1 Introduction
Prof. Luis Fariñas del Cerro wrote a bunch of articles reporting the results he obtained by designing logics for very interesting and specific purposes. The elegance of the underlying ideas and the presentation form is out of discussion. Many researchers in logic would like to have his ability to extract logic from facts and their relationships, building new judgments. One of the authors of this article ever tried to be able to have this resulting research. When we were invited to contribute to prof. Luis Fariñas del Cerro Festschrifft this become the opportunity to report to the master maybe the only research that we conduct on the lines of defining a logic for a specific formalization. In this article, we report our results in the last seven years in the designing logics for legal ontologies.
Classical First Order Logic has been widely used as a basis for ontology creation and reasoning in many domains. These domains naturally include Legal Knowledge and Jurisprudence. As we expect, consistency is an important issue for legal ontologies. However, due to their inherently normative feature, coherence (consistency) in legal ontologies is more subtle than in other domains. Consistency, or absence of logical contradictions, seems more difficult to maintain when more than one law system can judge a case, what we call a conflict of laws. There are some legal mechanisms to solve these conflicts such as stating privileged fori or other ruling jurisdiction. In most of the cases, the conflict is solved by adopting a law hierarchy or precedence, rather better, ordering on laws. Even under these precedence mechanisms, coherence is still a major issue in legal systems. Each layer in this legal hierarchy has to be consistent. Since consistency is a direct consequence of how one deals with the logical negation, negation is also a main concern in legal systems. Deontic Logic, here considered as an extension of Classical Logic, has been widely used to formalize the normative aspects of the legal knowledge. There is some disagreement on using deontic logic, and any of its variants, to this task. Since a seminal paper by Alchourron and Martino Alchourron Martino (1990), the propositional aspect of laws has been under discussion. In Alchourron Martino (1990), the authors argue that laws are not to be considered as propositions, in full agreement with Hans Kelsen jurisprudence. The Kelsenian approach to Legal Ontologies considers the term “ontologies on laws” more appropriate than “law ontology”. In previous works, we showed that Classical logic is not adequate to cope with a Kelsenian based Legal Ontology. Because of the popularity of Description Logic for expressing ontologies nowadays, we developed an Intuitionistic version of Description Logic particularly devised to express Legal Ontologies. This logic is called iALC. In this article, we show how iALC avoids some Contrarytoduty paradoxes, as Chisholm paradox and other paradoxes that appear in deontic logic, such as the good samaritan and the knower. For these paradoxes, we provide iALC models. Finally, we discuss the main role of the intuitionistic negation in this issue, finding out that its success may be a consequence of its paracomplete logical aspect. This investigation opens the use of other paracomplete logics in accomplishing a logical basis for Kelsenian legal ontologies, as a complementary solution to those based on paraconsistent logics, see M.E. Coniglio (2009).
2 A brief discussion on Kelsenian Jurisprudence and its logic
A very important task in jurisprudence (legal theory) is to make precise the use of the term “law”, the individuation problem, and it is one of the most fundamental open questions in jurisprudence. It requires firstly answering the question “What is to count as one complete law?” Raz (1972). There are two main approaches to answer this question. One approach is to consider “the law” as the result of a natural process that yields a set of norms responsible for stating perfect social behavior. Another approach is to consider “the law” as a set of individual legal statements, each of them created to enforce a positively desired behavior in the society. As a consequence, in the first approach, the norms say what are the best morally speaking accepted state of affairs in a particular society, while, in the second approach, each legal statement rules an aspect of the society that the legislature wants to enforce the behavior. The first is more related to what is called Natural Law and the last to Legal positivism. We can say that the Legal positivism is closer to the way modeling is taken in Computer Science. In the natural approach to the law, it is even harder to define a system of laws than in the legal positivism. The natural approach demands stronger knowledge of the interdependency between the underlying legal statements than legal positivism. Because of that, the natural approach, in essence, is harder to be shared with practical jurisprudence principles, since they firstly are concerned to justify the law, on an essentially moral basis. This justification is quite hard to maintain from a practical point of view.
