MINIMAL MODAL LOGICS, CONSTRUCTIVE MODAL LOGICS AND THEIR RELATIONS

Abstract

We present a family of minimal modal logics (namely, modal logics based on minimal propositional logic) corresponding each to a different classical modal logic. The minimal modal logics are defined based on their classical counterparts in two distinct ways: (1) via embedding into fusions of classical modal logics through a natural extension of the Gödel–Johansson translation of minimal logic into modal logic S4; (2) via extension to modal logics of the multi- vs. single-succedent correspondence of sequent calculi for classical and minimal logic. We show that, despite being mutually independent, the two methods turn out to be equivalent for a wide class of modal systems. Moreover, we compare the resulting minimal version of K with the constructive modal logic CK studied in the literature, displaying tight relations among the two systems. Based on these relations, we also define a constructive correspondent for each minimal system, thus obtaining a family of constructive modal logics which includes CK as well as other constructive modal logics studied in the literature.

1 Introduction

Although modal logics are usually defined as extensions of classical logic, significant attention has been also devoted to the analysis of modalities over non-classical basis, such as relevant [7, 17, 19, 39, 55, 56], linear [21, 40, 51] or other substructural logics [10, 30, 53]. In this context, a major role is played by intuitionistic logic, many modal extensions of which have been studied with motivations ranging from philosophical or legal reasoning to computer science applications.

A crucial difference between classical and intuitionistic modal logics is that, by analogy with intuitionistic connectives, in the latter systems the modalities $\Box $ and $\Diamond $ are assumed to be not interdefinable. This peculiarity allows for the definition of systems validating independent $\Box $ - and $\Diamond $ -principles, as well as with different interactions between the two modalities. Such a notable freedom has led to the question of how to define meaningful intuitionistic (or constructive) counterparts of a given classical modal logic. Clearly, the meaning of ‘intuitionistic counterpart’ is not univocal, and different interpretations have produced different intuitionistic modal systems. This situation is particularly evident in the case of intuitionistic correspondents of classical modal logic $\mathsf {K}$ , with a wide variety of counterparts of it studied in the literature. ¹ We mention here the three most relevant such systems: Intuitionistic K, denoted $\mathsf {I.K}$ [49, 57–59]; Wijesekera’s K (the propositional fragment of Wijesekera’s Constructive Concurrent Dynamic Logic [65, 66]), that we call $\mathsf {W.K}$ , and Constructive K, denoted $\mathsf {C.K}$ [6, 41, 48]. These systems have increasing strength, from the weakest $\mathsf {C.K}$ to the strongest $\mathsf {I.K}$ , and are definable axiomatically extending intuitionistic propositional logic ( $\mathsf {IPL}$ ) as follows:

In particular, $\mathsf {I.K}$ and $\mathsf {C.K}$ belong each to one of the two most studied families of modal logics based on $\mathsf {IPL}$ : so-called intuitionistic modal logics, that stem from the works of Fischer Servi [57, 58], Plotkin and Stirling [49], and Ewald [15], and so-called constructive modal logics, whose definition goes back to Fitch [18] and has been further developed by de Paiva, Mendler and Ritter [1, 41, 48] among others. The latter logics have found several applications in the constructivisation of description [42] and dynamic [66] logic, the formalisation of provability and consistency in Heyting arithmetic [34], contextual reasoning [41], and also serve as type languages for programming in structured knowledge bases and for computation stages [6, 14, 44].

Once an intuitionistic correspondent of a classical modal logic is defined, the question arises of how to define analogous counterparts of further classical modal logics. To this aim, general theories of intuitionistic modal logics have been proposed in the literature. An elegant characterisation of this type of systems was provided by Simpson [59] via reduction to validities in first-order intuitionistic logic ( $\mathsf {FOIL}$ ): Given a classical modal logic $\mathsf {L}$ , its intuitionistic counterpart $\mathsf {I.L}$ is defined as the set of formulas A such that

$$\begin{align*}A \in \mathsf{I.L} \text{ if and only if } \mathscr T\vdash_{\mathsf{FOIL}} \forall x(A^x),\end{align*}$$

where the superscript x denotes the standard translation of modal formulas into first-order sentences with respect to the variable x, and $\mathscr T$ is a set of first-order sentences that express the frame conditions corresponding to the modal axioms of $\mathsf {L}$ in the relational semantics of classical modal logics (e.g., reflexivity, transitivity, symmetry for ${T}$ , ${4}$ , ${B}$ ). The same logics have been also shown to be reducible to products of modal logics [20] via suitable translations of modal formulas [57, 67].

Furthermore, a general characterisation of $\mathsf {W.K}$ and related systems was proposed in [11] where so-called Wijesekera-style constructive modal logics were defined in two distinct but ultimately equivalent ways: (1) by restricting classical modal sequent calculi to sequents with at most one formula in the conclusion, thus extending to modal logics a relation that is known to hold between classical and intuitionistic sequent calculi since Gentzen [23]; (2) by means of a simple generalisation of the satisfaction clauses of modal formulas in relational models in order to fit with the preorder semantics of intuitionistic logic. The second way provides quite immediately a reduction of each system $\mathsf {W.L}$ , counterpart of the classical logic $\mathsf {L}$ , to the fusion of $\mathsf {S4}$ and $\mathsf {L}$ [20] via a natural extension of Gödel’s translation of $\mathsf {S4}$ into $\mathsf {IPL}$ [24].

By contrast, no such uniform characterisation of constructive modal logics has been proposed so far in the literature. Constructive counterparts for the whole modal cube from $\mathsf {K}$ to $\mathsf {S5}$ obtained by the combinations of the axioms ${D}$ , ${T}$ , ${4}$ , ${B}$ and ${5}$ have been presented in [1, 2, 43] and endowed with nested sequent calculi [2] and some of them also with relational semantics [1, 41], sequent calculi [6, 34, 36] and 2-sequent calculi [43]. These logics are defined based on their axiomatic systems by extending $\mathsf {C.K}$ with pairs of corresponding $\Box $ - and $\Diamond $ -axioms, like ${T_{\Box }}$ $\Box A \supset A$ , ${T_{\Diamond }}$ $A \supset \Diamond A$ and ${4_{\Box }}$ $\Box A \supset \Box \Box A$ , ${4_{\Diamond }}$ $\Diamond \Diamond A \supset \Diamond A$ , that are both needed due to the loss of duality between $\Box $ and $\Diamond $ (the axiom ${D}$ is an exception as it is taken in the usual formulation $\Box A \supset \Diamond A$ only). However, the validity of corresponding $\Box $ - and $\Diamond $ -principles in constructive modal logics does not hold in general, so that it is not clear how to extend this family of systems from a purely axiomatical point of view, especially when it comes to logics weaker than $\mathsf {K}$ . For instance, $\mathsf {C.K}$ validates $\Box A \land \Box B \supset \Box (A \land B)$ and the necessitation rule $A/\Box A$ , but does not validate the corresponding $\Diamond $ -principles $\Diamond (A \lor B) \supset \Diamond A \lor \Diamond B$ and $\neg \Diamond \bot $ . Moreover, $\Box $ - and $\Diamond $ -versions also exist for the axiom ${D}$ , namely $\neg (\Box A \land \Box \neg A)$ and $\Diamond A \lor \Diamond \neg A$ , but neither of them is derivable in $\mathsf {C.KD}$ .

In this paper, we address this problem by presenting a systematisation of constructive modal logics that includes the systems $\mathsf {C.K}$ , $\mathsf {C.KD}$ and $\mathsf {C.KT}$ already studied in the literature as well as new constructive counterparts of further classical modal logics. The starting point of our analysis is the observation that constructive modal logics share some similarities with minimal propositional logic ( $\mathsf {MPL}$ ) and possible modal extensions thereof, both from a semantical and from a proof-theoretical perspective.

Our approach toward the systematisation of constructive modal logics can be summarised as follows. First of all, we introduce a family of minimal modal logics $\mathsf {M.L}$ . These logics correspond each to a different classical modal logic, and are defined following an approach similar to the one of [11] for the definition of Wijesekera-style modal logics: we define the logics $\mathsf {M.L}$ by means of a reduction to fusions of classical modal logics of the form $\mathsf {S4}\oplus \mathsf {L}$ through a natural extension of the Gödel–Johansson translation of $\mathsf {MPL}$ into $\mathsf {S4}$ (in turn, this translation is a combination of Johansson’s translation of $\mathsf {MPL}$ into $\mathsf {IPL}$ [29] and Gödel’s translation of $\mathsf {IPL}$ into $\mathsf {S4}$ ). The logics $\mathsf {M.L}$ will coincide with the sets of modal formulas A such that $A^{t} \in \mathsf {S4}\oplus \mathsf {L}$ , where t is the aforementioned translation. To this aim, we provide the minimal modal logics with a modular semantic characterisation. We then show that the same minimal modal logics can be equivalently obtained by restricting sequent calculi for classical modal logics to sequents with exactly one formula in the succedent. This extends to modal logics a relation that holds between sequent calculi for classical and MPL firstly observed by Johansson [29] (see [61] for an extended presentation of sequent calculi for $\mathsf {CPL}$ , $\mathsf {IPL}$ and $\mathsf {MPL}$ and corresponding bounds on the cardinality of succedents of sequents). To our knowledge, this is the first study of modal logics based on MPL.

Furthermore, we observe that $\mathsf {M.K}$ , the resulting minimal counterpart of $\mathsf {K}$ , is strictly related to constructive $\mathsf {C.K}$ . In particular, the two systems share exactly the same modal principles, despite over a different propositional base. This means that $\mathsf {C.K}$ coincides with the extension of $\mathsf {M.K}$ with the principle of ex falso quodlibet $\bot \supset A$ (exactly as $\mathsf {IPL} = \mathsf {MPL} + \bot \supset A$ ). We show that equally tight relations between $\mathsf {M.K}$ and $\mathsf {C.K}$ can be also observed in terms of their semantics and sequent calculi. More precisely, $\mathsf {C.K}$ can be semantically characterised by means of a simple restriction of the models of $\mathsf {M.K}$ that ensures the validity of $\bot \supset A$ , and the sequent calculus for $\mathsf {C.K}$ defined in [6] can be re-obtained by adding the modal sequent rules for $\mathsf {M.K}$ to an intuitionistic sequent calculus. By extending these relations to the other minimal systems, we then define a constructive correspondent for each minimal and classical modal logic, thus obtaining an entire family of constructive modal logics with corresponding semantics and sequent calculi.

In short, the aim of this paper is threefold: (1) provide a uniform characterisation of some constructive modal logics already existing in the literature, (2) extend this family of logics by defining constructive counterparts of further classical modal logics coherently with our characterisation, (3) display the tight relations between minimal and constructive modalities.

The paper is organised as follows. In Sections 1.1 and 1.2, we present some preliminary notions needed throughout the paper. In Section 2, we define $\mathsf {M.K}$ , our minimal counterpart of $\mathsf {K}$ , via a reduction into $\mathsf {S4}\oplus \mathsf {K}$ , and also provide an adequate semantics for it. In Section 3, we apply the single-succedent restriction to a sequent calculus for $\mathsf {K}$ and show that the resulting logic coincides with $\mathsf {M.K}$ . In Section 4, we show how the constructive logic $\mathsf {C.K}$ relates to our $\mathsf {M.K}$ , both from the point of view of the semantics and of the sequent calculi. In Section 5, we apply the same two methods to further classical modal logics, thus obtaining an entire family of corresponding minimal systems. Moreover, by extending to these systems the relations just observed between $\mathsf {M.K}$ and $\mathsf {C.K}$ , we define in Section 6 an analogous family of constructive modal logics. Finally, Section 7 contains some discussion of the results.

1.1 Syntactic preliminaries

Given a countable set $\mathit {Atm} = \{p_0, p_1, p_2,\ldots \}$ of propositional variables and a finite set $\mathbb {M}$ of unary modal operators, the language $\mathcal L_{\mathbb {M}}^{\mathit {Atm}}$ is defined by the following BNF grammar, where $p\in \mathit {Atm}$ , $\circ \in \{\land , \lor , \supset \}$ , and $\heartsuit \in \mathbb {M}$ :

$$\begin{align*}A ::= p \mid \bot \mid A \circ A \mid \heartsuit A.\end{align*}$$

In the following, we use p, q, r as metavariables for elements of $\mathit {Atm}$ , and A, B, C, D as metavariables for formulas. Moreover, we define $\top := \bot \supset \bot $ , $\neg A := A \supset \bot $ , and $A \supset \subset B:=(A \supset B) \land (B \supset A)$ . Minimal modal logics will be defined in a language $\mathcal L_{\{\Box , \Diamond \}}^{\mathit {Atm}}$ containing the modalities $\Box $ and $\Diamond $ . For the sake of simplicity, we denote $\mathcal L_{\{\Box , \Diamond \}}^{\mathit {Atm}}$ as $\mathcal L$ .

We consider the following axiomatisation for $\mathsf {MPL}$ , formulated in $\mathcal L$ (see e.g., [54]) ² :

As usual, we can define $\mathsf {IPL}$ as the extension of $\mathsf {MPL}$ with ex falso quodlibet ³ :

$$\begin{align*}\mathsf{IPL} := \mathsf{MPL} + \bot\supset A,\end{align*}$$

and classical propositional logic ( $\mathsf {CPL}$ ) as the extension of $\mathsf {IPL}$ with excluded middle:

$$\begin{align*}\mathsf{CPL} := \mathsf{IPL} + A \lor \neg A.\end{align*}$$

In this paper, we shall define minimal and constructive counterparts of classical modal logics. Classical modal logics are defined as usual as extensions of $\mathsf {CPL}$ , formulated in $\mathcal L$ , with modal axioms and rules. For instance, the logic $\mathsf {K}$ is defined extending $\mathsf {CPL}$ with

and $\mathsf {S4}$ is defined extending $\mathsf {K}$ with ${T_{\Box }} \ \Box A \supset A$ and ${4_{\Box }} \ \Box A \supset \Box \Box A$ . ⁴ Further classical modal logics will be introduced in Section 5.

For any logic defined in the following (no matter whether based on classical, intuitionistic or minimal logic), we consider the standard notions of derivability: Given a logic $\mathsf {L}$ formulated in $\mathcal L_{\mathbb M}^{\mathit {Atm}}$ and formulas $A, B_1, \ldots , B_n$ of $\mathcal L_{\mathbb M}^{\mathit {Atm}}$ , the rule $B_1, \ldots , B_n / A$ is derivable in $\mathsf {L}$ if there is a finite sequence of formulas ending with A in which every formula is an (instance of an) axiom of $\mathsf {L}$ , or it belongs to $\{B_1, \ldots , B_n\}$ , or it is obtained from previous formulas by the application of a rule of $\mathsf {L}$ . A formula A is derivable in $\mathsf {L}$ , written $\mathsf {L} \vdash A$ , if the rule $\emptyset / A$ is derivable in $\mathsf {L}$ . Finally, A is (locally) derivable in $\mathsf {L}$ from a set of formulas $\Phi $ of $\mathcal L_{\mathbb M}^{\mathit {Atm}}$ , written $\Phi \vdash _{\mathsf {L}} A$ , if there is a finite set $\{B_1, \ldots , B_n\}\subseteq \Phi $ such that $\mathsf {L} \vdash B_1 \land \cdots \land B_n \supset A$ .

1.2 Semantic preliminaries

We shall define semantics for minimal modal logics by suitably extending relational models for MPL. We consider to this purpose relational models for MPL as defined in [54]: A minimal relational model is a tuple $\mathcal M = \langle \mathcal W, \leq, \mathbb {F}, \mathcal V\rangle $ , where $\mathcal W$ is a non-empty set of worlds, $\leq $ is a reflexive and transitive binary relation on $\mathcal W$ , $\mathbb {F}\subseteq \mathcal W$ is a $\leq $ -upward closed set (that is, if $w\in \mathbb {F}$ and $w\leq v$ , then $v\in \mathbb {F}$ ) of so-called fallible worlds, and $\mathcal V : \mathit {Atm} \longrightarrow \mathcal P(\mathcal W)$ is a hereditary valuation function (that is, if $w\in \mathcal V(p)$ and $w\leq v$ , then $v\in \mathcal V(p)$ ). We write $v\geq w$ for $w \leq v$ . The forcing relation $\mathcal M, w\Vdash A$ is inductively defined as follows:

Given a formula A, we say that A is valid in a model $\mathcal M$ , written $\mathcal M\models A$ , if $\mathcal M, w \Vdash A$ for all worlds $w$ of $\mathcal M$ . The same definition of validity applies to all kinds of models considered in this paper. In the following, we simply write $w \Vdash A$ when $\mathcal M$ is clear from the context.

