited range of expansion, whereas when $\psi $ is added to the inputs, embellishing $\varphi \wedge \psi $ might take us topically further afield. This further embellishment licenses the inference to $\eta $ in $X^{\varphi \wedge \psi }\eta $ , an inference that would not have been licensed simply by embellishing $\varphi $ without additional inputs. Where non-inferential embellishment permits topic-expansion, it need not permit such expansions arbitrarily, in contrast to adding an explicit imaginative input.