1 Introduction and Background
A significant part of the literature in deontic logic revolves around the discussions of puzzles and paradoxes which show that certain logical systems are not acceptable—typically, this happens with deontic , i.e., Standard Deontic Logic ()—or which suggest that obligations and permissions should enjoy some desirable properties.
One wellknown puzzle is the the socalled Free Choice Permission paradox, which was originated by the following remark by von Wright in [23, p. 21]:
“On an ordinary understanding of the phrase ‘it is permitted that’, the formula ‘’ seems to entail ‘’. If I say to somebody ‘you may work or relax’ I normally mean that the person addressed has my permission to work and also my permission to relax. It is up to him to choose between the two alternatives.”
Usually, this intuition is formalised by the following schema:
(FCP) 
Many problems have been discussed in the literature around FCP: for a comprehensive overview, discussion, and some solutions, see [14, 11, 20].
Three basic difficulties can be identified, among the others [11, p. 43]:

Problem 2: Closure under Logical Equivalence Problem – “In its classical form FCP entails that classically equivalent formulas can be substituted to the scope of a permission operator. This is also implausible: It is permitted to eat an apple or not iff it is permitted to sell a house or not”;

Problem 3: Resource Sensitivity Problem – “Many deontic logics become resourceinsensitive in the presence of FCP. They validate inferences of the form ‘if the patient with stomach trouble is allowed to eat one cookie then he is allowed to eat more than one’ ”.
In this paper, we focus on another basic problem: how, and if, FCP can be accepted when we have incompatibilities between certain varieties of permissions and prohibitions/obligations. The issue is that since Problem 1 licenses the derivation that anything is permitted provided that something is permitted, no prohibition/obligation is allowed, otherwise we get an inconsistency [20]. In doing so, we will offer simple logics that take two of the three problems above into account.
The layout of the paper is as follows. The remainder of this section briefly comments on the three major problems mentioned above: the Permission Explosion Problem (Section 1.1), the Closure under Logical Equivalence Problem (Section 1.2), and the Resource Sensitivity Problem (1.3). Section 2 illustrates the theoretical intuitions and assumptions that we adopt to analyse free choice permission. In particular, we assume the distinction between norms and obligations/permissions, and we study the role of deontic incompatibilities, the duality principle, and why free choice permission is strong permission. Section 3 reviews in some detail two related works that have direct implications for our proposal. Finally, Section 4 presents some minimal deontic systems, six Hilbertstyle deontic systems admitting guarded variants of FCP: the systems that we present are not too weak from the inferential viewpoint, as far as permission is concerned, and do not commit to weakening any specific logic for obligations. Some conclusions end the paper. An appendix offers proofs of the formal properties of the proposed systems presented in Section 4.
1.1 Problem 1: Permission Explosion Problem
One of the most acute problems springing from FCP is obtained in , where, if at least one obligation is true, then by necessitation and propositional logic, we get . Since axiom is in , i.e is valid, we trivially obtain , thus, assuming the Duality principle
(Duality) 
we derive through FCP that . Hence, licenses that, if something is obligatory, then everything is permitted.
However, a careful analysis shows that this undesired result is not strictly due to as such, but to adopting any monotonic modal deontic logic [10], i.e. any system just equipped with inference rule :
(RM) 
or, alternatively with
(RE) 
plus the following axiom schema
(M) 
Indeed, assume Classical Propositional Logic (), FCP, and RM for ^{1}^{1}1I.e.,
In this context, it is enough if we have that is true to derive that any other permission , i.e., for any . Whenever is accepted, such a problem strictly depends on the characteristic schemas and inference rules of monotonic modal logics, as the above derivation—or a simple semantic analysis—shows. Hence, permission explosion is not a problem of , but of any weaker modal deontic logic which is at least closed under classical implication or which is closed under logical equivalence and allows for the distribution of over implication. Notice that Duality plays no substantial role. Accordingly, we can have that is valid for permission, if and are duals and the logic for is a monotonic modal logic, or is independent of and is assumed for .
In conclusion, if we want not to completely reject the intuition behind FCP, we have two nonexclusive options to be explored in order to avoid the Permission Explosion Problem:
 NoCPL:

abandon and adopt suitable nonclassical logical connectives;
 NoRM:
Our paper aims at exploring under what conditions NoCPL can be avoided by accepting at least a restricted version of FCP. Hence, it seems that NoRM thesis must be accepted.
1.2 Problem 2: Closure under Logical Equivalence Problem
In the previous section we mentioned that must be weakened. Hence, we can also drop and keep axiom schema . This choice could look satisfactory for those who consider problematic the fact that the logic for is closed under logical equivalence.
We take here another route. Incidentally, one can argue that the implausibility of “It is permitted to eat an apple or not iff it is permitted to sell a house or not” does not depend on RE, but rather on the fact that “It is permitted to eat an apple or not” is
, which looks quite odd. However, besides this problem—which would lead us to commit to specific philosophical views—dropping
has in general two controversial technical side effects:
it rejects standard semantics for modal logics, since the class of all neighbourhood frames validate RE: [10] argued in fact that classical systems (i.e., containing RE but not RM) are the minimal modal logics;

it fails to make, for instance, and logically incompatible under the Duality Principle (while and of course are); similarly, and , or and , are not incompatible too (while they of course should be).
In conclusion, we standardly assume that holds both for permissions and obligations, which means that any logic for free choice permission must be a classical system of deontic logic in [10]’s sense, i.e., any modal deontic logic closed under logical equivalence and not under logical consequence.
1.3 Problem 3: Resource Sensitivity Problem
It has been noted [17] that from “You may eat an apple or a pear”, one can infer “You may eat an apple and that You may eat a pear”, but not “You may eat an apple and a pear” [7, p. 2].
We simply observe that the systems proposed in Section 4 do not license in general the inference above. However, a thoughtful treatment of this problem—the Resource Sensitivity Problem—goes beyond the scope of this paper. In fact, it has been widely discussed in the literature that it is strictly related to considerations from action theory, which have often found solutions shifting from CPL to nonclassical logics such as the substructural ones [see, among others, 7, 4, 11].
In conclusion, we do not commit here to find any suitable solution to such a problem.
2 Three Basic Intuitions
We are going to present some deontic systems that accommodate restricted variants of FCP. This is done under some minimal philosophical assumptions, which can in principle be compatible with several deontic theories. Of course, our approach is not neutral. In this section, we illustrate our fundamental intuitions and assumptions.
2.1 The Distinction between Norms and Obligations
We assume in the background a conceptual distinction between norms, on one side, and obligations and permissions, on the other side. The general idea of norms is that they describe conditions under which some behaviours are deemed as ‘legal’. In the simplest case, a behaviour can be qualified by an obligation (or a prohibition, or a permission), but often norms additionally specify the consequences of not complying with them, and what sanctions follow from violations and whether such sanctions compensate for the violations. The scintilla for this idea is the very influential contribution [1], which is complementary to the (modal) logicbased approaches to deontic logic. The key feature of this approach is that norms are dyadic constructs connecting applicability conditions to a deontic consequence. A large number of such pairs would constitute an interconnected system called a normative system [for more recent proposals in this direction, see 18, 19, 12, 13].
To be clear, this paper does not present any logic of norms, but our proposal for a logic of obligations and permissions—with restricted variants of FCP—can be better understood if one keeps in mind some intuitions about how norms should logically behave and about the relation between the logic of norms and deontic logic. In particular, our assumptions are:

obligations and permissions exist because norms generate them when applicable;

once obligations and permissions are generated from norms—which requires us to reason about norms—we can still perform some reasoning with the resulting obligations and permissions—this is the task of deontic logic in a strict sense, i.e., the logic of obligations and permissions;