Minimal relational models are a generalisation of the well-known intuitionistic relational models firstly introduced by Kripke [33]. In particular, a minimal relational model is an intuitionistic relational model if $\mathbb {F} = \emptyset $ . We point out that an alternative semantics for $\mathsf {IPL}$ can be obtained from minimal relational models by preserving the fallible worlds but assuming the condition $\mathbb {F}\subseteq \mathcal V(p)$ for all $p\in \mathit {Atm}$ which ensures the validity of ex falso quodlibet $\bot \supset A$ . We will consider this latter kind of restriction in Section 4.1.

2 Minimal K via bimodal companion

Our first approach toward the definition of minimal modal logics is based on reductions into fusions of classical modal logics. We consider to this purpose the following notions of fusion, Gödel–Johansson translation and bimodal companion.

Definition 2.1 (Fusion).

Let $\mathsf {L}_1$ and $\mathsf {L}_2$ be classical modal logics respectively defined in the languages $\mathcal L_{\{\Box _1, \Diamond _1\}}^{\mathit {Atm}}$ and $\mathcal L_{\{\Box _2, \Diamond _2\}}^{\mathit {Atm}}$ sharing the same propositional variables and propositional connectives but with disjoint sets of modalities. The fusion $\mathsf {L}_1\oplus \mathsf {L}_2$ of $\mathsf {L}_1$ and $\mathsf {L}_2$ is the smallest logic in the language $\mathcal L_{\{\Box _1, \Diamond _1,\Box _2, \Diamond _2\}}^{\mathit {Atm}}$ containing $\mathsf {L}_1 \cup \mathsf {L}_2$ and closed under the rules of $\mathsf {L}_1$ and $\mathsf {L}_2$ .

Definition 2.2 (Extended Gödel–Johansson translation).

Let $\mathit {Atm}' = \mathit {Atm} \cup \{f\}$ , with $f\notin \mathit {Atm}$ , and $\mathcal L_{1,2}'$ denote $\mathcal L_{\{\Box _1, \Diamond _1,\Box _2, \Diamond _2\}}^{\mathit {Atm}'}$ . The extended Gödel–Johansson translation $t: \mathcal L\longrightarrow \mathcal L_{1,2}'$ is inductively defined as follows:

Definition 2.3 (Bimodal companion).

For any logic $\mathsf {M}$ formulated in the language $\mathcal L$ , we say that a fusion of classical modal logics $\mathsf {L}_1\oplus \mathsf {L}_2$ in the language $\mathcal L_{1,2}'$ is the Gödel–Johansson bimodal companion (or just bimodal companion) of $\mathsf {M}$ if it holds:

$$\begin{align*}\mathsf{M} \vdash A\ \text{if and only if}\ \mathsf{L}_1\oplus\mathsf{L}_2 \vdash A^{t}.\end{align*}$$

The above translation t is based on Gödel’s [24] reduction of IPL into $\mathsf {S4}$ . The clauses for the modal formulas extend the translation in the trivial way, and are considered for instance in [16, 67], while the reduction of $\bot $ into a distinguished propositional constant f goes back to Johansson [29]. A similar translation that employs Johannson’s solution was already applied for the embedding of a constructive modal logic into a classical multimodal logic in [16].

Given a classical modal logic $\mathsf {L}$ , we shall define its minimal counterpart $\mathsf {M.L}$ by considering modal companions of the form $\mathsf {S4}\oplus \mathsf {L}$ .

Definition 2.4 (Minimal counterpart of a classical logic).

Given a classical modal logic $\mathsf {L}$ , the minimal counterpart of $\mathsf {L}$ is the logic $\mathsf {M.L}$ in $\mathcal L$ such that $\mathsf {S4}\oplus \mathsf {L}$ is the bimodal companion of $\mathsf {M.L}$ .

In other words, the minimal counterpart $\mathsf {M.L}$ of $\mathsf {L}$ is the solution of the equation

The solution of ( $\ast $ ), if it exists, is unique (modulo equivalent axiomatisations). Indeed, if both $\mathsf {M.L}$ and $\mathsf {M.L}'$ are solutions to ( $\ast $ ), then $\mathsf {M.L} \vdash A$ iff $\mathsf {S4}\oplus \mathsf {L} \vdash A^{t}$ iff $\mathsf {M.L}' \vdash A$ , hence $\mathsf {M.L} = \mathsf {M.L}'$ . We start by presenting the minimal counterpart of the classical modal logic $\mathsf {K}$ .

Definition 2.5 (Minimal $\mathsf {K}$ ).

The minimal modal logic $\mathsf {M.K}$ is defined extending $\mathsf {MPL}$ with the following axioms and rule:

In order to prove that $\mathsf {M.K}$ is our minimal counterpart of $\mathsf {K}$ (that is, the solution of $(\ast )$ for $\mathsf {L}$ replaced with $\mathsf {K}$ ), we first provide a semantics for $\mathsf {M.K}$ , which is defined by suitably extending minimal relational models for $\mathsf {MPL}$ (cf. Section 1.2) with an additional relation dealing with the modalities.

Definition 2.6 (Minimal birelational semantics).

A minimal birelational model is a tuple $\mathcal M = \langle \mathcal W, \leq , \mathbb {F}, \mathcal R, \mathcal V\rangle $ , where $\langle \mathcal W, \leq , \mathbb {F}, \mathcal V\rangle $ is a minimal relational model, and $\mathcal R$ is a binary relation on $\mathcal W$ . The forcing relation $\mathcal M, w\Vdash A$ is inductively defined extending the clauses for $p, \bot , \land , \lor , \supset $ in Section 1.2 with the following clauses for the modalities:

The semantics in Definition 2.6 essentially coincides with Wijesekera’s semantics for $\mathsf {W.K}$ [65] with the only difference of the addition of the fallible worlds, which is due to the fact that the base models are minimal rather than intuitionistic.

The generalisation of the standard clauses for $\Box , \Diamond $ in the relational semantics to all $\leq $ -successors is the simplest way to preserve the hereditary property of minimal relational models.

Proposition 2.1 (Hereditary property).

Given a minimal birelational model $\mathcal M$ and a formula A of $\mathcal L$ , for every worlds $w$ and $v$ of $\mathcal M$ it holds: If $w \Vdash A$ and $w \leq v$ , then $v\Vdash A$ .

Proof. Immediate by induction on the construction of A.

Theorem 2.2 (Soundness).

For all $A\in \mathcal L$ , if A is derivable in $\mathsf {M.K}$ , then A is valid in every minimal birelational model.

Proof. By showing that all the axioms and rules of $\mathsf {M.K}$ are valid, respectively validity preserving, in every model for $\mathsf {M.K}$ . We consider the modal principles.

• ( ${K_{\Box }}$ ) Suppose that $w \Vdash \Box (A \supset B)$ and $w \Vdash \Box A$ . Then for all $v, u$ , if $w \leq v$ and $v \mathcal R u$ , then $u \Vdash A \supset B$ and $u \Vdash A$ , hence $u \Vdash B$ . Thus, $w \Vdash \Box B$ . Therefore $\mathcal M\models \Box (A \supset B)\supset (\Box A\supset \Box B)$ .

• ( ${K_\Diamond }$ ) Suppose that $w \Vdash \Box (A \supset B)$ and $w \Vdash \Diamond A$ . Then for all v, if $w \leq v$ , then there is u such that $v \mathcal R u$ and $u \Vdash A$ . Moreover, $u \Vdash A \supset B$ . Thus $u \Vdash B$ , hence $w \Vdash \Diamond B$ . Therefore $\mathcal M\models \Box (A \supset B)\supset (\Diamond A\supset \Diamond B)$ .

• ( ${nec}$ ) Suppose that $\mathcal M\models A$ . Then for all $w, v, u$ , if $w \leq v$ and $v \mathcal R u$ , then $u\Vdash A$ , hence $w \Vdash \Box A$ . Therefore $\mathcal M\models \Box A$ .

We now present a completeness proof for $\mathsf {M.K}$ with respect to the minimal birelational semantics by means of the canonical model technique. The proof adapts the completeness proof for $\mathsf {W.K}$ by Wijesekera [65] with the addition of the impossible worlds. For every logic $\mathsf {L}$ in $\mathcal L$ , we call $\mathsf {L}$ -full any set $\Phi $ of formulas of $\mathcal L$ such that if $\Phi \vdash _{\mathsf {L}} A$ , then $A\in \Phi $ (closure under derivation), and if $A\lor B\in \Phi $ , then $A\in \Phi $ or $B\in \Phi $ (disjunction property). Moreover, for every set of formulas $\Phi $ , we denote ${\Box }^-\Phi $ the set $\{A \mid \Box A\in \Phi \}$ . The following holds.

Lemma 2.3 (Lindenbaum).

For every set $\Phi $ of formulas of $\mathcal L$ , there is a $\mathsf {M.K}$ -full set $\Psi $ such that $\Phi \subseteq \Psi $ . Moreover, if $\Phi \not \vdash _{\mathsf {M.K}} A$ , then there is a $\mathsf {M.K}$ -full set $\Psi $ such that $\Phi \subseteq \Psi $ and $A\notin \Psi $ .

Proof. The proof extends the one of [54] for $\mathsf {MPL}$ and $\mathsf {IPL}$ to the modal language $\mathcal L$ without essential modifications. Consider an enumeration $B_0, B_1, B_2, \ldots $ of all formulas of $\mathcal L$ . The set $\Psi $ is constructed as follows: $\Phi _0 = \Phi $ ; $\Phi _{n+1}=\Phi _n\cup \{B_n\}$ if $\Phi _n\not \vdash _{\mathsf {M.K}} B_n \supset A$ , and $\Phi _{n+1}=\Phi _n$ otherwise; $\Psi = \bigcup _{n \in \mathbb {N}}\Phi _n$ . It follows that $\Psi \not \vdash _{\mathsf {M.K}} A$ , otherwise there are $C_1,.., C_k\in \Psi $ such that $\vdash _{\mathsf {M.K}} C_1 \land \cdots \land C_k\supset A$ , which means that there is $n\in \mathbb {N}$ such that $C_1,.., C_k\in \Phi _n$ , hence $\Phi _n\vdash _{\mathsf {M.K}} A$ , against the construction hypothesis. Then, $A\notin \Psi $ (given the validity of $A \supset A$ in $\mathsf {MPL}$ ). Moreover, $\Psi $ is deductively closed: suppose $\Psi \vdash _{\mathsf {M.K}} C$ and $C\notin \Psi $ . Then $C=B_n$ for some $n\in \mathbb {N}$ , and, by construction, $\Phi _n \vdash _{\mathsf {M.K}} C\supset A$ . Since $\Phi _n\subseteq \Psi $ , it follows $\Psi \vdash _{\mathsf {M.K}} C\supset A$ , hence $\Psi \vdash _{\mathsf {M.K}} A$ , giving a contradiction. Finally, $\Psi $ satisfies the disjunction property: suppose $C \lor D\in \Psi $ , $C\notin \Psi $ and $D\notin \Psi $ . Then $C = B_i$ and $D = B_j$ for some $i,j \in \mathbb {N}$ . Let $n = max\{i,j\}$ . Then by construction, $\Phi _n \vdash _{\mathsf {M.K}} C \supset A$ and $\Phi _n \vdash _{\mathsf {M.K}} D \supset A$ , thus $\Phi _n \vdash _{\mathsf {M.K}} C \lor D \supset A$ . It follows $\Psi \vdash _{\mathsf {M.K}} C \lor D \supset A$ , and since $C \lor D \in \Psi $ , we have $\Psi \vdash _{\mathsf {M.K}} A$ , therefore $A \in \Psi $ , giving a contradiction.

Definition 2.7. For every logic $\mathsf {L}$ in $\mathcal L$ , an $\mathsf {L}$ -relational segment, or just segment, is a pair $(\Phi , \mathscr {U})$ , where $\Phi $ is an $\mathsf {L}$ -full set, and $\mathscr {U}$ is a set of $\mathsf {L}$ -full sets such that:

• • if $\Box A\in \Phi $ , then for all $\Psi \in \mathscr {U}$ , $A\in \Psi $ ; and

• • if $\Diamond A\in \Phi $ , then there is $\Psi \in \mathscr {U}$ such that $A\in \Psi $ .

The following holds.

Lemma 2.4. For every $\mathsf {M.K}$ -full set $\Phi $ , there exists an $\mathsf {M.K}$ -relational segment $(\Phi , \mathscr {U})$ .

Proof. Given a $\mathsf {M.K}$ -full set $\Phi $ , we define $\mathscr {U} = \{\Psi \ \mathsf {M.K}\text {-full} \mid {\Box }^-\Phi \subseteq \Psi \text { and } B\in \Psi \text { for some } \Diamond B \in \Phi \}$ . Then by definition, for all $\Box A\in \Phi $ and all $\Psi \in \mathscr {U}$ , $A\in \Psi $ . Moreover, suppose that $\Diamond A \in \Phi $ . By Lemma 2.3, there is an $\mathsf {M.K}$ -full set $\Psi $ such that ${\Box }^-\Phi \cup \{A\}\subseteq \Psi $ , then $A\in \Psi $ and $\Psi \in \mathscr {U}$ . Hence $(\Phi ,\mathscr {U})$ is an $\mathsf {M.K}$ -segment.

Definition 2.8. For every logic $\mathsf {L}$ in $\mathcal L$ , the canonical birelational model for $\mathsf {L}$ is the tuple $\mathcal M = \langle \mathcal W, \leq , \mathbb {F}, \mathcal R, \mathcal V \rangle $ , where:

• • $\mathcal W$ is the set of all $\mathsf {L}$ -relational segments;

• • for all $(\Phi , \mathscr {U}), (\Psi , \mathscr {V})\in \mathcal W$ , $(\Phi , \mathscr {U}) \leq (\Psi , \mathscr {V})$ if and only if $\Phi \subseteq \Psi $ ;

• • for all $(\Phi , \mathscr {U})\in \mathcal W$ , $(\Phi ,\mathscr {U})\in \mathbb {F}$ if and only if $\bot \in \Phi $ ;

• • for all $(\Phi , \mathscr {U}), (\Psi , \mathscr {V})\in \mathcal W$ , $(\Phi , \mathscr {U}) \mathcal R (\Psi , \mathscr {V})$ if and only if $\Psi \in \mathscr {U}$ ;

• • for all $(\Phi , \mathscr {U})\in \mathcal W$ , $(\Phi ,\mathscr {U})\in \mathcal V(p)$ if and only if $p\in \Phi $ .

It is easy to see that the canonical birelational model for $\mathsf {M.K}$ is a minimal birelational model (Definition 2.6). We prove the following lemma.

Lemma 2.5. Let $\mathcal M = \langle \mathcal W, \leq , \mathbb {F}, \mathcal R, \mathcal V \rangle $ be the canonical birelational model for $\mathsf {M.K}$ . Then for all $(\Phi , \mathscr {U})\in \mathcal W$ and all $A \in \mathcal L$ , $(\Phi , \mathscr {U})\Vdash A$ if and only if $A\in \Phi $ .

Proof. By induction on the construction of A. For the base case $A = p$ and the inductive cases $A = B \land C, B\lor C$ the proof is immediate. We show the other cases, writing $\vdash $ for $\vdash _{\mathsf {M.K}}$ .

• ( $A = \bot $ ) $(\Phi , \mathscr {U})\Vdash \bot $ iff $(\Phi , \mathscr {U})\in \mathbb {F}$ iff, by definition, $\bot \in \Phi $ .

• ( $A = B \supset C$ ) Suppose that $B\supset C\in \Phi $ , and assume $(\Phi , \mathscr {U})\leq (\Psi , \mathscr {V})$ and $(\Psi , \mathscr {V})\Vdash B$ . By definition, $\Phi \subseteq \Psi $ , thus $B\supset C\in \Psi $ . Moreover by i.h., $B\in \Psi $ , hence $C\in \Psi $ , thus by i.h., $(\Psi , \mathscr {V})\Vdash C$ . Therefore $(\Phi , \mathscr {U})\Vdash B \supset C$ . Now suppose that $B\supset C\notin \Phi $ . Then $\Phi \not \vdash B\supset C$ , thus $\Phi \cup \{B\}\not \vdash C$ . By Lemma 2.3, there is $\Psi $ $\mathsf {M.K}$ -full such that $\Phi \cup \{B\}\subseteq \Psi $ and $C\notin \Psi $ . Then by Lemma 2.4 and Definition 2.8, there is an $\mathsf {M.K}$ -segment $(\Psi , \mathscr {V})\in \mathcal W$ . Thus by definition, $(\Phi , \mathscr {U})\leq (\Psi , \mathscr {V})$ , and by i.h., $(\Psi , \mathscr {V})\Vdash B$ and $(\Psi , \mathscr {V})\not \Vdash C$ . Therefore $(\Phi , \mathscr {U})\not \Vdash B\supset C$ .