The coherence of “the law” in both approaches is essential. A debate on whether coherence is builtin by the restrictions induced by Nature in an evolutionary way, or whether coherence should be an object of knowledge management, seems to be a long debate. Despite that, legal positivism seems to be more suitable to Legal Artificial Intelligence. From the logical point of view, the
natural approach is also harder to deal with than the positivist one. When describing a morally desired stateofaffairs, the logical statements take the form of propositions that has as a model best of the moral worlds. Deontic logic is suitable to be used to fulfill this task. However, a legal statement (“a law”) is essentially an individual sentence that can also be seen as an order (mandatory command), and hence, it is not a proposition at all. As a consequence, deontic logic is not appropriate to be used in knowledge bases. Besides that, Valente (1995) shows that deontic logic does not properly distinguish between the normative status of a situation from the normative status of a norm (rule). We think that the best jurisprudence basis for Legal ontologies and reasoning is Legal positivism. Thus, we will be talking a legal ontology as an ontology about (individual) laws, and not an ontology on “the law”.Hans Kelsen initialized the Legal positivism tradition in 1934, for a contemporary reference see Kelsen (1991). He used this positive aspect of the legislature to define a theory of pure law and applied it to the problem of transfer citizen’s rights and obligations from one country to other when crossing boarders. He produces a quite good understanding of what nowadays we denominate Private International Law. This achievement was so important that in many references on international law, Kelsen jurisprudence is the basis for discussions on conflictoflaws derived from different statements coming from different fori. ^{1}^{1}1“It is one of Kelsen’s frequently repeated doctrines that conflict of norms, in the absence of a normative procedure for resolving the conflict, shatters the concept of a unified system”, is highly emphasized in Hughes Hughes (1971), for example, and it is one of the principles most cited when Kelsen jurisprudence is presented.
In what follows we introduce the main terminology and concepts of Kelsenian jurisprudence that we use in this article. We can summarize Kelsen theory of pure law in three principles:

According to what was discussed above, individually valid legal statements are the firstclass citizens of our ontology. Thus, only inhabitants of the Legal knowledge base are individual laws, see (Kelsen, 1967, supra note 5, pp 910) ^{2}^{2}2Kelsen takes norms and valid norms as synonyms. To say that a legal norm is valid is to say that it exists, is affirmed by Kelsen. For example, if it is the case that Maria is married with John, and, this was legally celebrated, then “MariamarriedwithJohn” is an individually valid legal statement, and hence, it is a member of the Legal Ontology;

Kelsen also says that that the validity of a legal norm can only be provided concerning the validity of another, and higher, one. So, , a norm, is legally valid if, and only if, it was created or promulgated in agreement with other, and higher, legally valid norm, . This justification induces a precedence relationship between norms that is transitive, that is, if precedes , and, precedes , then recedes ; ^{3}^{3}3See (Kelsen, 1967, supra note 5, p. 1967). This can be also found in (Kelsen, 1946, supra note 5, p. 1101)

There is a mechanism for relating laws from one Legal system to another, the socalled “choiceoflaw rule”. This mechanism is very important to the development of a concept of International Law. Assume that MaryismarriedwithJohn is an individual legal statement in legal system . Assume also that Mary is a citizen of a country adopting legal system . Is there any legal statement in ensuring that Mary is married in ? Well, this depends on itself, but there is a way to connected the individual law MaryismarriedwithJohn in to MaryismarriedwithJohn in . In some legal systems, this is accomplished by what Kelsen denominated “a connection”. As shown in the following quotation from (Kelsen, 1946, page 247), the connection between the laws of and is made by reference, but, in fact, each law belongs to its respective legal system. In this specific case we can consider MaryismarriedwithJohn in system is connected to MaryismarriedwithJohn in legal system the connection Lex Loci Celebrationis.