norms can be in conflict—without being inconsistent— but this does not hold for obligations and permissions.
Hence, we distinguish two levels of analysis: a normlogic level and a resulting deonticlogic level. This paper only technically deals with the second level of analysis.
Assume for example that we have two norms and , where is any ifthen suitable logical relation connecting applicability conditions of norms and their deontic effects. We can indeed have them—for example, in a legal system—but the point is what obligations/permissions we can obtain from them. A rather standard assumption is that in order to correctly derive deontic conclusions we need to solve the conflict between and . Specifically, our general view is prudent (or skeptical, as one says in nonmonotonic logics), because, unless we know how to solve the conflict (typically, by establishing that is stronger than or vice versa), we do not know if or holds. Since we do not accept that both can hold, it is pointless to consider at the deontic level that and are true—while any logic of norms can have both and .
In conclusion, we impose deontic consistency at the deonticlogic level, i.e., .
2.2 Deontic Incompatibilities, Duality, and Fcp
With the above said, the issue is whether FCP is an appropriate principle to adopt for normative reasoning. Our view is that this principle in general is not, even when Problem 1 and 2 above are solved. We provide below a simple counterexample to it, which considers the interplay between free choice permissions and prohibitions.
Example 2.1
When you have dinner with guests the etiquette allows you to eat or to have a conversation with your fellow guests. However, it is forbidden to speak while eating.
The full representation of the example is that each choice is permitted when one refrains from exercising the other one. In a situation when one eats, there is the prohibition to speak, while when one speaks, there is the prohibition to eat. Hence, it means that we can detach any single permission only if the content of such permission is not forbidden.
We will return in Section 2.3
to the logical import of the above scenario in a classical system of deontic logic. For the moment, taking stock of the example we just notice that
FCP could be reformulated as follows:(1) 
However, assuming Duality, is equivalent to , thus (1) reduces to
(2) 
(2) is a propositional tautology. Thus, (1) does not extend the expressive power of the logic unless one assumes a logic where obligation and permission are not the duals.
2.3 Strong Permission, Classical Systems, and Fcp
When permission is no longer the dual of obligation, we enter the territory of strong permission [22, 2, 3]^{3}^{3}3Besides von Wright’s theory [22], there is another sense in the literature of strong permission [15]. We will briefly return on this in Section 3.2.. As is wellknown, while it is sufficient to show that is not the case to argue that is weakly permitted, this does not hold for strong permission, for which the normative system explicitly says that there exists at least one norm permitting [3, p. 353–355].
In order to keep track of these two cases at the deonticlogic level, we can standardly distinguish in the deontic language two permission operators, for weak permission (such that ) and for strong permission (where Duality does not hold).
What is the minimal logic of strong permission at the deontic level in which some reasonable version of free choice permission can be accepted?
We mentioned that RM must be rejected. In fact, besides the Permission Explosion Problem, one may also argue that it is reasonable not to derive from any because we could have in the background that the normative system consists just of an explicit norm . If we have that, in presence of some version of free choice permission, you may also detach , which is against the abovementioned intuition that the strong permission should follow from explicit norms, or from combinations of them in normative systems where all disjuncts are explicitly considered [see, e.g., the discussion in 3, p. 354–355].
Second, as said above, deontic consistency should be ensured:
()  
() 
Notice that is the standard axiom of Standard Deontic Logic establishing the so called external consistency of obligations that, in turn, implies consistency among obligations and (weak) permissions. From we obtain, as expected, that strong permission entails weak permission [see, e.g., 3, p. 354], but not the other way around:
This is reasonable because the fact that at the normlevel we derive that is permitted using an explicit permissive norm means that no prohibitive norm (forbidding ) successfully applies or prevails over .
What about free choice permission? Coupling Assumptions 1 and 2 with the distinction between weak and strong permission allows us to identify a guarded variant of FCP for strong permission, consisting of two schemata:
()  
() 
These schemata take stock of what we said: you can detach from a disjunctive strong permission any single strong permission only if this last is weakly permitted.
The idea of the combination of the two axioms is that from repeated applications of and from a disjunctive permission, we can obtain the maximal subdisjunction such that no element is forbidden, and then, the application of the allows us to derive the individual strong permissions that are not forbidden. Notice that we cannot assume the following formula as the axiom for free choice permission.
The problem is that we do not know in advance how many elements of the disjunctive permission are (individually) forbidden. Consider for example, a theory consisting of the following formulas:
Here, one could use the conjunction to obtain , and , but then we have a contradiction from and (from axiom ). Notice, that in general, we are not able to use to detach a single (strong) permission, but a disjunction corresponding to the “remainder” of the disjunction, that is, in the case above, . Then, we can use the to “lift” the remaining elements from weak permissions to strong permissions. The only case when we can obtain an individual strong permission from a permissive disjunction is when the remainder is a singleton; but this means, that all the other elements of the permissive disjunction were forbidden. This further means that a disjunctive strong permission holds if at least one of its elements can be legally exercised. Going back to the example, if one extends the theory with , then we can derive .
Let us consider again the situation described in Example 2.1. The scenario can be formalised as follows (where and stand, respectively for “to eat” and “to speak”):
In a logic endorsing the unrestricted version of free choice permission, we have and . This means that as soon as one exercises one of the choices, we get that the other choice is at the same time permitted and forbidden, a situation that is either paradoxical or contradictory. Thus, the only way to avoid this kind of conflict is to refrain from exercising any of the two choices. However, this means that one is not really free to choose between the two options. Accordingly, either one has to adopt a restricted version of the free choice permission or abandon it. Notice, that axiom allows us to conclude that given , is forbidden (), and thus that is permitted (); similarly, one gets from , which implies .
Consider . One may argue why, in symmetry with , we cannot rather have
() 
Technically, it is obvious that implies but not the other way around, so both options are available. The variant is more prudent in that it licenses the detachment of an individual strong permission only if the normative system explicitly deals with that specific disjunct, while the second allows for the derivation in a slightly more relaxed way. So, if one wants to strictly reframe the structure of standard in a guarded version but does not want , then is the right option.
We should notice that the above schemata for free choice permission do not necessarily require the technical idea of deontic consistency, unless we assume—but we don’t—that obligation implies strong permission, and despite the fact that the consistency problem can occur if we endorse —as we do— and so that strong permission implies weak permission. In fact, if we do not validate RM, we would need anyway to model the following scenario
(3) 
as a true instance of our intuition, despite the fact that and are not inconsistent. One may rather argue that any state exercising the permission –i.e., a state where holds– does not comply with . If so, the condition that detached permissions are compliant with obligations is a rational requirement for free choice permission, which is not technically needed in classical modal systems for ensuring standard logical consistency between deontic statements.^{4}^{4}4Indeed, it is a trivial result in classical systems that the inference of from is not in general valid. Semantically, it is also immediate to build a neighbourhood model which falsifies that inference or which, in a similar perspective, admits the truth of both and . (This holds if is treated as an independent operator with no further conditions.)
Therefore, if one wants to consider (3) as an instance of free choice permission in logics that do not satisfy , we need to replace and with the following inference rules:
() 
() 
Of course, the same remark we made for and holds here, too, so we may have the following alternative:
() 
In the discussion leading to the formulation of the inference rules above, we provided the intuition and an example for . Let us now examine a few interesting cases for and . For the first case, we consider the following instance