• ( $A = \Box B$ ) Suppose that $\Box B\in \Phi $ . Then for all $(\Psi , \mathscr {V})\geq (\Phi ,\mathscr {U})$ , $\Box B\in \Psi $ . Moreover by definition, if $(\Psi , \mathscr {V})\mathcal R(\Theta , \mathscr {Z})$ , then $\Theta \in \mathscr {V}$ , hence by definition of segment, $B\in \Theta $ . Then by i.h., $(\Theta , \mathscr {Z}) \Vdash B$ , therefore $(\Phi , \mathscr {U})\Vdash \Box B$ . Now suppose that $\Box B\notin \Phi $ . Then ${\Box }^-\Phi \not \vdash B$ (indeed, if ${\Box }^-\Phi \vdash B$ , then $\vdash C_1 \land \cdots \land C_n \supset B$ for some $\Box C_1, \ldots , \Box C_n\in \Phi $ , then by ${nec}$ , $\vdash \Box (C_1 \land \cdots \land C_n \supset B)$ , and by ${K_{\Box }}$ , $\vdash \Box (C_1 \land \cdots \land C_n) \supset \Box B$ ; since $\vdash \Box C_1 \land \cdots \land \Box C_n \supset \Box (C_1 \land \cdots \land C_n)$ , we have $\vdash \Box C_1 \land \cdots \land \Box C_n \supset \Box B$ , hence $\Phi \vdash \Box B$ , therefore $\Box B\in \Phi $ , against the assumption). By Lemma 2.3, there is $\Psi $ $\mathsf {M.K}$ -full such that ${\Box }^-\Phi \subseteq \Psi $ and $B\notin \Psi $ . We define $\mathscr {V} = \{\Psi \} \cup \{\Theta \ \mathsf {M.K}\text {-full} \mid {\Box }^-\Phi \subseteq \Theta \text { and } C\in \Theta \text { for some } \Diamond C \in \Phi \}$ . Given that, by Lemma 2.3, such a set $\Theta $ exists for every $\Diamond C \in \Phi $ , we have that $(\Phi , \mathscr {V})$ is an $\mathsf {M.K}$ -segment. Furthermore, by Lemma 2.4, there exists an $\mathsf {M.K}$ -segment $(\Psi , \mathscr {Z})$ , hence since $\Psi \in \mathscr {V}$ , by definition, $(\Phi , \mathscr {V})\mathcal R(\Psi , \mathscr {Z})$ . Moreover, since $B\notin \Psi $ , by i.h., $(\Psi , \mathscr {Z})\not \Vdash B$ , then since $(\Phi , \mathscr {U})\leq (\Phi , \mathscr {V})$ , we have $(\Phi , \mathscr {U})\not \Vdash \Box B$ .

• ( $A = \Diamond B$ ) Suppose that $\Diamond B\in \Phi $ . Then for all $(\Psi , \mathscr {V})\geq (\Phi ,\mathscr {U})$ , $\Diamond B\in \Psi $ . Thus by Definition 2.7, there is $\Theta \in \mathscr {V})$ such that $B\in \Theta $ , and by Lemma 2.4, there is a segment $(\Theta , \mathscr {Z})\in \mathcal M$ . Moreover, by definition, $(\Psi , \mathscr {V})\mathcal R(\Theta , \mathscr {Z})$ , and by i.h., $(\Theta , \mathscr {Z})\Vdash B$ . It follows that $(\Phi ,\mathscr {U})\Vdash \Diamond B$ . Now suppose that $\Diamond B\notin \Phi $ . Then for every $\Diamond C\in \Phi $ , ${\Box }^-\Phi \cup \{C\}\not \vdash B$ (indeed, if ${\Box }^-\Phi \cup \{C\}\vdash B$ , then $\vdash D_1 \land \cdots \land D_n \land C \supset B$ for some $\Box D_1, \ldots , \Box D_n\in \Phi $ , thus $\vdash D_1 \land \cdots \land D_n \supset (C \supset B)$ , hence $\vdash \Box (D_1 \land \cdots \land D_n) \supset \Box (C \supset B)$ , then by ${K_\Diamond }$ and valid principles, $\vdash \Box D_1 \land \cdots \land \Box D_n \supset (\Diamond C \supset \Diamond B)$ , so $\vdash \Box D_1 \land \cdots \land \Box D_n \land \Diamond C \supset \Diamond B$ , which implies $\Phi \vdash \Diamond B$ , hence, finally, $\Diamond B\in \Phi $ , against the assumption). We define $\mathscr {V} = \{\Psi \ \mathsf {M.K}\text {-full} \mid {\Box }^-\Phi \subseteq \Psi , B \notin \Psi \text { and } C\in \Psi \text { for some } \Diamond C \in \Phi \}$ . By Lemma 2.3, such a set $\Psi $ exists for every $\Diamond C \in \Psi $ . It is easy to see that $(\Phi , \mathscr {V})$ is a $\mathsf {M.K}$ -segment, hence $(\Phi , \mathscr {V})\in \mathcal W$ . Moreover, by definition, for all $(\Psi , \mathscr {Z})$ such that $(\Phi , \mathscr {V})\mathcal R(\Psi , \mathscr {Z})$ , $B\notin \Psi $ , thus by i.h., $(\Psi , \mathscr {Z})\not \Vdash B$ . Given that $(\Phi , \mathscr {U})\leq (\Phi , \mathscr {V})$ , we obtain $(\Phi , \mathscr {U})\not \Vdash \Diamond B$ .

Theorem 2.6 (Completeness).

For all $A\in \mathcal L$ , if A is valid in every minimal birelational model, then A is derivable in $\mathsf {M.K}$ .

Proof. Suppose that $\mathsf {M.K}\not \vdash A$ . Then by Lemma 2.3, there is an $\mathsf {M.K}$ -full set $\Psi $ such that $A\notin \Psi $ , and by Lemma 2.4, there exists an $\mathsf {M.K}$ -segment $(\Psi ,\mathscr {U})$ . By Definition 2.8, $(\Psi ,\mathscr {U})$ belongs to the canonical model $\mathcal M$ for $\mathsf {M.K}$ , then by Lemma 2.5, $(\Psi ,\mathscr {U})\not \Vdash A$ . Since $\mathcal M$ is a minimal birelational model, we conclude that it is not the case that A is valid in all models for $\mathsf {M.K}$ .

Based on this semantic characterisation, we now show that $\mathsf {M.K}$ is the solution to the equation ( $\ast $ ) for $\mathsf {L}$ replaced with $\mathsf {K}$ , hence, according to our criterion, it is the minimal counterpart of classical $\mathsf {K}$ .

Theorem 2.7. For all $A\in \mathcal L$ , A is derivable in $\mathsf {M.K}$ if and only if $A^{t}$ is derivable in $\mathsf {S4}\oplus \mathsf {K}$ .

Proof. We recall that $\mathsf {S4}\oplus \mathsf {K}$ is sound and complete with respect to the class of all classical birelational models $\langle \mathcal W, \mathcal R_1, \mathcal R_2, \mathcal V\rangle $ , where $\mathcal R_1$ and $\mathcal R_2$ are binary relations on $\mathcal W$ and $\mathcal R_1$ is reflexive and transitive.

( $\Rightarrow $ ) Suppose that $\mathsf {S4}\oplus \mathsf {K}\not \vdash A^{t}$ . Then there are a model $\mathcal M = \langle \mathcal W, \mathcal R_1, \mathcal R_2, \mathcal V\rangle $ for $\mathsf {S4}\oplus \mathsf {K}$ and a world $w$ such that $\mathcal M, w\not \Vdash A^{t}$ . We define $\mathcal M' = \langle \mathcal W, \leq , \mathbb {F}, \mathcal R, \mathcal V'\rangle $ over the same set $\mathcal W$ of $\mathcal M$ , where $\leq \ = \mathcal R_1$ , $\mathcal R = \mathcal R_2$ , for all $p\in \mathit {Atm}$ , $\mathcal V'(p) = \{v \mid \text {for all } u, v\mathcal R_1 u \text { implies } u\in \mathcal V(p)\}$ , and $\mathbb {F} = \{v \mid \text {for all } u, v\mathcal R_1 u \text { implies } u\in \mathcal V(f)\}$ . It is easy to verify that $\mathcal M'$ is a minimal birelational model, in particular $\mathcal V(p)$ and $\mathbb {F}$ are $\leq $ -upward closed. We show that for all $v\in \mathcal W$ and all $B\in \mathcal L$ it holds:

$$\begin{align*}\mathcal M', v \Vdash B\ \text{if and only if}\ \mathcal M, v \Vdash B^{t},\end{align*}$$

from which it follows that $\mathcal M', w \not \Vdash A$ , therefore $\mathsf {M.K}\not \vdash A$ . The proof is by induction on the construction of B. The cases $B = C \land D$ and $B = C \lor D$ are immediate by i.h. We consider the other cases.

• ( $B = p$ ) $\mathcal M', v \Vdash p$ iff $v\in \mathcal V'(p)$ iff (by definition of $\mathcal V'$ ) for all u, $v\mathcal R_1 u$ implies $u\in \mathcal V(p)$ ; iff for all u, $v\mathcal R_1 u$ implies $\mathcal M, u\Vdash p$ ; iff $\mathcal M, v \Vdash \Box _1 p$ .

• ( $B = \bot $ ) $\mathcal M', v \Vdash \bot $ iff $v\in \mathbb {F}$ iff (by definition of $\mathbb {F}$ ) for all u, $v\mathcal R_1 u$ implies $u\in \mathcal V(f)$ ; iff for all u, $v\mathcal R_1 u$ implies $\mathcal M, u\Vdash f$ ; iff $\mathcal M, v \Vdash \Box _1 f$ .

• ( $B = C \supset D$ ) $\mathcal M', v \Vdash C \supset D$ iff for all $u\geq v$ , $\mathcal M', u \Vdash C$ implies $\mathcal M', u \Vdash D$ ; iff (by definition of $\leq $ and i.h.) for all u, if $v \mathcal R_1 u$ , then $\mathcal M, u \Vdash C^{t}$ implies $\mathcal M, u \Vdash D^{t}$ ; iff for all u, if $v \mathcal R_1 u$ , then $\mathcal M, u \Vdash C^{t} \supset D^{t}$ ; iff $\mathcal M, v \Vdash \Box _1(C^{t} \supset D^{t})$ .

• ( $B = \Box C$ ) $\mathcal M', v \Vdash \Box C$ iff for all $u\geq v$ , for all z, if $u \mathcal R z$ , then $\mathcal M', z \Vdash C$ ; iff (by definition of $\leq $ and $\mathcal R$ and i.h.) for all $u,z$ , if $v \mathcal R_1 u$ and $u \mathcal R_2 z$ , then $\mathcal M, z \Vdash C^{t}$ ; iff $\mathcal M, v \Vdash \Box _1\Box _2 C^{t}$ .

• ( $B = \Diamond C$ ) $\mathcal M', v \Vdash \Diamond C$ iff for all $u\geq v$ , there is z such that $u \mathcal R z$ and $\mathcal M', z \Vdash C$ ; iff (by definition of $\leq $ and $\mathcal R$ and i.h.) for all u, if $v \mathcal R_1 u$ , then there is z such that $u \mathcal R_2 z$ and $\mathcal M, z \Vdash C^{t}$ ; iff $\mathcal M, v \Vdash \Box _1\Diamond _2 C^{t}$ .

( $\Leftarrow $ ) Suppose that $\mathsf {M.K}\not \vdash A$ . Then there are a model $\mathcal M = \langle \mathcal W, \leq , \mathbb {F}, \mathcal R, \mathcal V\rangle $ for $\mathsf {M.K}$ and a world $w$ such that $\mathcal M, w\not \Vdash A$ . We define $\mathcal M" = \langle \mathcal W, \mathcal R_1, \mathcal R_2, \mathcal V"\rangle $ over the same set $\mathcal W$ of $\mathcal M$ , where $\mathcal R_1 = \ \leq $ , $\mathcal R_2 = \mathcal R$ , for all $p\in \mathit {Atm}$ , $\mathcal V"(p) = \mathcal V(p)$ , and $\mathcal V"(f) = \mathbb {F}$ . $\mathcal M"$ is a model for $\mathsf {S4}\oplus \mathsf {K}$ . We show that for all $v\in \mathcal W$ and all $B\in \mathcal L$ it holds:

$$\begin{align*}\mathcal M, v \Vdash B \text{ \ if and only if \ } \mathcal M", v \Vdash B^{t},\end{align*}$$

from which it follows that $\mathcal M", w \not \Vdash A$ , therefore $\mathsf {S4}\oplus \mathsf {K}\not \vdash A^{t}$ . The proof is by induction on the construction of B. The cases $B = C \land D$ and $B = C \lor D$ are immediate by i.h. We consider the other cases.

• ( $B = p$ ) $\mathcal M, v \Vdash p$ iff $v\in \mathcal V(p)$ iff (since $\mathcal V$ is $\leq $ -closed) for all $u \geq v$ , $u\in \mathcal V(p)$ ; iff (by definition of $\mathcal R_1$ and $\mathcal V"$ ) for all u, if $v\mathcal R_1 u$ , then $u\in \mathcal V"(p)$ ; iff $\mathcal M", v \Vdash \Box _1 p$ .

• ( $B = \bot $ ) $\mathcal M, v \Vdash \bot $ iff $v\in \mathbb {F}$ iff (since $\mathbb {F}$ is $\leq $ -closed) for all $u \geq v$ , $u\in \mathbb {F}$ ; iff (by definition of $\mathcal R_1$ and $\mathcal V"$ ) for all u, if $v\mathcal R_1 u$ , then $u\in \mathcal V"(f)$ ; iff $\mathcal M", v \Vdash \Box _1 f$ .

• ( $B = C \supset D$ ) $\mathcal M, v \Vdash C \supset D$ iff for all $u\geq v$ , $\mathcal M, u \Vdash C$ implies $\mathcal M, u \Vdash D$ ; iff (by definition of $\mathcal R_1$ and i.h.) for all u, if $v \mathcal R_1 u$ , then $\mathcal M", u \Vdash C^{t}$ implies $\mathcal M", u \Vdash D^{t}$ ; iff $\mathcal M", v \Vdash \Box _1(C^{t} \supset D^{t})$ .

• ( $B = \Box C$ ) $\mathcal M, v \Vdash \Box C$ iff for all $u\geq v$ , for all z, if $u \mathcal R z$ , then $\mathcal M, z \Vdash C$ ; iff (by definition of $\mathcal R_1$ and $\mathcal R_2$ and i.h.) for all $u,z$ , if $v \mathcal R_1 u$ and $u \mathcal R_2 z$ , then $\mathcal M", z \Vdash C^{t}$ ; iff $\mathcal M", v \Vdash \Box _1\Box _2 C^{t}$ .

• ( $B = \Diamond C$ ) $\mathcal M, v \Vdash \Diamond C$ iff for all $u\geq v$ , there is z such that $u \mathcal R z$ and $\mathcal M, z \Vdash C$ ; iff (by definition of $\mathcal R_1$ and $\mathcal R_2$ and i.h.) for all u, if $v \mathcal R_1 u$ , then there is z such that $u \mathcal R_2 z$ and $\mathcal M", z \Vdash C^{t}$ ; iff $\mathcal M", v \Vdash \Box _1\Diamond _2 C^{t}$ .

3 Minimal K via sequent calculus

From the point of view of the sequent calculi, classical and MPL stay in a clear and neat relation: given a suitable sequent calculus $\mathsf {SC}$ for $\mathsf {CPL}$ , a calculus for $\mathsf {MPL}$ can be obtained by restricting the rules of $\mathsf {SC}$ to single-succedent sequents, namely sequents with exactly one formula in the succedent. This relation is particularly evident in G1-style sequent calculi [61]. In this section, we extend this relation to modal logics and define a minimal version of $\mathsf {K}$ by restricting a standard G1-calculus for $\mathsf {K}$ to single-succedent sequents. Most importantly, we show that the resulting logic is precisely $\mathsf {M.K}$ introduced in the previous section. We consider the following standard definitions.

Definition 3.1. A sequent is a pair $\Gamma \Rightarrow \Delta $ , where $\Gamma $ and $\Delta $ (respectively, the antecedent and the succedent of the sequent) are finite, possibly empty multisets of formulas of $\mathcal L$ . A sequent $\Gamma \Rightarrow \Delta $ is interpreted as a formula of $\mathcal L$ via the formula interpretation $\iota $ as $\bigwedge \Gamma \supset \bigvee \Delta $ if $\Gamma $ is non-empty, and as $\bigvee \Delta $ if $\Gamma $ is empty, where $\bigvee \emptyset $ is interpreted as $\bot $ . A sequent calculus $\mathsf {SC}$ is a set of initial sequents and sequent rules. ⁵ A derivation of a sequent $\mathit S$ in a calculus $\mathsf {SC}$ is a tree where each node is labelled by a sequent, the root is labelled by $\mathit S$ , the leaves are labelled by initial sequents and each node is obtained by the immediate predecessor(s) by the application of a rule of $\mathsf {SC}$ . A sequent $\mathit S$ is derivable in a calculus $\mathsf {SC}$ if there is a derivation of $\mathit S$ in $\mathsf {SC}$ . A formula A is derivable in $\mathsf {SC}$ if the sequent $\Rightarrow A$ is derivable in $\mathsf {SC}$ . A sequent calculus $\mathsf {SC}$ is a calculus for a logic $\mathsf {L}$ if for every formula A, A is derivable in $\mathsf {SC}$ if and only if it is derivable in $\mathsf {L}$ .