… the law of one State prescribes the application of the law of another State, and the latter does not object or demand it. It has no right to do so since it is not really its own law which is applied by the other State. The latter applies norms of its own law. The fact that these norms have the same contents as corresponding norms of another State does not concern the latter…Since the specific technique of these norms consists in “referring” to the norms of another system and by so doing incorporating norms of identical contents into their own legal system, it would be more justifiable to call them “reference rules”… The reference rule, that is … the norm regulating the application of foreign law, may be distinguished from the norm to be applied, that is, the norm referred to. Only the former is a norm of private international law. But from a functional point of view, the one is essentially connected with the other.
Nowadays it is a common terminology in Private International Law the use of the connecting factors or legal connections between individual laws in a different legal system. Only to enumerate some of them: LexDomicilii, LexPatriae, Lexlocicontratum, Lexlocisolutionis, etc.
There is a philosophical problem with the principle 2 above. It demands the existence of basic laws. These basic laws do not have their validity/existence as a consequence of other more basic laws. Kelsen name these basic laws Grundnorms. Their validity is based on legislature acts and in a certain sense is derived from the sovereign of the State. It is out of the scope of this article to discuss such problem in Kelsen’s jurisprudence. We take as granted that Kelsen jurisprudence can adequately support most of the existent legal systems, a definitively not an unreal working hypothesis.
From the three principles above, we have some very simple ontological commitments:
 I

Individuals are laws;
 II

There is a transitive and reflexive relationship between individual laws that reflects the natural precedence relationship between laws;
 III

There are legal connections between individual laws in different legal systems or between different fori in the same broader legal system.
From these commitments, we derive the basic constructs of the logic . In the first place, our legal ontology relates concepts to legal systems. Description logics uses nominals to refer to individuals. So, an expression as , stands for is an individual law, belonging to the legal system , a concept.
From commitment 2 we consider an expression as standing for the individual law legally precedes individual law . The subsumption relationship , from description logic, denotes that is a legal subsystem of . One could interpret this relation as the inclusion relationship. ^{4}^{4}4In Classical ALC this is just the case, but we shown here that classical reasoning it is not a good choice for dealing with legal ontologies We discuss the implications of using negated contents together with Kelsenian jurisprudence in the following. This can be found in Haeusler . (2011); Haeusler, de Paiva Rademaker (2010); Haeusler, Paiva Rademaker (2010); Haeusler, de Paiva Rademaker (2010) too.
Under the classical setting, a negated concept denotes the set of all inhabitants of the domain that do not belong to the interpretation of . Under ontological commitment 2 there is no individual law that does not exist in, belong to, the domain. Since norms and laws are not propositions, it is a complete nonsense to negate a law. As we already seen, we can negate a concept on laws. Consider the collection of all Brazilian individual laws. Call it . In a classical setting is the universe of laws. Thus, any law that it is not in has to be a law outside , that is, belonging to . For example, if Peter is 17 years old, it is not liable according to the Brazilian law. Is PeterIsLiable a valid law at all? If so, it has to belong to . Using Kelsen in a classical setting, individual laws not belonging to a concept automatically belong to its complementary concept. The problem with this is that it is possible to create laws outside a jurisdiction or forum by the very simple act of considering or experimenting a legal situation. Nowadays in Brazil, the parliament is discussing the liability under the 16 years. By the simple fact of discussing the validity of their corresponding individual laws, we are forced to accept they exist outside the Brazilian legal system. We do not consider this feature appropriate to legal ontology definition. Dealing with negations every time we assume the existence of a law may bring unnecessary complexity to legal ontology definition. Because the precedence relationship between laws, cf. ontological commitment 2, there is a natural alternative to classical logic, the intuitionistic logic (IL). According to IL semantics, , iff, for each law , such that , it is not the case that . This semantics means that does not provide any legal support for any individual law belong to , which agrees with Kelsen jurisprudence on the hierarchy of individual laws.
Commitment 2 gives rise to expressions of the form LexLociCelebrationis , where is MaryIsMarriedWithJohn and LexLociCelebrationis is a legal connection. Thus, if is the concept that represents all laws in Portugal, then the concept represents the Brazilian individual laws stating that Portuguese marriage is valid in Brazil. The private international law of any country is a collection of laws stated in similar ways for every possible legal connection. In Haeusler, de Paiva Rademaker (2010) it is shown in detail a judicial case deriving that a renting contract is solving a conflict of laws in space through private international law.
