1 Introduction
Both intuitionistic modal logic and nonnormal modal logic have been studied for a long time. The study of modalities with an intuitionistic basis goes back to Fitch in the late 40s (Fitch [7]) and has led to an important stream of research. We can very schematically identify two traditions: socalled Intuitionistic modal logics versus Constructive modal logics. Intuitionistic modal logics have been systematised by Simpson [23], whose main goal is to define an analogous of classical modalities justified from an intuitionistic point of view. On the other hand, constructive modal logics are mainly motivated by their applications to computer science, such as the typetheoretic interpretations (Curry–Howard correspondence, typed lambda calculi), verification and knowledge representation,^{2}^{2}2For a recent survey see Stewart et al. [25] and references therein. but also by their mathematical semantics (Goldblatt [11]).
On the other hand, nonnormal modal logics have been strongly motivated on a philosophical and epistemic ground. They are called “nonnormal” as they do not satisfy all the axioms and rules of the minimal normal modal logic K. They have been studied since the seminal works of Scott, Lemmon, and Chellas ([22], [2], see Pacuit [21] for a survey), and can be seen as generalisations of standard modal logics. They have found an interest in several areas such as epistemic and deontic reasoning, reasoning about games, and reasoning about probabilistic notions such as “truth in most of the cases”.
Although the two areas have grown up seemingly without any interaction, it can be noticed that some intuitionistic or constructive modal logics investigated in the literature contain nonnormal modalities. The prominent example is the logic CCDL proposed by Wijesekera [27], whose propositional fragment (that we call CCDL) has been recently investigated by Kojima [13]. This logic has a normal modality and a nonnormal modality, where does not distribute over the , that is
is not valid. The original motivation by Wijesekera comes from Constructive Concurrent Dynamic Logic, but the logic has also an interesting epistemic interpretation in terms of internal/external observers proposed by Kojima. A related system is Constructive K (CK), that has been proposed by Bellin et al. [1] and further investigated by Mendler and de Paiva [19], Mendler and Scheele [20]. This system not only rejects , but also its nullary version (). In contrast all these systems assume a normal interpretation of so that
is always assumed. A further example is Propositional Lax Logic (PLL) by Fairtlough and Mendler [4], an intuitionistic monomodal logic for hardware verification where the modality does not validate the rule of necessitation.
Finally, all intuitionistic modal logics reject the interdefinability of the two operators:
and its boolean equivalents.
To the best of our knowledge, no systematic investigation of nonnormal modalities with an intuitionistic base has been carried out so far. Our aim is to lay down a general framework which can accommodate in a uniform way intuitionistic counterparts of the classical cube of nonnormal modal logics, as well as CCDL and CK mentioned above. As we shall see, the adoption of an intuitionistic base leads to a finer analysis of nonnormal modalities than in the classical case. In addition to the motivations for classical nonnormal modal logics briefly recalled above, an intutionistic interpretation of nonnormal modalities may be justified by more specific interpretations, of which we mention two:

The deontic interpretation: The standard interpretation of deontic operators (Obligatory), (Permitted) is normal: but it has been known for a long time that the normal interpretation is problematic when dealing for instance with “Contrary to duty obligations".^{3}^{3}3For a survey on puzzles related to a normal interpretation of the deontic modalities see McNamara [18]. One solution is to adopt a nonnormal interpretation, rejecting in particular the monotonicity principle (from is valid infer ). Moreover, a constructive reading of the deontic modalities would further reject their interdefinability: one may require that the permission of must be justified explicitly or positively (say by a proof from a corpus of norms) and not just established by the fact that is not obligatory (see for instance the distinction between weak and strong permissions in von Wright [31]).

The contextual interpretation: A contextual reading of the modal operators is proposed in Mendler and de Paiva [19]. In this interpretation is read as “ holds in all contexts” and as “A holds in some context”. This interpretation invalidates , while retaining the distribution of over conjunction (). But this contextual interpretation is not the only possible one. We can interpret as is “justified” (proved) in some context , no matter what is meant by a context (for instance a knowledge base), and as is “compatible” (consistent) with every context. With this interpretation both operators would be nonnormal as they would satisfy neither , nor .
As we said, our aim is to provide a general framework for nonnormal modal logics with an intuitionistic base. However, in order to identify and restrain the family of logics of interest, we adopt some criteria, which partially coincide with Simpson’s requirements (Simpson [23]):

The modal logics should be conservative extensions of IPL.

The disjunction property must hold.

The two modalities should not be interdefinable.