In a logic with we can derive from , since is a tautology. The inference rules and allow us to replicate this type of reasoning without being forced to derive , thus we can apply to the instance above to obtain .
The second situation is given by the following theory:
Since and are tautologies, the theory implies, by applications of , , , and .
It is easy to verify that in conjunction with any of the axiom , and makes the corresponding inference rules admissible. But, as we have seen in Section 2.3, the combination of FCP and leads to the Permission Explosion Problem. This is not the case when we replace FCP with (or or the corresponding inference rules), since the guarded version of free choice permission allows us to derive an individual strong permission from a disjunctive (strong) permission containing the individual element only when we have that the individual permission is also weakly permitted. In general, the axiom and inference schemata we have proposed do not suffer from the permission explosion, though the resulting logics have a “controlled” permission explosion in the sense that every weak permission is lifted to a strong permission in case a tautology is explicitly strongly permitted, as the following derivation shows:
The consequence of the controlled permission explosion is that whenever the logic for the strong permission is a normal modality, the notion of strong permission collapses to that of weak permission.^{1}^{1}todo: 1vogliamo dire qualcosa sulla concezione di Alchurron e Bulygin che, logicamente, c’e’ un solo tipo di permesso e la differenza e’ nell’uso?
NINO: secondo me è meglio di no perche’ noi assumiamo la differenza nelle modalita’. Non mi è chiara la ragione che lega le condizioni di collasso discusse qui e quanto detto da Alchourron e Bulygin. O meglio, la cosa e’ chiara—semanticamente normal modality significa perdere la distinzione tra norme (una sola relazione di accessibilita’), ma FCP è irrilevante, cio’ che contna è che siamo in un sistema normale. Io eviterei casini, a meno che tu abbia le idee piu’ chiare di me.
To avoid this issue one could either impose the axiom ^{2}^{2}todo: 2Inserire riferimenti? discussione? or reject ; however, the later seems to be more problematic (see the discussion in Section 1.2 or part of the discussion in Section 3.1 below).
3 Two Related Works
In this section we review in some detail two related works that have directly implications with respect to our proposal [6, 5]. In fact, even though they have a different philosophical backgrounds, they propose simple nonnormal axiomatisations for obligation and permission—as we do—which avoid, e.g., Problem 1 and which are based on the concept of free choice permission as strong permission or, anyway, as a type of permission without Duality.
3.1 Asher and Bonevac’s Analysis
Asher and Bonevac [6]’s analysis presents a deontic logic based on AndersonKanger reduction of obligations and permissions. Assume denotes as usual Sanction. Then, —where is an operator—is dropped by introducing a suitable and weak conditional logic for , such that
Obligations and permissions are defined through the concept of sanction, but Duality no longer holds, since Asher and Bonevac [6] argue that free choice permission is strong permission.
The logic for is not closed under logical equivalence in conditional antecedents, so the resulting and are not closed under logical equivalence. The overall system is nonmonotonic and can naturally handle cases such as the ones expressed in .
So, we share with [6] important assumptions. However, Asher and Bonevac’s proposal suffers from some drawbacks that we consider difficult to accept. Besides the fact that the closure under logical equivalence does not hold (see Section 1.2), consider the following scenario [6, p. 311]:
Example 3.1 (SoupEggroll scenario [6])
Assume that .
“Suppose we go to the Chinese restaurant. There it’s part of the context that while you may have soup or eggroll, you can’t have both. Thus,
holds. But while we now can derive defeasibly that you may have, e.g., an eggroll, you can’t have both eggroll and soup, at least without paying extra.”
The SoupEggroll scenario, as analysed by [6], is debatable. Indeed, the fact that you can’t have both soup and eggroll means that having both is forbidden. The point is that a prohibition amounts here to the fact that both strong permissions are false rather that having , namely, . This is due to the fact that, in the example, prohibition is the negation of strong permission and not the obligation of the opposite.
Consider the following example. In some card games, it is obligatory not to play a trump card when one is the first to play; this can be expressed as when one is the first to play it is forbidden to play a trump card. Intuitively those two statements seem to be equivalent, but their formal representation, namely