3 Some philosophical discussion on the ontological criteria taken on using Kelsen in legal ontologies
We base our work on two ontological criteria: ^{5}^{5}5see Quine’s “On What there is” article and http://plato.stanford.edu/entries/simplicity, for example to a primer ontological criteria 1 Ontological Commitment (due to W.Quine), our logical approach is ontologically committed to Valid Legal Statements only, as discussed in section 2. The only nominals occurring in our logic language are valid individual laws, and; 2Ontological Parsimony, which is strongly related to Quine’s ontological commitment too, with a mention of its stronger version also known as Occam’s Razor, here denoted as OR. The second criteria is based on: “One ‘easy’ case where OR can be straightforwardly applied is when a theory T, postulates entities which are explanatorily idle. Excising these entities from T produces a second theory, T*, which has the same theoretical virtues as T but a smaller set of ontological commitments. Hence, according to OR, it is rational to pick T* over T.”
We observe that nominals, representing individuals, denote only valid individual laws and nothing in the language described in the following section, is committed with nonvalid individual laws, according to the second ontological criterion above, we do not have to consider nonvalid individual laws. Technically speaking there is no element in the language able to denote an invalid individual law in any model of any theory. If something is a valid individual law regarded some legal system in some place in the world, then this individual belongs to our semantic universe.
This philosophical basis allows us to have only sets of valid individuals as semantics for theories. Thus, as the a reviewer have already observed, this implies that is the set of individual laws holding outside Brazil, and the classical negation is not adequate to denote this set. If we get meaning “individual laws that do not hold in Brazil”, the set of laws being a proper subset of the universe, and is the conjunctive property “laws + holds in ”. Then the complement, would be all elements of the universe which are either not a valid individual law or do not hold in Brazil. But there is no way to take the semantics in this way, for the semantics we get from the ontological commitment 2 from section 2 is given by “The individual valid laws holding outside of Brazil”.
Finally, concerning contradictory individual laws, they can coexist in the same universe, since they are there because they hold in distinct legal systems. In fact they are apparently contradictory. For example, “There is death penalty” and “Death sentence is not allowed” can coexist, since there are countries where each of these legal statement are valid. Concretely: “There is death penalty”:Iran and “Death sentence is not allowed”:Brazil.
4 The Logic
Classical Description Logic has been widely used as a basis for ontology creation and reasoning in many knowledge specific domains, including Legal AI.
An adequate intuitionistic semantics for negation in a legal domain comes to the fore when we take legally valid individual statements as the inhabitants of our legal ontology. This allows us to elegantly deal with particular situations of legal coherence, such as conflict of laws, as those solved by Private International Law analysis. In Haeusler, de Paiva Rademaker (2010, 2010); Haeusler, Paiva Rademaker (2010) we present an Intuitionistic Description Logic, called for Intuitionistic (for Attributive Language with Complements, the canonical classical description logic system). A labeled sequent calculus for based on a labeled sequent calculus for Rademaker (2012), was also presented. In these previous articles, we discussed the jurisprudence foundation of our system, and show how we can perform a coherence analysis of “Conflict of Laws in Space” by means of . This conflict happens when several laws can be applied, with different outcomes, to a case depending on the place where the case occurs. Typical examples are those ruling the rights of a citizen abroad.
In Haeusler, de Paiva Rademaker (2010), the semantics of is precisely provided and follows the framework for constructive modal logics presented by Simpson Simpson (1993) and adapted to description languages by Paiva de Paiva (2006). In the cited reference, we applied to the problem of formalizing legal knowledge.