In order to analyse the sequent calculi, we also consider the following standard notions regarding the sequent rules.

Definition 3.2. A rule R is admissible in a calculus $\mathsf {SC}$ if whenever the premisses of R are derivable in $\mathsf {SC}$ , the conclusion is also derivable in $\mathsf {SC}$ . A formula is principal in the application of a rule if it occurs in the conclusion and not in the premiss(es), while it is active if it occurs in (at least) one premiss and not in the conclusion. Structural rules are an exception to these definitions: the principal and active formula of a contraction (respectively, weakening) rule is the one formula A which has $n> 0$ occurrences in the conclusion and $n + 1$ (respectively, $n - 1$ ) occurrences in the premiss. All formulas which are neither principal nor active are the context.

The well-known G1-sequent calculi $\mathsf {G1\text {-}{\mathsf {CPL}}}$ and $\mathsf {G1\text {-}{\mathsf {MPL}}}$ [61] for $\mathsf {CPL}$ and $\mathsf {MPL}$ are displayed in Figure 1. It is easy to see that $\mathsf {G1\text {-}{\mathsf {MPL}}}$ corresponds to the single-succedent restriction of $\mathsf {G1\text {-}{\mathsf {CPL}}}$ . In particular, the initial sequent $\bot _{\mathsf L}^{c\ell }$ and the rule $\mathsf {ctr}_{\mathsf {R}}^{cl}$ are dropped in $\mathsf {G1\text {-}{\mathsf {MPL}}}$ as they have respectively no formula in the succedent, and two occurrences of the active formula in the succedent of the premiss. $\mathsf {wk}_{\mathsf {R}}^{cl}$ is also dropped as it requires either no formula in the succedent of the premiss or at least two formulas in the succedent of the conclusion. Concerning the other rules, the right context is removed from the sequents with an active or principal formula in the succedent, this is the case for instance of initial sequents $\mathsf {init}^m$ and of the rule $\land ^m_{\mathsf {R}}$ , as well as of the left premiss of the rule $\supset ^m_{\mathsf {L}}$ . In the other rules, the right context is converted from an arbitrary multiset $\Delta $ to a single formula C.

Figure 1 Sequent calculi $\mathsf {G1\text {-}{\mathsf {CPL}}}$ and $\mathsf {G1\text {-}{\mathsf {MPL}}}$ .

In order to extend the multi- vs. single-succedent relation to modal logics, we consider the G1-sequent calculus $\mathsf {G1\text {-}{\mathsf {K}}}$ for $\mathsf {K}$ , defined extending $\mathsf {G1\text {-}{\mathsf {CPL}}}$ with the modal rules $\mathsf K_{\Box }^{c\ell }$ and $\mathsf K_{\Diamond }^{c\ell }$ in Figure 2. In these rules and the following, given a multiset $\Gamma = A_1, \ldots , A_n$ , we denote $\Box \Gamma $ and $\Diamond \Gamma $ the multisets $\Box A_1, \ldots , \Box A_n$ and $\Diamond A_1, \ldots , \Diamond A_n$ , respectively. As for axiomatic systems, sequent calculi for $\mathsf {K}$ are more commonly defined in terms of $\Box $ only. Here we consider a formulation of the calculus with both $\Box $ and $\Diamond $ explicit in order to better display the relation with minimal modal logics, where $\Box $ and $\Diamond $ are not interdefinable. The rules $\mathsf K_{\Box }^{c\ell }$ and $\mathsf K_{\Diamond }^{c\ell }$ for $\mathsf {K}$ with explicit $\Box $ and $\Diamond $ can be found e.g., in [28].

Figure 2 Modal rules for $\mathsf {G1\text {-}{\mathsf {K}}}$ and $\mathsf {G1\text {-}{\mathsf {M.K}}}$ .

On the basis of $\mathsf {G1\text {-}{\mathsf {K}}}$ , we now define the calculus $\mathsf {G1\text {-}{\mathsf {M.K}}}$ as the single-succedent restriction of $\mathsf {G1\text {-}{\mathsf {K}}}$ . As a result, $\mathsf {G1\text {-}{\mathsf {M.K}}}$ contains the rules of $\mathsf {G1\text {-}{\mathsf {MPL}}}$ and the modal rules $\mathsf {K}_{\Box }^m$ and $\mathsf {K}_{\Diamond }^m$ . Indeed, in the rule $\mathsf K_{\Box }^{c\ell }$ , the succedent of the conclusion must have a $\Box $ -formula $\Box A$ and can have additional $\Diamond $ -formulas. Then, its single-succedent restriction only preserves $\Box A$ . Concerning $\mathsf K_{\Diamond }^{c\ell }$ / $\mathsf {K}_{\Diamond }^m$ , the consequent of the conclusion of $\mathsf K_{\Diamond }^{c\ell }$ has an arbitrary number of $\Diamond $ -formulas. Correspondingly, the consequent of the conclusion of $\mathsf {K}_{\Diamond }^m$ has exactly one $\Diamond $ -formula.

In the remaining part of this section, we show that $\mathsf {G1\text {-}{\mathsf {M.K}}}$ is equivalent to the logic $\mathsf {M.K}$ defined in the previous section. The proof is based on the following theorem, which entails that the addition of the cut rule to $\mathsf {G1\text {-}{\mathsf {M.K}}}$ does not extend the set of derivable sequents. To do its length, the proof of Theorem 3.1 is presented in the appendix.

Theorem 3.1. The following rule $\mathsf {cut}$ is admissible in $\mathsf {G1\text {-}{\mathsf {M.K}}}$ :

Proof. The proof is in the appendix.

Theorem 3.2. For all $A\in \mathcal L$ , A is derivable in $\mathsf {G1\text {-}{\mathsf {M.K}}}$ if and only if A is derivable in $\mathsf {M.K}$ .

Proof. ( $\Rightarrow $ ) $\iota (A \Rightarrow A) = A \supset A$ is derivable in $\mathsf {M.K}$ , moreover we can show that for all rules $\mathit S_1, \ldots , \mathit S_n/\mathit S$ of $\mathsf {G1\text {-}{\mathsf {M.K}}}$ , the rule $\iota (\mathit S_1), \ldots , \iota (\mathit S_n)/\iota (\mathit S)$ is derivable in $\mathsf {M.K}$ , where $\iota $ is the formula interpretation of sequents as defined in Definition 3.2. For the propositional rules the proof is standard. We show in Figure 3 the derivations of the modal rules, considering the representative cases where $\Sigma = C_1, C_2$ . The cases where $\Sigma $ contains less or more formulas are a simplification or a generalisation of these cases.

( $\Leftarrow $ ) The proof consists in showing that all axioms and rules of $\mathsf {M.K}$ are derivable, respectively admissible in $\mathsf {G1\text {-}{\mathsf {M.K}}}$ . We omit the derivations of the propositional axioms which are standard. The derivations of the modal axioms and rule are displayed in Figure 4.

Figure 3 Derivations in $\mathsf {M.K}$ .

Figure 4 Derivations in $\mathsf {G1\text {-}{\mathsf {M.K}}}$ .

4 Relating minimal K and constructive K

As one can easily notice from their axiomatisations, our minimal logic $\mathsf {M.K}$ is strictly related with the constructive logic $\mathsf {C.K}$ studied in the literature, in particular $\mathsf {C.K}$ coincides with the extension of $\mathsf {M.K}$ with ex falso quodlibet $\bot \supset A$ (exactly as $\mathsf {IPL}$ amounts to $\mathsf {MPL}$ + $\bot \supset A$ ). This means that the two systems share exactly the same modal principles, despite over a different propositional base. We now show that analogously tight relations between the two logics can be also observed based on their semantics and sequent calculi.

4.1 Semantics

As recalled in Section 1.2, disregarding the modalities, there are two ways to transform relational models for $\mathsf {MPL}$ into relational models for $\mathsf {IPL}$ : (1) assuming $\mathbb {F} = \emptyset $ , thus obtaining Kripke’s intuitionistic relational models, or (2) preserving the fallible worlds but ensuring the validity of ex falso quodlibet by assuming $\mathbb {F}\subseteq \mathcal V(p)$ for all $p\in \mathit {Atm}$ . Interestingly, the two ways are equivalent for propositional logic, as they both provide a semantics for $\mathsf {IPL}$ , but they are not equivalent in presence of the modalities. In particular, if applied to minimal birelational models, the restriction (1) gives relational models for $\mathsf {W.K}$ as defined in [65]. By contrast, a suitable adaptation of (2) which ensures the validity of $\bot \supset A$ also in presence of the modalities gives the following birelational models for $\mathsf {C.K}$ .

Definition 4.1 (Constructive birelational semantics).

A minimal birelational model $\mathcal M = \langle \mathcal W, \leq , \mathbb {F}, \mathcal R, \mathcal V\rangle $ is a constructive birelational model if for all $w \in \mathbb {F}$ it holds:

1. (i) $w \in \mathcal V(p)$ for all $p \in \mathit {Atm}$ ;

2. (ii) if $w \mathcal R v$ , then $v\in \mathbb {F}$ ;

3. (iii) there is $v$ such that $w \mathcal R v$ .

Analogous constructive birelational models for $\mathsf {C.K}$ were defined in [41], ⁶ and a completeness proof for $\mathsf {C.K}$ was also provided. We show here how our canonical model construction extends to these models.

Theorem 4.1. For all $A\in \mathcal L$ , if A is derivable in $\mathsf {C.K}$ , then A is valid in every constructive birelational model $\mathcal M$ . In particular, $\mathcal M\models \bot \supset B$ for every $B\in \mathcal L$ .

Proof. The proof extends the proof of Theorem 2.2 by showing that $\mathcal M\models \bot \supset A$ for every A. Suppose that $w \Vdash \bot $ . Then $w \in \mathbb {F}$ . We show by induction on the construction of A that $w \Vdash A$ . ( $A = p$ ) By Definition 4.1, item (i), $w\in \mathcal V(p)$ , then $w \Vdash p$ . ( $A = \bot $ ) By hypothesis. ( $A = B \land C, B \lor C$ ) Immediate by applying the i.h. ( $A = B \supset C$ ) Immediate by i.h. and $\leq $ -upward closure of $\mathbb {F}$ . ( $A = \Box B$ ) Suppose $w \leq v$ . Since $\mathbb {F}$ is $\leq $ -upward closed, $v \in \mathbb {F}$ . Then by Definition 4.1, item (ii), for all u such that $v \mathcal R u$ , $u\in \mathbb {F}$ . Then by i.h., $u \Vdash B$ , therefore $w \Vdash \Box B$ . ( $A = \Diamond B$ ) Suppose $w \leq v$ . Since $\mathbb {F}$ is $\leq $ -upward closed, $v \in \mathbb {F}$ . Then by Definition 4.1, item (iii), there is u such that $v \mathcal R u$ , and by item (ii), $u\in \mathbb {F}$ . Then by i.h., $u \Vdash B$ , therefore $w \Vdash \Diamond B$ .

We now prove that $\mathsf {C.K}$ is complete with respect to constructive birelational models. First, note that Lemmas 2.3 and 2.4 also hold for $\mathsf {C.K}$ (in particular, for Lemma 2.4 the proof is the same, uniformly replacing $\mathsf {M.K}$ with $\mathsf {C.K}$ ). We additionally prove the following lemma.

Lemma 4.2. Let $\mathcal M = \langle \mathcal W, \leq , \mathbb {F}, \mathcal R, \mathcal V \rangle $ be the canonical birelational model for $\mathsf {C.K}$ (Definition 2.8). Then for all $(\Phi , \mathscr {U})\in \mathcal W$ and all $A \in \mathcal L$ , $(\Phi , \mathscr {U})\Vdash A$ if and only if $A\in \Phi $ . Moreover, $\mathcal M$ is a constructive birelational model.

Proof. The first claim is proved exactly as Lemma 2.5. For the second claim, we show that $\mathcal M$ satisfies the conditions of Definition 4.1. Suppose that $(\Phi , \mathscr {U})\in \mathbb {F}$ . Then $\bot \in \Phi $ . Since $\Phi $ is closed under derivation, by ex falso quodlibet we obtain $\Phi = \mathcal L$ , which entails the following. (i) For all $p\in \mathit {Atm}$ , $p\in \Phi $ , hence by definition, $(\Phi , \mathscr {U})\in \mathcal V(p)$ . (ii) $\Box \bot \in \Phi $ , hence by Definition 2.7, $\bot \in \Psi $ for all $\Psi \in \mathscr {U}$ . Then $(\Phi ,\mathscr {U})\mathcal R(\Psi ,\mathscr {V})$ entails $\bot \in \Psi $ , thus $(\Psi , \mathscr {V})\in \mathbb {F}$ . (iii) $\Diamond \bot \in \Phi $ , hence by Definition 2.7, there is $\Psi \in \mathscr {U}$ such that $\bot \in \Psi $ . By Lemma 2.4 (which holds for $\mathsf {C.K}$ as well), there exists a $\mathsf {C.K}$ -segment $(\Psi ,\mathscr {V})$ . Then $(\Phi ,\mathscr {U})\mathcal R(\Psi ,\mathscr {V})$ and $(\Psi , \mathscr {V})\in \mathbb {F}$ .

As a consequence of the lemma, we obtain the completeness of $\mathsf {C.K}$ (cf. proof of Theorem 2.6).

Theorem 4.3. For all $A\in \mathcal L$ , A is derivable in $\mathsf {C.K}$ if and only if A is valid in every constructive birelational model.

4.2 Sequent calculus

We have considered in Section 3 the multi- vs. single-succedent correspondence between the sequent calculi $\mathsf {G1\text {-}{\mathsf {CPL}}}$ and $\mathsf {G1\text {-}{\mathsf {MPL}}}$ . A similar relation holds between sequent calculi for $\mathsf {CPL}$ and $\mathsf {IPL}$ . In particular, the calculus $\mathsf {G1\text {-}{\mathsf {IPL}}}$ for $\mathsf {IPL}$ can be defined by restricting $\mathsf {G1\text {-}{\mathsf {CPL}}}$ to sequents with at most one formula in the succedent (cf. [61]). The resulting calculus is displayed in Figure 5, where $0 \leq |\delta | \leq 1$ . If we apply the same restriction to $\mathsf {G1\text {-}{\mathsf {K}}}$ , we obtain the calculus

for $\mathsf {W.K}$ defined in [65]. By contrast, in order to obtain a calculus for $\mathsf {C.K}$ , we need to extend $\mathsf {G1\text {-}{\mathsf {IPL}}}$ with the minimal modal rules only:

$$\begin{align*}\mathsf{G1\text{-}{\mathsf{C.K}}} := \mathsf{G1\text{-}{\mathsf{IPL}}} + \mathsf{K}_{\Box}^m + \mathsf{K}_{\Diamond}^m. \end{align*}$$

In this way, we re-obtain the sequent calculus for $\mathsf {C.K}$ defined and proved to be cut-free in [6].

Figure 5 Sequent calculus $\mathsf {G1\text {-}{\mathsf {IPL}}}$ .

5 A family of minimal modal logics

We have seen that the two considered methods, respectively based on bimodal companion and sequent calculus restriction, define the same minimal counterpart of $\mathsf {K}$ . In this section, we show that this result is not a peculiarity of $\mathsf {K}$ only: we apply the same procedure to a family of 14 standard classical modal logics, and show that for each of them the two methods construct the same logic, thus obtaining a minimal counterpart for all these classical systems.

In order to apply our sequent-based approach, we need to restrict to classical modal logics enjoying standard cut-free Gentzen calculi (this restriction excludes well-known modal logics for which such calculi are not available, such as $\mathsf {S5}$ ⁷ ). We also require the logics to have a uniform semantic characterisation, we consider to this purpose a neighbourhood semantics that uniformly covers all considered systems, that include both normal and non-normal modal logics.

Specifically, we consider 14 classical modal logics that are axiomatically defined in the language $\mathcal L$ extending $\mathsf {CPL}$ , formulated in $\mathcal L$ , with the following modal axioms and rules from Figure 6:

Figure 6 Modal axioms and rules.