We do not consider systems containing the controversial .
Our starting point is the study of monomodal systems, which extend IPL with either or , but not both. We consider the monomodal logics corresponding to the classical cube generated by the weakest logic E extended with conditions M, N, C (with the exception of ). We give an axiomatic characterisation of these logics and equivalent cutfree sequent systems similar to the one by Lavendhomme and Lucas [15] for the classical case.
Our main interest is however in logics which contain both and , and allow some form of interaction between the two. Their interaction is always weaker than interdefinability. In order to define logical systems we take a prooftheoretical perspective: the existence of a simple cutfree system, as in the monomodal case, is our criteria to identify meaningful systems. A system is retained if the combination of sequent rules amounts to a cutfree system.
It turns out that one can distinguish three degrees of interaction between and , that are determined by answering to the question, for any two formulas and :
under what conditions and are jointly inconsistent?
Since there are three degrees of interaction, even the weakest classical logic E has three intuitionistic counterparts of increasing strength. When combined with M, N, C properties of the classical cube, we end up with a family of 24 distinct systems, all enjoying a cutfree calculus and, as we prove, an equivalent Hilbert axiomatisation. This shows that intuitionistic nonnormal modal logic allows for finer distinctions whence a richer theory than in the classical case.
The existence of a cutfree calculus for each of the logics has some important consequences: We can prove that all systems are indeed distinct, that all of them are “good” extensions of intuitionistic logic, and more importantly that all of them are decidable.
We then tackle the problem of giving a semantic characterisation of this family of logics. The natural setting is to consider an intuitionistic version of neighbourhood models for classical logics. Since we want to deal with the language containing both and , we consider neighbourhood models containing two distinct neighbourhood functions and . As in standard intuitionistic models, they also contain a partial order on worlds. Different forms of interaction between the two modal operators correspond to different (but natural) conditions relating the two neighbourhood functions. By considering further closure conditions of neighbourhoods, analogous to the classical case, we can show that this semantic characterises modularly the full family of logics. Moreover we prove, through a filtration argument, that most of the logics have the finite model property, thereby obtaining a semantic proof of their decidability.
It is worth noticing that in the (easier) case of intuitionistic monomodal logic with only a similar semantics and a matching completeness theorem have been given by Goldblatt [11]. More recently, Goldblatt’s semantics for the intuitionistic version of system E has been reformulated and extended to axiom T by Witczak [29].
But our neighbourhood models have a wider application than the characterisation of the family of logics mentioned above. We show that adding suitable interaction conditions between and we can capture CCDL as well as CK. We show this fact first directly by proving that both CCDL and CK are sound and complete with respect to our models satisfying an additional condition. We then prove the same result by relying on some preexisting semantics of these two logics and by transforming models. In case of CCDL, there exists already a characterisation of it in terms of neighbourhood models, given by Kojima [13], although the type of models is different, in particular Kojima’s models contain only one neighbourhood function.
The case of CK is more complicated, whence more interesting: this logic is characterised by a relational semantics defined in terms of Kripke models of a peculiar nature: they contain “fallible” worlds, i.e. worlds which force . We are able to show directly that relational models can be transformed into our neighbourhood models satisfying a specific interaction condition and vice versa.
All in all, we get that the wellknown CK can be characterised by neighbourhood models, after all rather standard structures, alternative to nonstandard Kripke models with fallible worlds. This fact provides further evidence in favour of our neighbourhood semantics as a versatile tool to analyse intuitionistic nonnormal modal logics.
2 Classical nonnormal modal logics
2.1 Hilbert systems
Classical nonnormal modal logics are defined on a propositional modal language based on a set of countably many propositional variables. Formulas are given by the following grammar, where ranges over :
.
We use as metavariables for formulas of . , and are abbreviations for, respectively, , and . We take both modal operators and as primitive (as well as all boolean connectives), as it will be convenient for the intuitionistic case. Their duality in classical modal logics is recovered by adding to any system one of the duality axioms or (Figure 1), which are equivalent in the classical setting.
The weakest classical nonnormal modal logic E is defined in language by extending classical propositional logic (CPL) with a duality axiom and rule , and it can be extended further by adding any combination of axioms , and . We obtain in this way eight distinct systems (Figure 2), which compose the family of classical nonnormal modal logics.
Equivalent axiomatisations for these systems are given by considering the modal axioms in the righthand column of Figure 1(). Thus, logic E could be defined by extending CPL with axiom and rule , and its extensions are given by adding combinations of axioms , and .
It is worth recalling that axioms , and are syntactically equivalent with the rules , and , respectively, and that axiom is derivable from and . As a consequence, we have that the top system MCN is equivalent to the weakest classical normal modal logic K.
2.2 Neighbourhood semantics
The standard semantics for classical nonnormal modal logics is based on the socalled neighbourhood (or minimal, or ScottMontague) models.
Definition 2.1.
A neighbourhood model is a triple , where is a nonempty set, is a neighbourhood function , and is a valuation function . A neighbourhood model is supplemented, closed under intersection, or contains the unit, if satisfies the following properties:
If and , then  (Supplementation); 