Description Logics are an important knowledge representation formalism, unifying and giving a logical basis to the well known AI framebased systems of the eighties. Description logics are very popular right now. Given the existent and proposed applications of the Semantic Web, there has been a fair amount of work into finding the most wellbehaved system of description logic that has the broadest application, for any specific domain. Description logics tend to come in families of logical systems, depending on which concept constructors you allow in the logic. Since description logics came into existence as fragments of firstorder logic chosen to find the best tradeoff possible between expressiveness and tractability of the fragment, several systems were discussed and in the taxonomy of systems that emerged the has come to be known as the canonical one. The basic building blocks of description logics are concepts, roles and individuals. Concepts are described as unary predicates in usual firstorder logic and roles as binary atomic predicates used to modify the concepts.
As discussed in de Paiva (2006), considering versions of constructive description logics makes sense, both from a theoretical and from a practical viewpoint. There are several possible and sensible ways of defining constructive description logics, whether your motivation is natural language semantics, de Paiva (2006), or Legal AI, Haeusler, de Paiva Rademaker (2010). As far as constructive description logics are concerned, Mendler and Scheele have worked out a very compelling system Mendler Scheele (2010), based on the constructive modal logic CK Bellin . (2001)), one possible choice for us. However in this note we follow a different path and describe a constructive version of , based on the framework for constructive modal logics developed by Simpson (the system IK) in his phd thesis Simpson (1993) (For a prooftheoretic comparison between the constructive modal logics and one can see Ranalter (2010)).
Our motivation, besides Simpson’s work, is the framework developed by Braüner and de Paiva in Braüner de Paiva (2006) for constructive Hybrid Logics. We reason that having already frameworks for constructive modal and constructive hybrid logics in the labelled style of Simpson, we might end up with the best style of constructive description logics, in terms of both solid foundations and ease of implementation. Since submitting this paper we have been told about the master thesis of Clément Clément (2008) which follows broadly similar lines. Clément proves soundness and completeness of this system and then provides a focused version of it, a very interesting development, as focused systems are, apparently, very useful for proof search.
Building up from the Simpson’s constructive modal logics (called here IML), in Braüner de Paiva (2006), it is introduced intuitionistic hybrid logics, denoted by IHL. Hybrid logics add to usual modal logics a new kind of propositional symbols, the nominals, and also the socalled satisfaction operators. A nominal is assumed to be true at exactly one world, so a nominal can be considered the name of a world. If is a nominal and is an arbitrary formula, then a new formula called a satisfaction statement can be formed. The satisfaction statement expresses that the formula is true at one particular world, namely the world denoted by . In hindsight one can see that IML shares with hybrid formalisms the idea of making the possibleworld semantics part of the deductive system. While IML makes the relationship between worlds (e.g., ) part of the deductive system, IHL goes one step further and sees the worlds themselves as part of the deductive system, (as they are now nominals) and the satisfaction relation itself as part of the deductive system, as it is now a syntactic operator, with modalitylike properties.
Our Sequent Calculus for was first presented in de Paiva . (2010) where we briefly described the immediate properties of this system and most importantly we discuss a case study of the use of in legal AI.
A very importante obsertation is that this article corrects and extends the presentation of appearing in all previous articles. It points out the difference between and the intuitionistic hybrid logic presented in de Paiva (2006). Completeness and soundness proofs are revised. A discussion on the computational complexity of is also taken.
5 Intuitionistic ALC
The logic is based on the framework for intuitionistic modal logic IK proposed in Simpson (1993); FischerServi (1984); Plotkin Stirling (1986). These modal logics arise from interpreting the usual possible worlds definitions in an intuitionistic metatheory. As we will see in the following paragraphs, ideas from Braüner de Paiva (2006) were also used, where the framework IHL, for intuitionistic hybrid logics, is introduced. concepts are described as:
where stands for concepts, for an atomic concept, for an atomic role. We could have used distinct symbols for subsumption of concepts and the subsumption concept constructor but this would blowup the calculus presentation. This syntax is more general than standard since it includes subsumption as a conceptforming operator. We have no use for nested subsumptions, but they do make the system easier to define, so we keep the general rules. Negation could be defined via subsumption, that is, , but we find it convenient to keep it in the language. The constant could also be omitted since it can be represented as .