We adopt the standard naming convention of monotonic non-normal modal logics (cf. e.g., [26, 38]): each classical modal logic is denoted $\mathsf {M\Sigma }$ , where $\mathsf \Sigma \subseteq \{\mathsf {C,N,P,D,T}\}$ corresponds to the list of axioms among ${C_{\Box }}$ , ${N_{\Box }}$ , ${P_{\Box }}$ , ${D}$ , ${T_{\Box }}$ extending $\mathsf {M}$ . The only exceptions to this notation are $\mathsf {K}$ , $\mathsf {KD}$ and $\mathsf {KT}$ for which we use the usual names. Note however that $\mathsf {K}$ amounts to $\mathsf {MCN}$ , this axiomatisation of $\mathsf {K}$ is equivalent to the more standard one with ${nec}$ and ${K_{\Box }}$ considered in Section 1.1 (cf. e.g., [9]). As usual, given the duality between $\Box $ and $\Diamond $ in classical logics, the above systems can be equivalently defined by replacing ${mon_{\Box }}$ , ${N_{\Box }}$ , ${C_{\Box }}$ , ${P_{\Box }}$ , and ${T_{\Box }}$ , with their $\Diamond $ -versions ${mon_{\Diamond }}$ , ${N_\Diamond }$ , ${C_\Diamond }$ , ${P_{\Diamond }}$ , and ${T_{\Diamond }}$ (Figure 6). ⁸ The systems $\mathsf {MCP}$ and $\mathsf {KP}$ are not listed above as they are respectively equivalent to $\mathsf {MCD}$ and $\mathsf {KD}$ ( ${P_{\Box }}$ and ${D}$ are interderivable given ${mon_{\Box }}$ and ${C_{\Box }}$ ). The resulting classical modal logics and their inclusion relations are displayed in Figure 7. In the following, we use $\mathsf {L}$ or $\mathsf {M\Sigma }$ , without specifying the set $\mathsf \Sigma $ , to denote any of the above classical logics.

Figure 7 Diagram of classical modal logics.

In the following subsections, we show that the two methods define, for each classical modal logic $\mathsf {L}$ , the following minimal counterpart $\mathsf {M.L}$ .

Definition 5.1 (Minimal modal logics).

Minimal modal logics are axiomatically defined in the language $\mathcal L$ extending $\mathsf {MPL}$ with the following modal axioms and rules from Figure 6:

It is worth observing immediately that not all logics contain both the $\Box $ - and the $\Diamond $ -versions of the characteristic modal axioms of their classical counterparts. In particular, as we will show later in this section, ${P_{\Box }}$ , ${N_\Diamond }$ and ${C_\Diamond }$ are not valid in the corresponding minimal systems, with the latter axiom that needs to be replaced with ${K_\Diamond }$ in order to obtain a complete axiomatisation. On the other hand, by means of routine axiomatic derivations one can show that the above definition of $\mathsf {M.K}$ is equivalent to the one of Definition 2.5 based on ${K_{\Box }}$ and ${nec}$ , in particular ${K_{\Box }}$ is derivable from ${mon_{\Box }}$ , ${C_{\Box }}$ and the axioms of $\mathsf {MPL}$ .

5.1 Minimal modal logics via bimodal companions

We prove that each minimal logic $\mathsf {M.L}$ above is the minimal counterpart of the classical logic $\mathsf {L}$ as defined in Definition 2.4 (that is, $\mathsf {M.L}\vdash A$ if and only if $\mathsf {S4}\oplus \mathsf {L}\vdash A^{t}$ ). As before, in order to prove this result, we first provide a semantics for minimal modal logics.

We start recalling the neighbourhood semantics for classical modal logics (cf. [9, 47]). A classical neighbourhood model is a tuple $\mathcal M = \langle \mathcal W, \mathcal N, \mathcal V\rangle $ , where $\mathcal W$ is a non-empty set of worlds, $\mathcal V : \mathit {Atm} \longrightarrow \mathcal P(\mathcal W)$ is a valuation function for propositional variables, and $\mathcal N$ is a function $\mathcal W\longrightarrow \mathcal P(\mathcal P(\mathcal W))$ , called neighbourhood function. Modal formulas are interpreted in classical neighbourhood models as $w\Vdash \Box B$ iff there is $\alpha \in \mathcal N(w)$ such that for all $v \in \alpha $ , $v \Vdash B$ ; and $w\Vdash \Diamond B$ iff for all $\alpha \in \mathcal N(w)$ , there is $v\in \alpha $ such that $v\Vdash B$ . Each classical modal logic $\mathsf {L}$ considered in this work is characterised by the class of all classical neighbourhood models satisfying the following condition (C), (N), (P), (D), or (T), for all $\alpha , \beta \subseteq \mathcal W$ , if $\mathsf {L}$ contains the axiom ${C_{\Box }}$ , ${N_{\Box }}$ , ${P_{\Box }}$ , ${D}$ , or ${T_{\Box }}$ , respectively:

We also remark that for each considered classical modal logic $\mathsf {L}$ , the fusion $\mathsf {S4} \oplus \mathsf {L}$ is characterised by the class of models $ \langle \mathcal W, \mathcal R, \mathcal N, \mathcal V\rangle $ , where $\mathcal R$ is a reflexive and transitive binary relation on $\mathcal W$ , and $\mathcal N$ is a neighbourhood function satisfying the conditions among (C), (N), (D), (P), (T) satisfied by the models for $\mathsf {L}$ . This characterisation of fusions $\mathsf {S4} \oplus \mathsf {L}$ can be easily proved by combining the completeness proofs by canonical models for $\mathsf {S4}$ and for $\mathsf {L}$ (see e.g., [9]).

By combining relational models for $\mathsf {MPL}$ and classical neighbourhood models, we now define minimal neighbourhood models for minimal modal logics as follows.

Definition 5.2 (Minimal neighbourhood semantics).

A minimal neighbourhood model is a tuple $\mathcal M = \langle \mathcal W, \leq , \mathbb {F}, \mathcal N, \mathcal V\rangle $ , where $\langle \mathcal W, \leq , \mathbb {F}, \mathcal V\rangle $ is a minimal relational model, and $\mathcal N$ is a neighbourhood function $\mathcal W\longrightarrow \mathcal P(\mathcal P(\mathcal W))$ . The forcing relation $\mathcal M, w\Vdash A$ is inductively defined extending the clauses for $p, \bot , \land , \lor , \supset $ in Section 1.2 with the following clauses for the modalities:

where $\alpha \Vdash ^{\forall } B$ and $\alpha \Vdash ^{\exists } B$ are abbreviations for, respectively, ‘for all $u\in \alpha $ , $\mathcal M, u\Vdash B$ ’, and ‘there is $u\in \alpha $ such that $\mathcal M, u\Vdash B$ ’.

For each minimal modal logic $\mathsf {M.M\Sigma }$ , we say that a minimal neighbourhood model $\mathcal M$ is a model for $\mathsf {M.M\Sigma }$ (or it is a $\mathsf {M.M\Sigma }$ -model) if it satisfies the condition (X) above for all $\mathsf {X} \in \mathsf {\Sigma }$ . Note that $\mathsf {M.K}$ amounts to $\mathsf {M.MCN}$ , hence the corresponding models must satisfy both (C) and (N).

By an easy induction on the construction of formulas one can prove the following.

Proposition 5.1 (Hereditary property).

For every $A\in \mathcal L$ , every minimal neighbourhood model $\mathcal M$ , and every world $w$ of $\mathcal M$ , if $w\Vdash A$ and $w\leq v$ , then $v\Vdash A$ .

Now we prove that the logics $\mathsf {M.L}$ are sound and complete with respect to the corresponding classes of models.

Theorem 5.2. For all $A\in \mathcal L$ and all minimal modal logic $\mathsf {M.L}$ , if A is derivable in $\mathsf {M.L}$ , then A is valid in all minimal neighbourhood models for $\mathsf {M.L}$ .

Proof. We show that all modal axioms and rules of $\mathsf {M.L}$ are valid, respectively validity preserving, in every minimal neighbourhood model $\mathcal M$ for $\mathsf {M.L}$ .

• ( ${mon_{\Box }}$ ) Suppose that $\mathcal M\models A \supset B$ and $w \Vdash \Box A$ . Then for all $v \geq w$ , there is $\alpha \in \mathcal N(v)$ such that for all $z\in \alpha $ , $z\Vdash A$ , thus $z\Vdash B$ , hence $w \Vdash \Box B$ . Therefore $\mathcal M\models \Box A \supset \Box B$ .

• ( ${mon_{\Diamond }}$ ) Suppose that $\mathcal M\models A \supset B$ and $w \Vdash \Diamond A$ . Then for all $v \geq w$ , for all $\alpha \in \mathcal N(v)$ , there is $z\in \alpha $ such that $z\Vdash A$ , thus $z\Vdash B$ , hence $w \Vdash \Diamond B$ . Therefore $\mathcal M\models \Diamond A \supset \Diamond B$ .

• ( ${N_{\Box }}$ ) For all w and all $v\geq w$ , by (N), there is $\alpha \in \mathcal N(v)$ . Since $z\Vdash \top $ for all $z\in \alpha $ , we have $w \Vdash \Box \top $ . Thus $\mathcal M\models \Box \top $ .

• ( ${C_{\Box }}$ ) Suppose that $w\Vdash \Box A\land \Box B$ . Then for all $v \geq w$ , there are $\alpha ,\beta \in \mathcal N(v)$ such that $\alpha \Vdash ^{\forall } A$ and $\beta \Vdash ^{\forall } B$ . By (C), $\alpha \cap \beta \in \mathcal N(v)$ , moreover $\alpha \cap \beta \Vdash ^{\forall } A \land B$ . Hence $\mathcal M\models \Box A \land \Box B \supset \Box (A\land B)$ .

• ( ${K_\Diamond }$ ) Suppose that $w \Vdash \Box (A \supset B)$ and $w \Vdash \Diamond A$ . Then for all $v \geq w$ , there is $\alpha \in \mathcal N(v)$ such that $\alpha \Vdash ^{\forall } A \supset B$ . Now, suppose that $\beta \in \mathcal N(v)$ . By (C), $\alpha \cap \beta \in \mathcal N(v)$ . Since $\alpha \cap \beta \subseteq \alpha $ , $\alpha \cap \beta \Vdash ^{\forall } A \supset B$ . Moreover, by $w\Vdash \Diamond B$ , $\alpha \cap \beta \Vdash ^{\exists } A$ . Thus $\alpha \cap \beta \Vdash ^{\exists } B$ , which implies $\beta \Vdash ^{\exists } B$ . Since this holds for every $\beta \in \mathcal N(v)$ , $w\Vdash \Diamond B$ . Therefore $\mathcal M\models \Box (A \supset B) \supset (\Diamond A \supset \Diamond B)$ .

• ( ${P_{\Diamond }}$ ) For all w and all $v\geq w$ , by (P), $\emptyset \notin \mathcal N(v)$ . Hence, for all $\alpha \in \mathcal N(v)$ , $\alpha \not =\emptyset $ , thus $\alpha \Vdash ^{\exists }\top $ . Then we have $w \Vdash \Diamond \top $ . Thus $\mathcal M\models \Diamond \top $ .

• ( ${D}$ ) Suppose that $w \Vdash \Box A$ . Then for all $v \geq w$ , there is $\alpha \in \mathcal N(v)$ such that $\alpha \Vdash ^{\forall } A$ . Now, suppose that $\beta \in \mathcal N(v)$ . By (D), there is $z\in \alpha \cap \beta $ . Then $z\Vdash A$ , hence $\beta \Vdash ^{\exists } A$ , therefore $w \Vdash \Diamond A$ . Hence $\mathcal M\models \Box A \supset \Diamond A$ .

• ( ${T_{\Box }}$ ) Suppose that $w \Vdash \Box A$ . Then for all $v \geq w$ , there is $\alpha \in \mathcal N(v)$ such that $\alpha \Vdash ^{\forall } A$ . Hence in particular there is $\alpha \in \mathcal N(w)$ such that $\alpha \Vdash ^{\forall } A$ . By (T), $w\in \alpha $ , then $w \Vdash A$ . Therefore $\mathcal M\models \Box A \supset A$ .

• ( ${T_{\Diamond }}$ ) Suppose that $w \Vdash A$ . By the hereditary property of minimal neighbourhood models, for all $v\geq w$ , $v\Vdash A$ . Moreover, by (T), for all $\alpha \in \mathcal N(v)$ , $v\in \alpha $ , hence $\alpha \Vdash ^{\exists } A$ . Thus $w \Vdash \Diamond A$ , therefore $\mathcal M\models \Diamond A$ .

The proof of completeness proceeds essentially as the one in Section 2. First, we observe that Lemma 2.3 also holds for all logics $\mathsf {M.L}$ . We consider the following definition of neighbourhood segment.

Definition 5.3. For every logic $\mathsf {L}$ in $\mathcal L$ , an $\mathsf {L}$ -neighbourhood segment, or just segment, is a pair $(\Phi , \mathscr {C})$ , where $\Phi $ is an $\mathsf {L}$ -full set, and $\mathscr {C}$ is a class of sets of $\mathsf {L}$ -full sets such that:

• • if $\Box A\in \Phi $ , then there is $\mathscr {U}\in \mathscr {C}$ such that for all $\Psi \in \mathscr {U}$ , $A\in \Psi $ ; and

• • if $\Diamond A\in \Phi $ , then for all $\mathscr {U}\in \mathscr {C}$ , there is $\Psi \in \mathscr {U}$ such that $A\in \Psi $ .

Moreover, if $\mathsf {L}$ contains the axiom ${C_{\Box }}$ , or the axiom ${D}$ , or the axiom ${T_{\Box }}$ , then the $\mathsf {L}$ -segments must satisfy the following corresponding condition:

Lemma 5.3. For every minimal modal logic $\mathsf {M.L}$ and every $\mathsf {M.L}$ -full set $\Phi $ ,

1. (i) there exists an $\mathsf {M.L}$ -neighbourhood segment $(\Phi , \mathscr {C})$ ;

2. (ii) if $\Box A\notin \Phi $ , then there exists an $\mathsf {M.L}$ -neighbourhood segment $(\Phi , \mathscr {C})$ such that for all $\mathscr {U}\in \mathscr {C}$ , there is $\Psi \in \mathscr {U}$ such that $A\notin \Psi $ ;

3. (iii) if $\Diamond A\notin \Phi $ , then there exists an $\mathsf {M.L}$ -neighbourhood segment $(\Phi , \mathscr {C})$ such that there is $\mathscr {U}\in \mathscr {C}$ such that for all $\Psi \in \mathscr {U}$ , $A\notin \Psi $ .

Proof.

1. (i) Given an $\mathsf {M.L}$ -full set $\Phi $ , we construct an $\mathsf {M.L}$ -segment $(\Phi , \mathscr {C})$ as follows. For all $\Box A\in \Phi $ , we define $\mathscr {U}_{A}^- = \{\Psi \ \mathsf {M.L}\text {-full} \mid A \in \Psi \text { and there is } \Diamond B\in \Phi \text { such that } B\in \Psi \}$ ; and $\mathscr {U}_{A} = \mathscr {U}_{A}^-$ if $\mathsf {M.L}$ does not contain ${T_{\Box }}$ , and $\mathscr {U}_{A} = \mathscr {U}_{A}^- \cup \{\Phi \}$ if $\mathsf {M.L}$ contains ${T_{\Box }}$ . Moreover, we define $\mathscr {C} = \{\mathscr {U}_{A} \mid \Box A \in \Phi \}$ . We show that $(\Phi , \mathscr {C})$ is an $\mathsf {M.L}$ -segment.

   • • If $\Box A\in \Phi $ , then by definition $\mathscr {U}_{A}\in \mathscr {C}$ . Moreover, if $\mathsf {M.L}$ does not contain ${T_{\Box }}$ , then $A\in \Psi $ for all $\Psi \in \mathscr {U}_{A}$ . If instead $\mathsf {M.L}$ contains ${T_{\Box }}$ , then for all $\Psi \in \mathscr {U}_{A}$ we have $A\in \Psi $ or $\Psi =\Phi $ , where, by ${T_{\Box }}$ and closure under derivation of $\mathsf {M.L}$ -full sets, $A\in \Phi $ .

   • • If $\Diamond A\in \Phi $ , then assume $\mathscr {U}\in \mathscr {C}$ . Then, by definition, $\mathscr {U} = \mathscr {U}_{B}$ for some $\Box B\in \Phi $ . By Lemma 2.3, there is an $\mathsf {M.L}$ -full set $\Psi $ such that $A,B\in \Psi $ , hence $\Psi \in \mathscr {U}_{B} = \mathscr {U}$ and $A\in \Psi $ .

   Moreover, the conditions (C-s), (D-s) and (T-s) are satisfied if $\mathsf {M.L}$ contains the axioms ${C_{\Box }}$ , ${D}$ , or ${T_{\Box }}$ , respectively:

   • (C-s) Suppose $\mathscr {U}, \mathscr {V}\in \mathscr {C}$ . Then $\mathscr {U} = \mathscr {U}_{A}$ and $\mathscr {V} = \mathscr {U}_{B}$ for some $\Box A, \Box B\in \Phi $ . Hence, given that $\mathsf {M.L}$ contains ${C_{\Box }}$ , by closure under derivation of $\mathsf {M.L}$ -full sets, we have $\Box (A \land B)\in \Phi $ , thus $\mathscr {U}_{A\land B}\in \mathscr {C}$ . Note also that for all $\mathsf {M.L}$ -full sets $\Psi $ it holds $A,B\in \Psi $ if and only if $A\land B\in \Psi $ . One can easily verify that this implies $\mathscr {U}_{A\land B} = \mathscr {U}_{A} \cap \mathscr {U}_{B}$ , therefore $\mathscr {U} \cap \mathscr {V} = \mathscr {U}_{A} \cap \mathscr {U}_{B} \in \mathscr {C}$ .