If , then  (Closure under intersection); 
for all  (Containing the unit). 
The forcing relation is defined inductively as follows:
iff  ;  
;  
iff  and ;  
iff  or ;  
iff  implies ;  
iff  ;  
iff  ; 
where denotes the set , called the truth set of .
We can also recall that in the supplemented case, the forcing conditions for modal formulas are equivalent to the following ones:
iff  there is s.t. ;  
iff  for all , . 
Theorem 2.1 (Chellas [2]).
Logic E(M,C,N) is sound and complete with respect to neighbourhood models (which in addition are supplemented, closed under intersection and contain the unit).
3 Intuitionistic nonnormal monomodal logics
Our definition of intuitionistic nonnormal modal logics begins with monomodal logics, that is logics containing only one modality, either or . We first define the axiomatic systems, and then present their sequent calculi.
Under “intuitionistic modal logics” we understand any modal logic L that extends intuitionistic propositional logic (IPL) and satisfies the following requirements:

L is conservative over IPL: its nonmodal fragment coincides with IPL.

L satisfies the disjunction property: if is derivable, then at least one formula between and is also derivable.
3.1 Hilbert systems
From the point of view of axiomatic systems, two different classes of intuitionistic nonnormal monomodal logics can be defined by analogy with the definition of classical nonnormal modal logics (cf. Section 2). Intuitionistic modal logics are modal extensions of IPL, for which we consider the following axiomatisation:
    

    
    