are not. Notice, also, that, since the first statement contains a negation and the logic for is not closed under logical equivalence, the proper translation is with the double negation, and as a consequence, and are not equivalent; while this might be acceptable in deontic logics based on noninvolutive multivalued logics, it seems counterintuitive when the underlying logic is the bivalent classical propositional logic.
In conclusion, while [6] has the great merit of identifying fundamental intuitions for free choice permission, it fails to frame those intuitions in a convincing general theory of obligations.
3.2 Open Reading of Permission and Obligation as Weakest Permission
A more recent proposal discussing free choice permission is due to [5], though the philosophical background of this work is significantly different from ours.
First of all, [5] works on the concept of open reading of permissions [16, 9]. Consider that “it is permitted to board the plane”——and assume that is an action type, i.e., something saying that “there are many, mutually exclusive action tokens of that type”. Hence, there “might be many ways to board a plane. There might be more than one gate to go through, there might be several times within a fixed period when one can proceed, etc.”, i.e., there are many action tokens [5, p. 808]. How to read [16]? The authors adopt the socalled open reading, according to which at least one token of type (but possibly not all) is acceptable according to the normative system.^{5}^{5}5Permission defined through the open reading is sometimes called strong permission [15], which is different from [22]’s notion [see 5, p. 808, fn. 1]. We can ignore this issue here, by generically assuming that strong permission is just not the dual of obligation.
Hence, rational obligations and permissions should be seen, respectively, as giving necessary and sufficient conditions for rational agents (typically, in gametheoretic settings) [5, sec. 2.3]. An action type is obligatory whenever it is exactly the normatively ideal action type: obligation as weakest permission. This means that “playing any action type that rules out being rational is forbidden. This is not the case in the logic of obligations as weakest permissions. There the unique obligation bearing on the players is to play a rational strategy” [11, p. 17].
We believe that this contribution may have a general import on the debate on permission, which goes beyond the philosophical discussion on the open reading.
Technically, the modal system of [5] is classical, too, in [10]’s sense, and is not the dual of , but an independent operator.
Conceptually, it is interesting in general to explore a deontic logic that allows for deriving a unique obligation [5, p. 817]. If we assume the distinction between norms and obligations, then this means that, given a normative system whose norms prescribe , then we can only have in the deontic logic but not, for example, any obligation (). This means two things: (a) is not valid thus leading to rejecting , (b) we exclude the possibility of having any norm supporting some other conclusion, which is made applicable by one single obligation.
As for point (b), while in several cases it is not harmful to only derive the entire obligative conjunctive content of a normative system—this is not a problem for checking compliance, for instance—there are cases where one may need to logically speak of single obligations. Consider a normative system, which states that some obligations are conjunctively typical only of a certain type of entities—as many legal systems do. For instance, assume that holds only for specific commercial entities, such as corporations. This means that
However, this cannot exclude that a subset of those obligations (e.g., and ) implies that your company is a partnership:
That this is not admitted in a deontic logic looks to us too restrictive if we go beyond the domain of the open reading of permission.
4 Six Minimal Deontic Axiomatisations with Guarded Free Choice Permission
Finally, we present some minimal deontic systems, six Hilbertstyle deontic systems admitting a guarded version of FCP. The systems that we present are not too weak from the inferential viewpoint, as far as permission is concerned, and do not commit to weakening any specific logic for obligations.
4.1 Language, Axioms and Inference Rules
The modal language and the concept of well formed formula are defined as usual [see 10, 8]. We just recall that we have three modal operators, two operators, for obligations and for strong permissions, and for weak permission. As usual, we assume to be an abbreviation for .
For convenience, let us synoptically recall below all relevant schemata and inference rules, where .
Inference Rules:



and

, and

, .
Schemata:







.
Given the discussion of Section 2, we can identify some deontic systems, as specified in Table 1. Notice that we consider also systems and , which are monotonic, so they contain . Strictly speaking, this is the limit which we cannot trespass, since we have three restricted forms of Permission Explosion. We will return on this in the concluding section of the paper.
Deontic System  Properties  Derivable 

,  
,  
,  
,  
,  
,  
,  
, 
4.2 Semantics and System Properties
Let us begin with standard concepts. Assume that is the set of atomic sentences.
Definition 4.1.
A deontic neighbourhood frame is a structure where

is a nonempty set of possible worlds;

and are functions .
Definition 4.2.
A deontic neighbourhood model is a structure where is a deontic neighbourhood frame and is an evaluation function .
Definition 4.3 (Truth in a model).
Let be a model and . The truth of any formula in is defined inductively as follows:

standard valuation conditions for the boolean connectives;

iff ,

iff ,

iff ,
where, as usual, is the truth set of wrt to :^{6}^{6}6Whenever clear from the context we drop the references to the model.
A formula is true at a world in a model iff ; true in a model , written iff for all worlds , ; valid in a frame , written iff it is true in all models based on that frame; valid in a class of frames, written , iff it is valid in all frames in the class. Analogously, an inference rule (where are the premises and the conclusion) is valid in a class of frames iff, for any , if then ^{7}^{7}7Of course, if any has the form then trivially means ..
We can now characterise different classes of deontic neighbourhood frames that are adequate of the deontic systems in Table 1.
Definition 4.4 (Frame Properties).
Let be a deontic neighbourhood frame.

supplementation: is supplemented, , iff for any and , ;

coherence: is coherent iff for any and , ;

coherence: is coherent iff for any and , ;

permission: is permitted iff for any and , ;

permission: is permitted iff for any and , ;

permission: is permitted iff for any and , ;

permission: is permitted iff for any and , ;

permission: is permitted iff for any and , ;

permission: is permitted iff for any and , .
Here below are some relevant characterisation results. The proofs are in the Appendix.
Lemma 4.1.
For any deontic neighbourhood frame ,

is valid in the class of coherent frames;

is valid in the class of coherent frames;

is valid in the class of permitted frames;

is valid in the class of permitted frames;

is valid in the class of permitted frames;

is valid in the class of permitted frames;

is valid in the class of permitted frames;

is valid in the class of permitted frames.
Completeness results for the three deontic systems are ensured: again see the Appendix for a proof.
Theorem 4.1.

the system is sound and complete w.r.t. the class of deontic neighbourhood frames;

the system is sound and complete w.r.t. the class of  and coherent frames;

the system is sound and complete w.r.t. the class of  and permitted frames;

the system is sound and complete w.r.t. the class of  and permitted frames;

the system is sound and complete w.r.t. the class of supplemented,  and permitted frames;

the system is sound and complete w.r.t. the class of  and permitted frames;

the system is sound and complete w.r.t. the class of  and permitted frames;

the system is sound and complete w.r.t. the class of supplemented,  and permitted frames.
Finally, a corollary showing the relative strength of the six deontic systems.
Corollary 4.1.

,
and
. 
Let , and let and be classes of frames adequate for and . If then .
5 Conclusions
In this paper we have investigated how, and if the notion of free choice permission is admissible in modal deontic logic. As is well known, several problems can be put forward in regard to this notion, the most fundamental of them being the socalled Permission Explosion Problem, according to which all systems containing FCP and closed under RM and RMP license the derivation of any arbitrary permission whenever at least one specific permission is true.
We argued (Section 1.1) that a plausible solution to this problem is to jump from monotonic into classical deontic logics, i.e., systems closed under RE but not RM. This solution does not necessarily mean that the resulting deontic system is very weak, as far as permission is concerned, if further schemata and inference rules are added (Sections 2.3 and 4.2).
The basic intuitions for extending classical deontic logics are the following:

We assume in background the distinction between norms and obligations/permissions. While we conceptually accept that the normative system may contain conflicting norms, it is logically inadmissible that such norms generate actual conflicting obligations/permissions since conflicts must be rationally solved, otherwise no obligation/permission can be obtained; hence, we validate schemata and ;

Free choice permission is strong permission, meaning that it is a permission generated by explicit permissive norms;

The possibility of detaching single strong permissions from disjunctive strong permissions, i.e., from strictly depends on the fact that is not the case.
Taking the above points into account, we thus proposed different guarded variants of FCP that significantly increase the inferential power of the logic. In particular, six Hilbertstyle classical deontic systems were presented.
We observed that four of these systems are classical modal systems, while we can have other two acceptable systems which are monotonic. In fact, the fact that those two systems are closed under RM does not lead to full Permission Explosion, but only to a “controlled” version of it: indeed, in systems like any permission is obtainable via free choice permission only if it is not incompatible with existing prohibitions.
Some directions for future work can be identified. In particular:

It is still an open issue to fully discuss the Resource Sensitivity Problem in our setting. In fact, while we argued that this problem goes beyond our paper, there are scenarios where our intuitions are relevant for this problem as well. For example, suppose that there is a fruit basket in the kitchen containing a banana and an apple. Bob and Alice are permitted to eat the banana or the apple and Alice first eats the former. Bob cannot do anything but take the apple. However, if Bob is allergic of apples, so no permission can be reasonably derived because it is forbidden for him to eat the apple.

Our idea of free choice permission relies on the fact that no strong permission can be detached from a disjunctive permissive expression if another norm allows for deriving a conflicting obligation. Hence a full understanding of schemata such as or may benefit for an explicit logical treatment of the logic of norms adopting defeasible reasoning [11].
Acknowledgments
This work was partially supported by the EU H2020 research and innovation programme under the Marie SkłodowskaCurie grant agreement No. 690974 for the project MIREL: MIning and REasoning with Legal texts.
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Appendix A Basic Properties of the Deontic Systems
Let us start by proving Lemma 4.1.
See 4.1
Proof.
The proof for case (1) is straightforward. The proof of (2) is trivial and standard. Both are omitted. ^{3}^{3}todo: 3Allora, dire che una formula è valida in una classe di frames, significa dire che per ogni mondo, e per tutti i modelli la formula è vera nel mondo, . Quindi mi sembra che ci sia una sola direzione della dimostrazione, e ho commentato la direzione
Case (3) – Consider any frame that is permitted but such that . This means that there exists a model based on such that , i.e., there is a world where
(4)  
(5) 
By construction, from (5) we have , while from (4) we have and , so is not permitted.
Case (6) – As usual in these cases, the proof must show that, on the class of all permitted frames, for any model based on and for any world in it,
The definitions of some basic notions and of canonical model for the classical bimodal logic (just consisting of for and ) are standard.
In the rest of this section when we refer to a Deontic System we mean one the logic axiomatised in Section 4.
Definition A.1 (maximality).
A set is maximal iff it is consistent and for any formula , either , or .
Lemma A.1 (Lindenbaum’s Lemma).
For any Deontic System , any consistent set of formulae can be extended to an maximal set .
Definition A.2 (Canonical Model [10, 21]).
A canonical neighbourhood model for any system in our language (where ) is defined as follows:

is the set of all the maximal sets.

For any propositional letter , , where .

If , let where for each world , .
Thus, we have as usual basic completeness result for . To cover the other systems, it is enough to prove that all frame properties for the relevant schemata and rules are canonical.
Lemma A.3.
The frame properties of Definition 4.4 are canonical.
Proof.
The proofs for supplementation, coherence, and coherence are standard.
permission – Let us consider a canonical model for , any world in it, and any truth sets such that and . Clearly, . Since is valid (Lemma 4.1), then . By construction, this means that , thus the model is permitted.
permission and permission– Similar to the case above.
permission – Let us consider a canonical model for , any world in it, and any truth sets such that , , and . Clearly, . Also, assume and . Since is valid (Lemma 4.1) then . By construction, this means that and , thus the model is permitted.
permission and permission– Similar to the case above.
∎
Hence, the following result is ensured.
See 4.1
Finally, let us prove Corollary 4.1.
See 4.1
Proof.
Case (i) – For we first notice that for every formula , ; hence, axioms and can be considered as simple instances of and respectively.
To show that the inclusion is strict the model below provides an permitted model that does not validate .
Let , where:

;

, and ;

; and

.
It is easy to verify that the model is permitted, and are true in : and , and clearly . However, .
For it is enough to prove the following showing that and are derivable from and and . ( is valid in every classical modal logic containing [10].)
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