   • (D-s) Suppose $\mathscr {U}, \mathscr {V}\in \mathscr {C}$ . Then $\mathscr {U} = \mathscr {U}_{A}$ and $\mathscr {V} = \mathscr {U}_{B}$ for some $\Box A, \Box B\in \Phi $ . Given that $\mathsf {M.L}$ contains ${D}$ , by closure under derivation of $\mathsf {M.L}$ -full sets, we have $\Diamond A, \Diamond B\in \Phi $ . By Lemma 2.3, there is an $\mathsf {M.L}$ -full set $\Psi $ such that $A, B\in \Psi $ . Then by definition, $\Psi \in \mathscr {U}_{A}$ and $\Psi \in \mathscr {U}_{B}$ , hence $\Psi \in \mathscr {U}_{A}\cap \mathscr {U}_{B}$ , therefore $\mathscr {U} \cap \mathscr {V} = \mathscr {U}_{A} \cap \mathscr {U}_{B} \not =\emptyset $ .

   • (T-s) By definition, for all $\mathscr {U}\in \mathscr {C}$ , $\Phi \in \mathscr {U}$ .

2. (ii) For all $\Box B\in \Phi $ , we define $\mathscr {U}_{B}^- = \{\Psi \ \mathsf {M.L}\text {-full} \mid B \in \Psi \text { and there is } \Diamond C\in \Phi \text { such that } C\in \Psi \} \cup \{\Psi \ \mathsf {M.L}\text {-full} \mid B \in \Psi \text { and } A \notin \Psi \}$ ; and $\mathscr {U}_{B} = \mathscr {U}_{B}^-$ if $\mathsf {M.L}$ does not contain ${T_{\Box }}$ , and $\mathscr {U}_{B} = \mathscr {U}_{B}^- \cup \{\Phi \}$ if $\mathsf {M.L}$ contains ${T_{\Box }}$ . Moreover, we define $\mathscr {C} = \{\mathscr {U}_{B} \mid \Box B \in \Phi \}$ . We can show that $(\Phi , \mathscr {C})$ is an $\mathsf {M.L}$ -segment as in item (i). Now, suppose that $\mathscr {U}\in \mathscr {C}$ . Then $\mathscr {U} = \mathscr {U}_{B}$ for some $\Box B\in \Phi $ . Thus, since $\Box A\notin \Phi $ , $\{B\}\not \vdash A$ (otherwise $\vdash B \supset A$ , and by ${mon_{\Box }}$ , $\vdash \Box B\supset \Box A$ , hence by closure under derivation, $\Box A\in \Phi $ ). Then by Lemma 2.3, there is an $\mathsf {M.L}$ -full set $\Psi $ such that $B\in \Psi $ and $A \notin \Psi $ , and by definition, $\Psi \in \mathscr {U}_{B} = \mathscr {U}$ .

3. (iii) (iii.i) $\mathsf {M.L}$ does not contain ${C_{\Box }}$ , ${K_\Diamond }$ . We define $\mathscr {U}^- = \{\Psi \ \mathsf {M.L}\text {-full} \mid A \notin \Psi \text { and there is } \Diamond B\in \Phi \text { such that } B\in \Psi \}$ , and for all $\Box C\in \Phi $ , we define $\mathscr {U}_{C}^- = \{\Psi \ \mathsf {M.L}\text {-full} \mid C \in \Psi \text { and there is } \Diamond B\in \Phi \text { such that } B\in \Psi \}$ . Moreover, we define $\mathscr {U} = \mathscr {U}^-$ , $\mathscr {U}_{C} = \mathscr {U}_{C}^-$ if $\mathsf {M.L}$ does not contain ${T_{\Box }}$ , and $\mathscr {U} = \mathscr {U}^- \cup \{\Phi \}$ , $\mathscr {U}_{C} = \mathscr {U}_{C}^- \cup \{\Phi \}$ if $\mathsf {M.L}$ contains ${T_{\Box }}$ . Finally, we define $\mathscr {C} = \{\mathscr {U}\}\cup \{\mathscr {U}_{C} \mid \Box C \in \Phi \}$ . Note that $\mathscr {U}$ satisfies the condition of the lemma, in particular if ${T_{\Box }}$ belongs to $\mathsf {M.L}$ , then $A\notin \Phi $ , since if $A\in \Phi $ , then by ${T_{\Diamond }}$ , $\Diamond A\in \Phi $ , against the assumption. We can show that $(\Phi , \mathscr {C})$ is an $\mathsf {M.L}$ -segment. First, the conditions of $\mathsf {M.L}$ -segments for any $\Box B\in \Phi $ and for any $\Diamond B\in \Phi $ can be shown to be satisfied similarly to item (i). Moreover, the property (T-s) of $\mathsf {M.L}$ -segments for $\mathsf {M.L}$ containing ${T_{\Box }}$ follows immediately from the definition. We show that (D-s) is satisfied if $\mathsf {M.L}$ contains the axiom ${D}$ : Suppose that $\mathscr {V}, \mathscr {Z}\in \mathscr {C}$ . If $\mathscr {V} = \mathscr {U}_{C}, \mathscr {Z} = \mathscr {U}_{D}$ for some $\Box C, \Box D \in \Phi $ , the proof is analogous to the one of (D-s) in item (i). Now suppose $\mathscr {V} = \mathscr {U}_{C}$ for some some $\Box C \in \Phi $ and $\mathscr {Z} = \mathscr {U}$ . Then by axiom ${D}$ , $\Diamond C\in \Phi $ . Thus we have $\{C\}\not \vdash A$ , otherwise we would have $\vdash C \supset A$ , and by ${mon_{\Diamond }}$ , $\vdash \Diamond C \supset \Diamond A$ , hence $\Diamond A \in \Phi $ , against the assumption. By Lemma 2.3, there is an $\mathsf {M.L}$ -full set $\Psi $ such that $C\in \Psi $ and $A\notin \Psi $ . By definition, $\Psi \in \mathscr {U}$ , moreover $\Psi \in \mathscr {U}_{C}$ (since $\Diamond C\in \Phi $ ), hence $\Psi \in \mathscr {U}\cap \mathscr {U}_{C}=\mathscr {V}\cap \mathscr {Z}$ , therefore $\mathscr {V}\cap \mathscr {Z}\not =\emptyset $ .(iii.ii) $\mathsf {M.L}$ contains ${C_{\Box }}$ , ${K_\Diamond }$ . We define $\mathscr {U}^- = \{\Psi \ \mathsf {M.L}\text {-full} \mid A \notin \Psi , \text { and } {\Box }^-\Phi \subseteq \Psi , \text { and } B\in \Psi \text { for some } \Diamond B\in \Phi \}$ , and $\mathscr {U} = \mathscr {U}^-$ if $\mathsf {M.L}$ does not contain ${T_{\Box }}$ , and $\mathscr {U} = \mathscr {U}^- \cup \{\Phi \}$ $\mathsf {M.L}$ it contains ${T_{\Box }}$ . Moreover, we define $\mathscr {C} = \{\mathscr {U}\}$ . Clearly, $\mathscr {U}$ satisfies the claim of the lemma (in particular, if $\mathsf {M.L}$ contains ${T_{\Box }}$ , then $A\notin \Phi $ ). We show that $(\Phi , \mathscr {C})$ is an $\mathsf {M.L}$ -segment. First, observe that for any $\Diamond B\in \Phi $ , ${\Box }^-\Phi \cup \{B\}\not \vdash A$ . Indeed, if ${\Box }^-\Phi \cup \{B\}\vdash A$ , then there are $C_1, \ldots , C_n\in {\Box }^-\Phi $ such that $\vdash C_1\land \cdots \land C_n\land B \supset A$ , hence $\vdash C_1\land \cdots \land C_n\supset (B \supset A)$ , then by ${mon_{\Box }}$ , $\vdash \Box (C_1\land \cdots \land C_n)\supset \Box (B \supset A)$ , thus by ${C_{\Box }}$ (n times) and ${K_\Diamond }$ , $\vdash \Box C_1\land \cdots \land \Box C_n\supset (\Diamond B \supset \Diamond A)$ , which gives $\vdash \Box C_1\land \cdots \land \Box C_n\land \Diamond B \supset \Diamond A$ , therefore $\Box C_1, \ldots , \Box C_n, \Diamond B \vdash \Diamond A$ ; since $\Box C_1, \ldots , \Box C_n, \Diamond B\in \Phi $ , this entails $\Diamond A\in \Phi $ , against the assumption. We then have: the condition for any $\Box B\in \Phi $ follows immediately from the definition. If $\Diamond B\in \Phi $ , then ${\Box }^-\Phi \cup \{B\}\not \vdash A$ , thus by Lemma 2.3, there is an $\mathsf {M.L}$ -full set $\Psi $ such that ${\Box }^-\Phi \subseteq \Psi $ , $B\in \Psi $ and $A \notin \Psi $ , hence $\Psi \in \mathscr {U}$ . By the same argument, the property (D-s) is satisfied for $\mathsf {M.L}$ containing the axiom ${D}$ given that $\Diamond \top \in \Phi $ entails the existence of such an $\mathsf {M.L}$ -full set $\Psi $ , hence $\mathscr {U}\not =\emptyset $ . Moreover, (C-s) is trivial, and (T-s) for $\mathsf {M.L}$ containing ${T_{\Box }}$ follows immediately from the definition.

Definition 5.4. For every logic $\mathsf {L}$ in $\mathcal L$ , the canonical neighbourhood model for $\mathsf {L}$ is the tuple $\mathcal M = \langle \mathcal W, \leq , \mathbb {F}, \mathcal N, \mathcal V \rangle $ , where:

• • $\mathcal W$ is the set of all $\mathsf {L}$ -neighbourhood segments;

• • for all $(\Phi , \mathscr {C}), (\Psi , \mathscr {D})\in \mathcal W$ , $(\Phi , \mathscr {C}) \leq (\Psi , \mathscr {D})$ if and only if $\Phi \subseteq \Psi $ ;

• • for all $(\Phi , \mathscr {C})\in \mathcal W$ , $(\Phi ,\mathscr {C})\in \mathbb {F}$ if and only if $\bot \in \Phi $ ;

• • for all sets $\mathscr {U}$ of $\mathsf {M.L}$ -full sets, $\alpha _{\mathscr {U}} = \{(\Phi , \mathscr {C}) \mid \Phi \in \mathscr {U}\}$ ;

• • for all $(\Phi , \mathscr {C})\in \mathcal W$ , $\alpha _{\mathscr {U}}\in \mathcal N((\Phi , \mathscr {C}))$ if and only if $\mathscr {U}\in \mathscr {C}$ ;

• • for all $(\Phi , \mathscr {C})\in \mathcal W$ , $(\Phi ,\mathscr {C})\in \mathcal V(p)$ if and only if $p\in \Phi $ .

Lemma 5.4. For every minimal modal logic $\mathsf {M.L}$ , the canonical neighbourhood model $\mathcal M$ for $\mathsf {M.L}$ is a minimal neighbourhood model for $\mathsf {M.L}$ .

Proof. It is easy to very that $\mathcal M$ is a minimal neighbourhood model. We show that $\mathcal M$ satisfies the conditions among (C), (N), (P), (D), (T) associated with the axioms of $\mathsf {M.L}$ .

1. (C) Suppose that $\alpha , \beta \in \mathcal N((\Phi , \mathscr {C}))$ . Then, by definition, $\alpha =\alpha _{\mathscr {U}}$ and $\beta = \alpha _{\mathscr {V}}$ for some $\mathscr {U}, \mathscr {V}\in \mathscr {C}$ . By the property (C-s) of $\mathsf {M.L}$ -segments, $\mathscr {U}\cap \mathscr {V}\in \mathscr {C}$ , thus $\alpha _{\mathscr {U}\cap \mathscr {V}}\in \mathcal N((\Phi , \mathscr {C}))$ , where $\alpha _{\mathscr {U}\cap \mathscr {V}} = \{(\Phi , \mathscr {C}) \mid \Phi \in \mathscr {U}\cap \mathscr {V}\} = \{(\Phi , \mathscr {C}) \mid \Phi \in \mathscr {U}\} \cap \{(\Phi , \mathscr {C}) \mid \Phi \in \mathscr {V}\} = \alpha _{\mathscr {U}} \cap \alpha _{\mathscr {V}} = \alpha \cap \beta $ .

2. (N) For all $\mathsf {M.L}$ -full sets $\Phi $ , $\Box \top \in \Phi $ , then for all $\mathsf {M.L}$ -segments $(\Phi , \mathscr {C})$ , $\mathscr {C}\not =\emptyset $ , thus $\mathcal N((\Phi , \mathscr {C}))\not =\emptyset $ .

3. (P) For all $\mathsf {M.L}$ -full sets $\Phi $ , $\Diamond \top \in \Phi $ , then for all $\mathsf {M.L}$ -segments $(\Phi , \mathscr {C})$ and all $\mathscr {U}\in \mathscr {C}$ , $\mathscr {U}\not =\emptyset $ , thus for all $\alpha _{\mathscr {U}}\in \mathcal N((\Phi , \mathscr {C}))$ , $\alpha _{\mathscr {U}}\not =\emptyset $ , that is, $\emptyset \notin \mathcal N((\Phi , \mathscr {C}))$ .

4. (D) Suppose that $\alpha , \beta \in \mathcal N((\Phi , \mathscr {C}))$ . Then $\alpha =\alpha _{\mathscr {U}}$ and $\beta = \alpha _{\mathscr {V}}$ for some $\mathscr {U}, \mathscr {V}\in \mathscr {C}$ . By (D-s), $\mathscr {U}\cap \mathscr {V}\not =\emptyset $ , which implies $\alpha _{\mathscr {U}} \cap \alpha _{\mathscr {V}} = \alpha \cap \beta \not =\emptyset $ .

5. (T) Suppose that $\alpha \in \mathcal N((\Phi , \mathscr {C}))$ . Then $\alpha =\alpha _{\mathscr {U}}$ for an $\mathscr {U}\in \mathscr {C}$ . By (T-s), $\Phi \in \mathscr {U}$ , thus $(\Phi , \mathscr {C})\in \alpha _{\mathscr {U}} = \alpha $ .

Lemma 5.5. Let $\mathsf {M.L}$ be a minimal modal logic and $\mathcal M = \langle \mathcal W, \leq , \mathbb {F}, \mathcal N, \mathcal V \rangle $ be the canonical neighbourhood model for $\mathsf {M.L}$ . Then for all $(\Phi , \mathscr {C})\in \mathcal W$ and all $A \in \mathcal L$ , $(\Phi , \mathscr {C})\Vdash A$ if and only if $A\in \Phi $ .

Proof. By induction on the construction of A. For the cases $A = p, \bot , B\land C, B \lor C, B \supset C$ the proof is exactly as the proof of Lemma 2.5. We consider the inductive cases $A = \Box B, \Diamond B$ .

• ( $A = \Box B$ ) Suppose that $\Box B\in \Phi $ . Then for all $(\Psi ,\mathscr {D})\geq (\Phi ,\mathscr {C})$ , $\Box B\in \Psi $ . By definition of segment, there is $\mathscr {U}\in \mathscr {D}$ such that for all $\Theta \in \mathscr {U}$ , $B\in \Theta $ . Then, by definition of canonical model, $\alpha _{\mathscr {U}}\in \mathcal N((\Psi , \mathscr {D}))$ , and by i.h., $(\Theta ,\mathscr {E})\Vdash B$ for all $(\Theta ,\mathscr {E})\in \alpha _{\mathscr {U}}$ . Therefore $(\Phi , \mathscr {C})\Vdash \Box B$ . Now suppose that $\Box B\notin \Phi $ . By Lemma 5.3 (ii), there is an $\mathsf {M.L}$ -segment $(\Phi , \mathscr {D})$ such that for all $\mathscr {U}\in \mathscr {D}$ , there is $\Psi \in \mathscr {U}$ such that $B\notin \Psi $ . By definition, $(\Phi , \mathscr {D})\in \mathcal W$ and $(\Phi , \mathscr {C}) \leq (\Phi , \mathscr {D})$ . Moreover, assume $\alpha \in \mathcal N((\Phi , \mathscr {D}))$ . Then $\alpha = \alpha _{\mathscr {U}}$ for some $\mathscr {U}\in \mathscr {D}$ . Thus, there is $\Psi \in \mathscr {U}$ such that $B\notin \Psi $ . By Lemma 5.3 (i), there is an $\mathsf {M.L}$ -segment $(\Psi , \mathscr {E})$ , thus by definition, $(\Psi , \mathscr {E})\in \alpha _{\mathscr {U}}$ , and by i.h., $(\Psi , \mathscr {E})\not \Vdash B$ . Hence $\alpha = \alpha _{\mathscr {U}} \not \Vdash ^{\forall } B$ , therefore $(\Phi , \mathscr {C})\not \Vdash \Box B$ .