  efq  
  mp 
We define over IPL two families of intuitionistic nonnormal monomodal logics, that depend on the considered modal operator, and are called therefore the  and the family. The family is defined in language by adding to IPL the rule and any combination of axioms , and . The family is instead defined in language by adding to IPL the rule and any combination of axioms and . It is work remarking that we don’t consider intuitionistic nonnormal modal logics containing axiom . We denote the resulting logics by, respectively, IE and IE, where E replaces any system of the classical cube (for logics, any system non containing ).
Notice that, having rejected the definability of the lacking modality,  and logics are distinct, as and behave differently. Moreover, as a consequence of the fact that the systems in the classical cube are pairwise nonequivalent, we have that the family contains eight distinct logics, while the family contains four distinct logics (something not derivable in a classical system is clearly not derivable in the corresponding intuitionistic system). It is also worth noticing that, as it happens in the classical case, axioms , and are interderivable, respectively, with rules , and , and that is derivable from and (as the standard derivations are intuitionistically valid).
3.2 Sequent calculi
We now present sequent calculi for intuitionistic nonnormal monomodal logics. The calculi are defined as modal extensions of a given sequent calculus for IPL. We take as base calculus (Figure 4), and extend it with suitable combinations of the modal rules in Figure 5. The rules can be compared with the rules given in Lavendhomme and Lucas [15], where sequent calculi for classical nonnormal modal logics are presented. However, our rules are slightly different as (i) they have a single formula in the righthand side of sequents; and (ii) contexts are added to the lefthand side of sequents appearing in the conclusion. Restriction (i) is adopted in order to have singlesuccedent calculi (as is), while with (ii) we implicitly embed weakening in the application of the modal rules. We consider the sequent calculi to be defined by the modal rules that are added to . The calculi are the following.
G.IE  :=  G.IEC  :=  
G.IM  :=  G.IMC  :=  
G.IEN  :=  +  G.IECN  :=  + 
G.IMN  :=  +  G.IMCN  :=  + 
G.IE  :=  
G.IM  :=  
G.IEN  :=  +  
G.IMN  :=  + 
Notice that  as in Lavendhomme and Lucas [15]  axiom doesn’t have a corresponding sequent rule, but it is captured by modifying the rules and . In particular, these rules are replaced by and , respectively, that are the generalisations of and with principal formulas (instead of just one) in the lefthand side of sequents. Observe that and are nonstandard, as they introduce an arbitrary number of modal formulas with a single application, and that has in addition an arbitrary number of premisses. An other way to look at and is to see them as infinite sets of rules, each set containing a standard rule for any . Under the latter interpretation the calculi are anyway nonstandard as they are defined by infinite sets of rules.
We now prove the admissibility of some structural rules, and then show the equivalence between the sequent calculi and the Hilbert systems.
Proposition 3.1.
The following weakening and contraction rules are heightpreserving admissible in any monomodal calculus:
.
Proof.
By induction on , we show that whenever the premiss of an application of , or has a derivation of height , then its conclusion has a derivation of the same height. As usual, the proof considers the last rule applied in the derivation of the premiss (when the premiss is not an initial sequent). For rules of the proof is standard. For modal rules, left and right weakening are easily handled. For istance, the premiss of is necessarily derived by . Then contains a formula that is principal in the application of , which in turn has as premiss. By a different application of to we can derive for any .
The proof is also immediate for contraction, where the most interesting case is possibly when both occurrences of in the premiss of are principal in the last rule applied in its derivation. In this case, the last rule is either or . If it is , then for some , and the sequent is derived from for some in . By i.h. we can apply to the last sequent and obtain , and then by derive sequent , which is the conclusion of (the proof is analogous for ). ∎
We now show that the cut rule
is admissible in any monomodal calculus. The proof is based on the following notion of weight of formulas:
Definition 3.1 (Weight of formulas).
Function assigning to each formula its weight is defined as follows: ; ; for ; and .
Observe that, given the present definition, has a smaller weight than and . Although irrelevant to the next theorem, this will be used in Section 4 for the proof of cut elimination in bimodal calculi.
Theorem 3.2.
Rule is admissible in any monomodal calculus.
Proof.
Given a derivation of a sequent with some applications of , we show how to remove any such application and obtain a derivation of the same sequent without . The proof is by double induction, with primary induction on the weight of the cut formula and subinduction on the cut height. We recall that, for any application of , the cut formula is the formula which is deleted by that application, while the cut height is the sum of the heights of the derivations of the premisses of .
We just consider the cases in which the cut formula is principal in the last rule applied in the derivation of both premisses of . Moreover, we treat explicitly only the cases in which both premisses are derived by modal rules, as the nonmodal cases are already considered in the proof of cut admissibility for , and because modal and nonmodal rules don’t interact in any relevant way.
(; ). Let and . We have the following situation:
… …
The proof is converted as follows, with several applications of with as cut formula, hence with a cut formula of smaller weight. First we derive
Then for any , we derive
Finally we can apply as follows
… …
(; ) is analogous to (; ). (; ) and (; ) are the particular cases where .
(; ). Let . The situation is as follows:
…
The proof is converted as follows, with an application of on a cut formula of smaller weight.
…
(; ) is analogous to (; ). (; ) and (; ) are the particular cases where .
(; ) and (; ) are analogous to (; ) and (; ), respectively.
(; ). We have
which become
(; ) is analogous to (; ). ∎
As a consequence of the admissibility of we obtain the equivalence between the sequent calculi and the axiomatic systems.
Proposition 3.3.
Let L be any intuitionistic nonnormal monomodal logic. Then calculus G.L is equivalent to system L.
Proof.
The axioms and rules of L are derivable in G.L. For the axioms of IPL and mp we can consider their derivations in , as G.L enjoys admissibility of . Here we show that any modal rule allows us to derive the corresponding axiom:
Moreover, the rules of G.L are derivable in L. As before, it suffices to consider the modal rules. The derivations are in most cases straightforward, we just consider the following.
If L contains , then is derivable. Assume . Then by (which is equivalent to ), .
If L contains , then is derivable. Assume . Since , by , . Then , and, since , we have .
If L contains , then is derivable. Assume and for all . Then . By , . In addition, by several applications of , . Therefore . ∎
4 Intuitionistic nonnormal bimodal logics
In this section we present intuitionistic nonnormal modal logics with both and . In this case we first present their sequent calculi, and then give equivalent axiomatisations.
A simple way to define intuitionistic nonnormal bimodal logics would be by considering the fusion of two monomodal logics that belong respectively to the  and to the family. Given two logics IE and IE, their fusion in language is the smallest bimodal logics containing IE and IE (for the sake of simplicity we can assume that and share the same set of propositional variables, and differ only with respect to and ). The resulting logic is axiomatised simply by adding to IPL the modal axioms and rules of IE, plus the modal axioms and rules of IE.
It is clear, however, that in the resulting systems the modalities don’t interact at all, as there is no axiom involving both and . On the contrary, finding suitable interactions between the modalities is often the main issue when intuitionistic bimodal logics are concerned. In that case, by reflecting the fact that in IPL connectives are not interderivable, it is usually required that and are not dual. We take the lacking of duality as an additional requirement for the definition of intuitionistic nonnormal bimodal logics:

and are not interdefinable.
In order to define intuitionistic nonnormal bimodal logics by the axiomatic systems, we would need to select the axioms between a plethora of possible formulas satisfying (R3). If we look for instance at the literature on intuitionistic normal modal logics, we see that many different axioms have been considered, and the reasons for the specific choices are varied. We take therefore a different way, and define the logics starting with their sequent calculi. In particular we proceed as follows.

Intuitionistic nonnormal bimodal logics are defined by their sequent calculi. The calculi are conservative extensions of a given calculus for IPL, and have as modal rules some characteristic rules of intuitionistic nonnormal monomodal logics, plus some rules connecting and . In addition, we require that the rule is admissible. As usual, this means that adding rule to the calculus does not extend the set of derivable sequents.

To the purpose of defining the basic systems, we consider only interactions between and that can be seen as forms of “weak duality principles”. In order to satisfy (R3), we require that these interactions are strictly weaker than and , in the sense that and must not be derivable in any corresponding system.

We will distinguish logics that are monotonic and logics that are nonmonotonic. Moreover, the logics will be distinguished by the different strength of interactions between the modalities.
The above points are realised in practice as follows. As before, we take (Figure 4) as base calculus for intuitionistic logics. This is extended with combinations of the characteristic rules of intuitionistic nonnormal monomodal logics in Figure 5. The difference is that now the calculi contain both some rules for and some rules for . In order to distinguish monotonic and nonmonotonic logics, we require that the calculi contain either both and (in this case the corresponding logic will be nonmonotonic), or both and (corresponding to monotonic logics). In addition, the calculi will contain some of the interaction rules in Figure 6. Since the logics are also distinguished according to the different strenghts of the interactions between the modalities, we require that the calculi contain either both and , or both and , or .
In the following we present the sequent calculi for intuitionistic nonnormal bimodal logics obtained by following our methodology. After that, for each sequent calculus we present an equivalent axiomatisation.
4.1 Sequent calculi
In the first part, we focus on sequent calculi for logics containing only axioms between , , and (that is, we don’t consider axiom ). The calculi are obtained by adding to (Figure 4) suitable combinations of the modal rules in Figures 5 and 6. Although in principle any combination of rules could define a calculus, we accept only those calculi that satisfy the restrictions explained above. This entails in particular the need of studying cut elimination. As usual, the first step to do towards the study of cut elimination is to prove the admissibility of the other structural rules.
Proposition 4.1.
Proof.
We can now examine the admissibility of . As it is stated by the following theorem, following our methodology we obtain 12 sequent calculi for intuitionistic nonnormal bimodal logics.
Theorem 4.2.
We let the sequent calculi be defined by the set of modal rules which are added to . The rule is admissible in the following calculi:
G.IE  :=  + + + 

G.IE  :=  + + + 
G.IE  :=  + + 
G.IM  :=  + + 
Moreover, letting be any of the previous calculi, is admissible in
N  :=  + 
N  :=  + + 
Proof.
The structure of the proof is the same as the proof of Theorem 3.2. Again, we consider only the cases where the cut formula is principal in the last rule applied in the derivation of both premisses, with the further restriction that the last rules are modal ones.
The combinations between rules, or between rules, have been shown in the proof of Theorem 3.2. Here we consider the possible combinations of  or rules with rules for interaction.
(; ). We have
which become
(; ). We have
which become
(; ). We have
which become
(; ). We have
which become
(; ). We have:
which is converted into the following derivation:
Observe that the former derivation has two application of , both of them with a cut formula of smaller weight as, in particular, (cf. Definition 3.1).
(; ) is analogous to the next case (; ).
(; ). We have:
which is converted into the following derivation:
(; ). We have
which become
∎
It is worth noticing that all cutfree calculi containing rule also contain rule . In fact, combinations of rules containing and not would give calculi where the rule is not admissible. This is due to the form of the interaction rules, that for instance allow us to derive the sequent using and . A possible derivation is the following:
Instead, sequent doesn’t have any cutfree derivation where is not applied, as no rule different from has in the conclusion. We will consider in Section 7 a calculus containing and not
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