• ( $A = \Diamond B$ ) Suppose that $\Diamond B\in \Phi $ . Then for all $(\Psi ,\mathscr {D})\geq (\Phi ,\mathscr {C})$ , $\Diamond B\in \Psi $ . By definition of segment, for all $\mathscr {U}\in \mathscr {D}$ , there is $\Psi \in \mathscr {U}$ such that $B\in \Psi $ . Now, assume $\alpha \in \mathcal N((\Psi ,\mathscr {D}))$ . By definition, $\alpha = \alpha _{\mathscr {U}}$ for some $\mathscr {U}\in \mathscr {D}$ . Then there is $\Psi \in \mathscr {U}$ such that $B\in \Psi $ . By Lemma 5.3 (i), there is an $\mathsf {M.L}$ -segment $(\Psi , \mathscr {E})$ , thus by definition, $(\Psi , \mathscr {E})\in \alpha _{\mathscr {U}}$ , and by i.h., $(\Psi , \mathscr {E})\Vdash B$ , which implies $\alpha = \alpha _{\mathscr {U}} \Vdash ^{\exists } B$ . Since this holds for every $\alpha \in \mathcal N((\Psi ,\mathscr {D}))$ , we have $(\Phi , \mathscr {C})\Vdash \Diamond B$ . Now suppose that $\Diamond B\notin \Phi $ . By Lemma 5.3 (iii), there is an $\mathsf {M.L}$ -segment $(\Phi , \mathscr {D})$ and a $\mathscr {U}\in \mathscr {D}$ such that for all $\Psi \in \mathscr {U}$ , $B\notin \Psi $ . By definition, $(\Phi , \mathscr {D})\in \mathcal W$ , $(\Phi , \mathscr {C}) \leq (\Phi , \mathscr {D})$ , and $\alpha _{\mathscr {U}}\in \mathcal N((\Phi , \mathscr {D}))$ . Moreover, for all $(\Psi , \mathscr {E})\in \alpha _{\mathscr {U}}$ , $B\notin \Psi $ , then by i.h., $(\Psi , \mathscr {E})\not \Vdash B$ . Hence, $\alpha _{\mathscr {U}} \not \Vdash ^{\exists } B$ , therefore $(\Phi , \mathscr {C})\not \Vdash \Diamond B$ .

As a consequence of these lemmas, we obtain the following completeness result (cf. proof of Theorem 2.6).

Theorem 5.6 (Completeness).

For all $A\in \mathcal L$ and all minimal modal logics $\mathsf {M.L}$ , if A is valid in every minimal neighbourhood model for $\mathsf {M.L}$ , then A is derivable in $\mathsf {M.L}$ .

Finally, on the basis of this semantic characterisation of logics $\mathsf {M.L}$ , we can show that, for each classical logic $\mathsf {L}$ , the fusion $\mathsf {S4}\oplus \mathsf {L}$ is the bimodal companion of the corresponding minimal logic $\mathsf {M.L}$ .

Theorem 5.7. For all $A\in \mathcal L$ and all classical modal logics $\mathsf {L}$ , A is derivable in $\mathsf {M.L}$ if and only if $A^{t}$ is derivable in $\mathsf {S4}\oplus \mathsf {L}$ .

Proof. ( $\Rightarrow $ ) Suppose that $\mathsf {S4}\oplus \mathsf {L}\not \vdash A^{t}$ . Then there are a model $\mathcal M = \langle \mathcal W, \mathcal R, \mathcal N, \mathcal V\rangle $ for $\mathsf {S4}\oplus \mathsf {L}$ and a world $w$ such that $\mathcal M, w\not \Vdash A^{t}$ . We define $\mathcal M' = \langle \mathcal W, \leq , \mathbb {F}, \mathcal N, \mathcal V'\rangle $ over the same $\mathcal W$ and $\mathcal N$ , where $\leq \ = \mathcal R$ , for all $p\in \mathit {Atm}$ , $\mathcal V'(p) = \{v \mid \text {for all } u, v\mathcal R u \text { implies } u\in \mathcal V(p)\}$ , and $\mathbb {F} = \{v \mid \text {for all } u, v\mathcal R u \text { implies } u\in \mathcal V(f)\}$ . By the properties of $\mathcal N$ in $\mathcal M$ , it immediately follows that $\mathcal M'$ is a minimal neighbourhood model for $\mathsf {M.L}$ . We show that for all $v\in \mathcal W$ and all $B\in \mathcal L$ , $\mathcal M', v \Vdash B$ if and only if $\mathcal M, v \Vdash B^{t}$ , which implies that $\mathcal M', w \not \Vdash A$ , therefore $\mathsf {M.L}\not \vdash A$ . The proof is by induction on the construction of B. The cases $B = p, \bot , C\land D, C\lor D, C\supset D$ are as in the proof of Theorem 2.7, case ( $\Rightarrow $ ). We show the cases $B = \Box C, \Diamond C$ .

• ( $B = \Box C$ ) $\mathcal M', v \Vdash \Box C$ iff for all $u\geq v$ , there is $\alpha \in \mathcal N(u)$ such that for all $z\in \alpha $ , $\mathcal M', z \Vdash C$ ; iff (by definition of $\leq $ and i.h.) for all u, if $v \mathcal R u$ , then there is $\alpha \in \mathcal N(u)$ such that for all $z\in \alpha $ , $\mathcal M, z \Vdash C^{t}$ ; iff for all u, if $v \mathcal R u$ , then $\mathcal M, u \Vdash \Box _2 C^{t}$ ; iff $\mathcal M, v \Vdash \Box _1\Box _2 C^{t}$ .

• ( $B = \Diamond C$ ) $\mathcal M', v \Vdash \Diamond C$ iff for all $u\geq v$ , for all $\alpha \in \mathcal N(u)$ , there is $z\in \alpha $ such that $\mathcal M', z \Vdash C$ ; iff (by definition of $\leq $ and i.h.) for all u, if $v \mathcal R u$ , then for all $\alpha \in \mathcal N(u)$ , there is $z\in \alpha $ such that $\mathcal M, z \Vdash C^{t}$ ; iff for all u, if $v \mathcal R u$ , then $\mathcal M, u \Vdash \Diamond _2 C^{t}$ ; iff $\mathcal M, v \Vdash \Box _1\Diamond _2 C^{t}$ .

( $\Leftarrow $ ) Suppose that $\mathsf {M.L}\not \vdash A$ . Then there are a minimal neighbourhood model $\mathcal M = \langle \mathcal W, \leq , \mathbb {F}, \mathcal N, \mathcal V\rangle $ for $\mathsf {M.L}$ and a world $w$ such that $\mathcal M, w\not \Vdash A$ . We define $\mathcal M" = \langle \mathcal W, \mathcal R, \mathcal N, \mathcal V"\rangle $ over the same $\mathcal W$ and $\mathcal N$ , where $\mathcal R = \leq $ , for all $p\in \mathit {Atm}$ , $\mathcal V"(p) = \mathcal V(p)$ , and $\mathcal V"(f) = \mathbb {F}$ . Then $\mathcal M"$ is a model for $\mathsf {S4}\oplus \mathsf {L}$ . We show that for all $v\in \mathcal W$ and all $B\in \mathcal L$ , $\mathcal M, v \Vdash B$ if and only if $\mathcal M", v \Vdash B^{t}$ , which implies that $\mathcal M", w \not \Vdash A$ , therefore $\mathsf {S4}\oplus \mathsf {L}\not \vdash A$ . The proof is by induction on the construction of B. The cases $B = p, \bot , C\land D, C\lor D, C\supset D$ are as in the proof of Theorem 2.7, case ( $\Leftarrow $ ). We show the cases $B = \Box C, \Diamond C$ .

• ( $B = \Box C$ ) $\mathcal M, v \Vdash \Box C$ iff for all $u\geq v$ , there is $\alpha \in \mathcal N(u)$ such that for all $z\in \alpha $ , $\mathcal M, z \Vdash C$ ; iff (by definition of $\mathcal R$ and i.h.) for all u, if $v \mathcal R u$ , then there is $\alpha \in \mathcal N(u)$ such that for all $z\in \alpha $ , $\mathcal M", z \Vdash C^{t}$ ; iff for all u, if $v \mathcal R u$ , then $\mathcal M", u \Vdash \Box _2 C^{t}$ ; iff $\mathcal M", v \Vdash \Box _1\Box _2 C^{t}$ .

• ( $B = \Diamond C$ ) $\mathcal M, v \Vdash \Diamond C$ iff for all $u\geq v$ , for all $\alpha \in \mathcal N(u)$ , there is $z\in \alpha $ such that $\mathcal M, z \Vdash C$ ; iff (by definition of $\mathcal R$ and i.h.) for all u, if $v \mathcal R u$ , then for all $\alpha \in \mathcal N(u)$ , there is $z\in \alpha $ such that $\mathcal M", z \Vdash C^{t}$ ; iff for all u, if $v \mathcal R u$ , then $\mathcal M", u \Vdash \Diamond _2 C^{t}$ ; iff $\mathcal M", v \Vdash \Box _1\Diamond _2 C^{t}$ .

As an additional remark, based on this semantics we can also show that the axioms ${P_{\Box }}$ , ${N_\Diamond }$ and ${C_\Diamond }$ are not valid in $\mathsf {M.MP}$ , $\mathsf {M.MN}$ and $\mathsf {M.MC}$ , respectively. For the first two axioms, consider a model $\mathcal M = \langle \mathcal W, \leq , \mathbb {F}, \mathcal N, \mathcal V \rangle $ where $\mathcal W = \{w, v\}$ , $\leq \ = \{(w,w), (v,v)\}$ , $\mathbb {F} = \{v\}$ and $\mathcal N(w) = \mathcal N(v) = \{\mathbb {F}\}$ . $\mathcal M$ satisfies both conditions (P) and (N). Moreover, since $\mathbb {F}\Vdash ^{\forall } \bot $ and $\mathbb {F}\Vdash ^{\exists } \bot $ , we have $w \Vdash \Box \bot $ and $w \Vdash \Diamond \bot $ . However, $w \not \Vdash \bot $ , therefore $\mathcal M\not \models \neg \Box \bot $ and $\mathcal M\not \models \neg \Diamond \bot $ . For ${C_\Diamond }$ , consider a model $\mathcal M$ where $\mathcal W = \{w, v, u\}$ , $\leq \ = \{(w,v), (w,u),(w,w), (v,v), (u,u)\}$ , $\mathcal V(p) = \{v\}$ , $\mathcal V(q) = \{u\}$ , $\mathcal N(w) = \mathcal N(v) = \{\{v\}\}$ and $\mathcal N(u) = \{\{u\}\}$ , that trivially satisfies (C). One can easily verify that $w \Vdash \Diamond (p \lor q)$ but $w \not \Vdash \Diamond p$ and $w \not \Vdash \Diamond q$ , therefore $\mathcal M \not \models \Diamond (p \lor q)\supset \Diamond p\lor \Diamond q$ .

5.2 Minimal modal logics via sequent calculi

Figure 8 Modal rules for classical sequent calculi $\mathsf {G1\text {-}{\mathsf {L}}}$ .

G1-style sequent calculi for the considered classical modal logics are defined extending $\mathsf {G1\text {-}{\mathsf {CPL}}}$ (Figure 1) with the following modal rules from Figure 8:

These calculi are studied and shown to be cut-free complete in [27, 35, 38, 46]. By applying the single-succedent restriction to the classical calculi $\mathsf {G1\text {-}{\mathsf {L}}}$ , we obtain the corresponding calculi $\mathsf {G1\text {-}{\mathsf {M.L}}}$ which extend $\mathsf {G1\text {-}{\mathsf {MPL}}}$ (Figure 1) with the following modal rules from Figure 9:

As before, the rules containing sequents with an empty succedent or with two active/principal formulas in the succedent are dropped (namely, $\mathsf {mnc}_{\mathsf {M}}^{c\ell }$ , $\mathsf {mem}_{\mathsf {M}}^{c\ell }$ , $\mathsf {mnc}_{\mathsf {C}}^{c\ell }$ , $\mathsf {mem}_{\mathsf {C}}^{c\ell }$ , $\mathsf N_{\Diamond }^{c\ell }$ , $\mathsf P_{\Box }^{c\ell }$ , $\mathsf D_{\Box }^{c\ell }$ and $\mathsf D_{\Diamond }^{c\ell }$ ), while the remaining rules preserve only one formula in the consequent of sequents (note in particular that the modal context $\Diamond \Pi $ is removed from $\mathsf {C}_{\Box }^m$ , $\mathsf {K}_{\Box }^m$ and $\mathsf {K}_{\Diamond }^m$ ). Observe also that the single-succedent restriction applied to $\mathsf C_{\Diamond }^{c\ell }$ and $\mathsf K_{\Diamond }^{c\ell }$ produces the same rule $\mathsf {K}_{\Diamond }^m$ . Finally, the calculi $\mathsf {G1\text {-}{\mathsf {M.MD}}}$ and $\mathsf {G1\text {-}{\mathsf {M.MND}}}$ contain the rule $\mathsf {P}_{\Diamond }^m$ that corresponds to the restriction of the rule $\mathsf D_{\Diamond }^{c\ell }$ in the particular case where $A = B$ ( $\mathsf P_{\Diamond }^{c\ell }$ is derivable in $\mathsf {G1\text {-}{MD}}$ and $\mathsf {G1\text {-}{MND}}$ from $\mathsf D_{\Diamond }^{c\ell }$ and $\mathsf {ctr}_{\mathsf {L}}^{cl}$ ).

Figure 9 Modal rules for minimal sequent calculi $\mathsf {G1\text {-}{\mathsf {M.L}}}$ .

We now show that the rule $\mathsf {cut}$ is admissible in the calculi $\mathsf {G1\text {-}{\mathsf {M.L}}}$ . As a consequence of this result, we prove that the calculi $\mathsf {G1\text {-}{\mathsf {M.L}}}$ are equivalent to the corresponding axiomatic systems $\mathsf {M.L}$ .

Theorem 5.8. For every calculus $\mathsf {G1\text {-}{\mathsf {M.L}}}$ , the rule $\mathsf {cut}$ is admissible in $\mathsf {G1\text {-}{\mathsf {M.L}}}$ .

Proof. The proof is in the appendix.

Theorem 5.9. For every calculus $\mathsf {G1\text {-}{\mathsf {M.L}}}$ , for all $A\in \mathcal L$ , A is derivable in $\mathsf {G1\text {-}{\mathsf {M.L}}}$ if and only if A is derivable in $\mathsf {M.L}$ .

Proof. ( $\Rightarrow $ ) For every modal rule $\mathit S_1, \ldots , \mathit S_n/\mathit S$ of $\mathsf {G1\text {-}{\mathsf {M.L}}}$ , we show that the rule $\iota (\mathit S_1), \ldots , \iota (\mathit S_n)/\iota (\mathit S)$ is derivable in $\mathsf {M.L}$ . ( $\mathsf {M}_{\Box }^m$ ) From $A \supset B$ , by ${mon_{\Box }}$ we get $\Box A \supset \Box B$ . ( $\mathsf {M}_{\Diamond }^m$ ) From $A \supset B$ , by ${mon_{\Diamond }}$ we get $\Diamond A \supset \Diamond B$ . ( $\mathsf {N}_{\Box }^m$ ) From A we get $\top \supset A$ , then by ${mon_{\Box }}$ , $\Box \top \supset \Box A$ , hence with $\Box \top $ we obtain $\Box A$ . ( $\mathsf {P}_{\Diamond }^m$ ) From A we get $\top \supset A$ , then by ${mon_{\Diamond }}$ , $\Diamond \top \supset \Diamond A$ , hence with $\Diamond \top $ we obtain $\Diamond A$ . ( $\mathsf {D}^m$ ) From $A \supset B$ , by ${mon_{\Box }}$ , $\Box A \supset \Box B$ , then with $\Box B \supset \Diamond B$ we get $\Box A \supset \Diamond B$ . ( $\mathsf {CD}^m$ ) Assume $\Sigma = A_1 \land \cdots \land A_n$ . Then from $A_1 \land \cdots \land A_n \supset B$ , by ${mon_{\Box }}$ we get $\Box (A_1 \land \cdots \land A_n) \supset \Box B$ . From ${C_{\Box }}$ we have $\Box A_1 \land \cdots \land \Box A_n \supset \Box (A_1 \land \cdots \land A_n)$ , then with $\Box B \supset \Diamond B$ we obtain $\Box A_1 \land \cdots \land \Box A_n \supset \Diamond B$ . ( $\mathsf {T}_{\Box }^m$ ) From $\bigwedge \Gamma \land A \supset B$ , with $\Box A \supset A$ we get $\bigwedge \Gamma \land \Box A \supset B$ . ( $\mathsf {T}_{\Diamond }^m$ ) From $\bigwedge \Gamma \supset A$ , with $A \supset \Diamond A$ we get $\bigwedge \Gamma \supset \Diamond A$ . For $\mathsf {C}_{\Box }^m$ , $\mathsf {K}_{\Box }^m$ and $\mathsf {K}_{\Diamond }^m$ see the derivations in Figure 3, replacing consecutive applications of ${nec}$ and ${K_{\Box }}$ with one application of ${mon_{\Box }}$ .

( $\Leftarrow $ ) For the other direction, it is easy to see that the modal axioms and rules of $\mathsf {M.L}$ are derivable, respectively admissible, in $\mathsf {G1\text {-}{\mathsf {M.L}}}$ . We show as examples the derivations of ${C_{\Box }}$ and ${mon_{\Box }}$ .

6 Constructive modal logics

On the basis of the relations between $\mathsf {M.K}$ and $\mathsf {C.K}$ observed in Section 4, we now define a constructive counterpart for each logic $\mathsf {M.L}$ . First, the constructive modal logics $\mathsf {C.L}$ are defined extending $\mathsf {M.L}$ with ex falso quodlibet $\bot \supset A$ .

Definition 6.1 (Constructive modal logics).

For every minimal modal logic $\mathsf {M.L}$ , the corresponding constructive modal logic $\mathsf {C.L}$ is defined as $\mathsf {M.L} + \bot \supset A$ .

We show that each logic $\mathsf {C.L}$ is semantically characterised by neighbourhood models obtained by suitably restricting the minimal neighbourhood models for the corresponding system $\mathsf {M.L}$ . The restriction is analogous to the one of Definition 4.1, with the difference that the neighbourhood function is now involved.

Definition 6.2 (Constructive neighbourhood semantics).

For every constructive modal logic $\mathsf {C.L}$ , a constructive neighbourhood model for $\mathsf {C.L}$ is any minimal neighbourhood model $\mathcal M = \langle \mathcal W, \leq , \mathbb {F}, \mathcal N, \mathcal V\rangle $ for the corresponding minimal logic $\mathsf {M.L}$ such that the following hold for all $w\in \mathbb {F}$ :

1. (i) $w\in \mathcal V(p)$ for all $p\in \mathit {Atm}$ ;

2. (ii) there is $\alpha \in \mathcal N(w)$ such that $\alpha \subseteq \mathbb {F}$ ;

3. (iii) for all $\alpha \in \mathcal N(w)$ , $\alpha \cap \mathbb {F}\not =\emptyset $ .

Theorem 6.1. For all $A\in \mathcal L$ and all constructive modal logics $\mathsf {C.L}$ , A is valid in all constructive neighbourhood models for $\mathsf {C.L}$ if and only if A is derivable in $\mathsf {C.L}$ .

Proof. ( $\Rightarrow $ ) The proof extends the one of Theorem 5.2 by showing that $\bot \supset A$ is valid in every constructive neighbourhood model. Suppose that $w \Vdash \bot $ . We show by construction on A that $w \Vdash A$ , considering only the cases $A = \Box B, \Diamond B$ (see the proof of Theorem 4.1 for $A = p, \bot , B\land C, B\lor C, B\supset C$ ). ( $A = \Box B$ ) Suppose $w \leq v$ . Since $\mathbb {F}$ is $\leq $ -upward closed, $v \in \mathbb {F}$ . Then by Definition 6.2, item (ii), there is $\alpha \in \mathcal N(v)$ such that $\alpha \subseteq \mathbb {F}$ . Hence $\alpha \Vdash ^{\forall }\bot $ , and by i.h., $\alpha \Vdash ^{\forall } B$ . Therefore $w \Vdash \Box B$ . ( $A = \Diamond B$ ) Suppose $w \leq v$ . Since $\mathbb {F}$ is $\leq $ -upward closed, $v \in \mathbb {F}$ . Then by Definition 6.2, item (iii), for all $\alpha \in \mathcal N(v)$ , $\alpha \cap \mathbb {F}\not =\emptyset $ . Hence for all $\alpha \in \mathcal N(v)$ , $\alpha \Vdash ^{\exists }\bot $ , then by i.h., $\alpha \Vdash ^{\exists } B$ . Therefore $w \Vdash \Diamond B$ .

( $\Leftarrow $ ) The proof extends the completeness proof of minimal modal logics by showing that the canonical neighbourhood model for $\mathsf {C.L}$ (Definition 5.4) satisfies the conditions (i), (ii), (iii) in Definition 6.2. Suppose that $(\Phi , \mathscr {C})\in \mathbb {F}$ . Then $\bot \in \Phi $ . Since $\Phi $ is closed under derivation, by ex falso quodlibet we obtain $\Phi = \mathcal L$ , which entails the following. (i) For all $p\in \mathit {Atm}$ , $p\in \Phi $ , hence by definition, $(\Phi , \mathscr {C})\in \mathcal V(p)$ . (ii) $\Box \bot \in \Phi $ , hence by Definition 5.3, there is $\mathscr {U}\in \mathscr {C}$ such that for all $\Psi \in \mathscr {U}$ , $\bot \in \Psi $ . Then by Definition 5.4, there is $\alpha _{\mathscr {U}}\in \mathcal N((\Phi , \mathscr {C}))$ such that for all $(\Psi , \mathscr {D})\in \alpha _{\mathscr {U}}$ , $\bot \in \Psi $ . Then for all $(\Psi , \mathscr {D})\in \alpha _{\mathscr {U}}$ , $(\Psi , \mathscr {D})\in \mathbb {F}$ , thus $\alpha _{\mathscr {U}}\subseteq \mathbb {F}$ . (iii) $\Diamond \bot \in \Phi $ , hence by Definition 5.3, for all $\mathscr {U}\in \mathscr {C}$ , there is $\Psi \in \mathscr {U}$ such that $\bot \in \Psi $ . Then by Definition 5.4 and Lemma 5.3 (which holds for $\mathsf {C.L}$ -full segments as well), for all $\alpha _{\mathscr {U}}\in \mathcal N((\Phi , \mathscr {C}))$ , there is $(\Psi , \mathscr {D})\in \alpha _{\mathscr {U}}$ such that $\bot \in \Psi $ , hence $(\Psi , \mathscr {D})\in \mathbb {F}$ , thus $\alpha _{\mathscr {U}}\cap \mathbb {F}\not =\emptyset $ .

Now, we show that for each logic $\mathsf {C.L}$ , a sequent calculus $\mathsf {G1\text {-}{\mathsf {C.L}}}$ can be obtained by extending $\mathsf {G1\text {-}{\mathsf {IPL}}}$ with the modal rules of the corresponding calculus $\mathsf {G1\text {-}{\mathsf {M.L}}}$ .

Definition 6.3. For every logic $\mathsf {C.L}$ , the sequent calculus $\mathsf {G1\text {-}{\mathsf {C.L}}}$ contains the rules of $\mathsf {G1\text {-}{\mathsf {IPL}}}$ (Figure 5) plus the modal rules of the corresponding minimal calculus $\mathsf {G1\text {-}{\mathsf {M.L}}}$ , except for $\mathsf {T}_{\Box }^m$ which is replaced by its intuitionistic version $\mathsf {T}_{\Box }^i$ , with $0 \leq |\delta | \leq 1$ (that is, with the succedent containing at most one formula):

Differently from the other modal rules, $\mathsf T_{\Box }^{c\ell }$ and $\mathsf T_{\Diamond }^{c\ell }$ are local and must therefore be treated like the propositional rules. Since $\mathsf T_{\Diamond }^{c\ell }$ has a principal formula in the succedent which is preserved by both kinds of sequent restrictions, this only impacts on $\mathsf {T}_{\Box }^i$ that requires an intuitionistic succedent containing zero or one formula.

Theorem 6.2. For every calculus $\mathsf {G1\text {-}{\mathsf {C.L}}}$ , the rule $\mathsf {cut}$ is admissible in $\mathsf {G1\text {-}{\mathsf {C.L}}}$ .

Proof. The proof is in the appendix.

Theorem 6.3. For every calculus $\mathsf {G1\text {-}{\mathsf {C.L}}}$ , for all $A\in \mathcal L$ , A is derivable in $\mathsf {G1\text {-}{\mathsf {C.L}}}$ if and only if A is derivable in $\mathsf {C.L}$ .

Proof. The derivations of the intuitionistic axioms and sequent rules are standard. For the derivations of the modal axioms and sequent rules we refer to the proof of Theorem 5.9.

7 Discussion and future work

7.1 A framework of minimal and constructive modal logics

The aim of this paper was to provide a uniform characterisation of constructive modal logics. Our approach went through the definition of a family of minimal modal logics obtained from their classical counterparts (1) by means of a reduction into fusions of classical modal logics via the extended Gödel–Johansson translation, (2) by restricting G1 sequent calculi for classical modal logics to single-succedent sequents. We have seen that the resulting minimal counterpart of $\mathsf {K}$ is strictly connected with the constructive modal logic $\mathsf {C.K}$ studied in the literature, as the two systems validate the same modal principles. Moreover, we have seen that $\mathsf {C.K}$ can be obtained from $\mathsf {M.K}$ (1) axiomatically, by extending $\mathsf {M.K}$ with ex falso quodlibet $\bot \supset A$ ; (2) semantically, by adding suitable conditions on the set of fallible worlds; (3) based on the sequent calculi, by adding the minimal modal rules to an intuitionistic sequent calculus. By extending these relations to the other minimal systems, we have defined a constructive analog for each minimal system, obtaining a corresponding family of constructive modal logics. This family contains the logics $\mathsf {C.K}$ , $\mathsf {C.KD}$ and $\mathsf {C.KT}$ which are the constructive counterparts of $\mathsf {K}$ , $\mathsf {KD}$ and $\mathsf {KT}$ already studied in the literature. In particular, the same axiomatisations were defined in [2, 43] ( $\mathsf {C.KT}$ also coincides with the propositional fragment of Fitch’s first-order intuitionistic modal logic [18]), moreover our sequent calculi $\mathsf {G1\text {-}{\mathsf {C.K}}}$ , $\mathsf {G1\text {-}{\mathsf {C.KD}}}$ , $\mathsf {G1\text {-}{\mathsf {C.KT}}}$ coincide with those of [6, 34, 36]. The remaining minimal and constructive logics are new. All in all, this work organises pre-existing constructive modal logics into a uniform framework and also extends this family with constructive counterparts of some non-normal modal logics, providing for each of them corresponding semantics and sequent calculi. At the same time, our approach offers an alternative view on constructive modal logics with respect to axiomatic systems; we observe in particular that a constructive modal logic does not necessarily contain both the $\Box $ - and the $\Diamond $ -version of the characteristic axioms of its classical counterpart, as one could expect when starting from the axiomatisation.

7.2 Comparison with Wijesekera-style constructive modal logics

Wijesekera’s logic $\mathsf {W.K}$ , often presented as $\mathsf {C.K}$ + ${N_\Diamond }$ , was defined (in a first-order formulation) and provided with a birelational semantics and a sequent calculus in [65]. Interestingly, the same models can be obtained from the birelational models for $\mathsf {C.K}$ (Definition 4.1) by dropping the fallible worlds, and the sequent calculus amounts to the restriction of $\mathsf {G1\text {-}{\mathsf {K}}}$ (Figure 2) to sequents with at most one formula in the succedent (vs. restricting to exactly one formula in the succedent, that provides $\mathsf {G1\text {-}{\mathsf {C.K}}}$ , cf. also [13]). We can now observe that analogous relations hold for all constructive modal logics $\mathsf {C.L}$ studied in this paper, considering the corresponding Wijesekera-style logic $\mathsf {W.L}$ defined in [11]. Above all, we point out that each system $\mathsf {W.L}$ (thus also $\mathsf {W.K}$ ) can be obtained extending the corresponding system $\mathsf {C.L}$ with the axiom $\neg (\Box A \land \Diamond \neg A)$ , which expresses one direction of the duality principle. This shows that the difference between $\mathsf {C.K}$ and $\mathsf {W.K}$ does not rely that much on a stronger $\Diamond $ of the latter, but rather on a different interaction of $\Box $ and $\Diamond $ in the two systems. In particular, the modalities in $\mathsf {C.L}$ systems are barely connected. The relations between minimal, constructive and Wijesekera-style modal logics are summarised in Figure 10.

Figure 10 Axiomatic, semantical and proof-theoretical relations between minimal and constructive modal logics.

7.3 Simpson’s requirements

Simpson [59] listed some requirements that are now a standard to evaluate whether an intuitionistic modal logic $\mathsf {I.L}$ can be understood as an intuitionistic counterpart of a classical modal logic $\mathsf {L}$ , among which we have that $\mathsf {I.L}$ should be a conservative extension of $\mathsf {IPL}$ , it should contain all axioms of $\mathsf {IPL}$ (over the whole language $\mathcal L$ ) and be closed under modus ponens, it should satisfy the disjunction property (if $A \lor B$ is derivable, then A is derivable or B is derivable), the modalities in $\mathsf {I.L}$ should be independent, the extension of $\mathsf {I.L}$ with $A \lor \neg A$ should coincide with $\mathsf {L}$ . It looks natural to adapt these requirements to pairs of minimal and constructive/intuitionistic modal logics. It is easy to verify that each logic $\mathsf {M.L}$ is a conservative extension of $\mathsf {MPL}$ , contains all axioms of $\mathsf {MPL}$ and modus ponens, satisfies the disjunction property and has independent modalities. Moreover, the extension of $\mathsf {M.L}$ with $\bot \supset A$ coincides with the corresponding logic $\mathsf {C.L}$ . In this sense, each pair of corresponding logics $\mathsf {M.L}$ and $\mathsf {C.L}$ constitutes a Simpsonian pair of modal logics.

7.4 Computational properties

In this work, we have not considered the computational properties of the logics $\mathsf {M.L}$ and $\mathsf {C.L}$ . However, we can observe that the equation ( $\ast $ ) is not only a definitorial property of the logics $\mathsf {M.L}$ , it is also a polynomial reduction of the derivability problem for $\mathsf {M.L}$ into the derivability problem for $\mathsf {S4}\oplus \mathsf {L}$ . Considering that the derivability problems for $\mathsf {S4}\oplus \mathsf {K}$ , $\mathsf {S4}\oplus \mathsf {KD}$ and $\mathsf {S4}\oplus \mathsf {KT}$ are known to be PSpace-complete [20], and that $\mathsf {M.K}$ is a conservative extension of $\mathsf {MPL}$ (with respect to the fragment of the language without the modalities) which is also PSpace-complete, we can conclude that the derivability problems for $\mathsf {M.K}$ , $\mathsf {M.KD}$ and $\mathsf {M.KT}$ are PSpace-complete. We conjecture that the same complexity bound applies to all logics $\mathsf {M.L}$ and $\mathsf {C.L}$ . In future work, we would like to address this problem by studying terminating sequent calculi and construction of finite models in the style of [12]. Moreover, we conjecture that PSpace-complexity can be proved for $\mathsf {M.M}$ by combining the translation t with the reduction of classical $\mathsf {M}$ into multi-modal $\mathsf {K}$ presented in [22, 32]. We would also like to study reductions for the constructive logics $\mathsf {C.L}$ along the lines of [16].

7.5 Scalability and limitations of our approach

We have restricted our analysis to modal logics characterised by non-iterative axioms, namely, axioms without modal operators occurring within the scope of other modal operators. The reason is technical: models generated by modal logic fusions do not contain any interaction between the different relations (or between relations and neighbourhood functions). On the other hand, some interaction is needed in order to validate basic iterative axioms such as ${4_{\Box }}$ $\Box A \supset \Box \Box A$ , as it is witnessed by the birelational semantics for $\mathsf {C.S4}$ of [1] based on a confluence property of the form $w\mathcal R v \ \& \ v \leq u \Rightarrow \exists z(w \leq z \ \& \ z\mathcal R u)$ . Interestingly, it can be proved that this property is satisfied by the canonical birelational model for $\mathsf {C.K}$ (Definition 2.8), which implies that this property is admissible in its birelational semantics. This remark suggests the possible applicability of our canonical model construction to minimal and constructive logics with iterative axioms. However, it not obvious to establish whether this is actually the case, and it is left to future work.

Concerning instead our proof-theoretical approach, we have considered pure label-free, Gentzen-style sequent calculi because of their simplicity and the fact that constructive modal logics allow for this kind of calculi (against intuitionistic modal logics, for which they seem not possible). Similar single-succedent restrictions of labels and nested sequent calculi have generated proof systems for intuitionistic modal logics [59, 60]. It would be interesting to apply a similar strategy starting from alternative types of calculi, such as hypersequent or 2-sequent calculi, in order to obtain constructive correspondents of additional classical logics as well as to better understand the relations between the properties of a calculus and the logics resulting from its restrictions.