A constructive interpretation of is a structure consisting of a nonempty set of entities in which each entity represents a partially defined individual; a refinement preordering on , i.e., a reflexive and transitive relation; and an interpretation function mapping each role name to a binary relation and atomic concept to a set which is closed under refinement, i.e., and implies . The interpretation is lifted from atomic concepts to arbitrary concepts via:
Following the semantics of IK, the structures are models for if they satisfy two frame conditions:
 F1

if and then and
 F2

if and then and
The above conditions are diagrammatically expressed as:
Our setting simplifies Mendler Scheele (2010), since satisfies (like classical ) and .
In contrast with the above mentioned approaches, ours assign a truth values to some formulas, also called assertions, they are not concepts as in Braüner de Paiva (2006), for example. Below we define the syntax of general assertions () and nominal assertions () for ABOX reasoning in . Formulas () also includes subsumption of concepts interpreted as propositional statements.
where and are nominals, is a role symbol and is a concept. In particular, this allows , which is a perfectly valid nominal assertion.
Definition 1 (outer nominal)
In a nominal assertion , is said to be the outer nominal of this assertion. That is, in an assertion of the form , is the outer nominal.
We write to abbreviate which means that entity satisfies concept in the interpretation ^{6}^{6}6In IHL, this is a world and this satisfaction relation is possible world semantics. Further, is a model of , written iff , we have that . Finally, means , we have that . All previous notions are extended to sets of concepts in the usual universal fashion. Given the hybrid satisfaction statements, the interpretation and semantic satisfaction relation are extended in the expected way. The statement holds, if and only if, , we have that, . In a similar fashion, holds ,if and only if, . That is, the evaluation of the hybrid formulas does not take into account only the world , but it has to be monotonically preserved. It can be observed that for every , if and , then is a property holding on this satisfaction relation.
In common reasoning tasks the interpretation and the entity in a verification goal such as are not given directly but are themselves axiomatized by sets of concepts and formulas. Usually we have a set ^{7}^{7}7Here we consider only acycled TBox with and . of formulas and the set of concepts. Accordingly:
Definition 2
We write if it is the case that:
(1) 
where
denotes a vector of variables
and is the vector of all outer nominals occurring in each nominal assertion of . is the only outer nominal of a nominal assertion , while a (pure) concept has no outer nominal.A Hilbert calculus for is provided following Plotkin Stirling (1986); Simpson (1993); FischerServi (1984). It consists of all axioms of intuitionistic propositional logic plus the axioms and rules displayed in Figure 1. The Hilbert calculus implements TBoxreasoning. That is, it decides the semantical relationship . has only formulas as members.
A Sequent Calculus for is also provided. The logical rules of the Sequent Calculus for are presented in Figure 2. ^{8}^{8}8The reader may want to read Proof Theory books, for example, Takeuti (2013); Buss (1998); Negri Von Plato (2008); Girard . (1989). The structural rules and the cut rule are omitted but they are as usual. The stands for concepts or assertions ( or ), and for concept and for role. is a set of formulas. In rules p and p, the syntax means , that is, all concepts in are universal quantified with the same role. The assertions in are kept unmodified. In the same way, in rule pN the addition of the nominal is made only in the concepts of (and in if that is a concept) keeping the assertions unmodified.
The propositional connectives () rules are as usual, the rule r is omitted. The rules are presented without nominals but for each of these rules there is a counterpart with nominals. For example, the rule r has one similar: nr
The main modification comes for the modal rules, which are now role quantification rules. We must keep the intuitionistic constraints for modal operators. Rule l has the usual condition that is not in the conclusion. Concerning the usual condition on the r rule, it is not the case in this system, for the interpretation of the a nominal assertion in a sequent is already implicitly universal (Definition 2).
r  l 
r  l 
r  l 
r  l 
r  l 
p  p 
pN 
Theorem 1
The completeness of our system is proved relative to the axiomatization of , shown in Figure 1. The proof is presented in Section 6.
The soundness of the system is proved directly from the semantics of including the ABOX, that is, including nominals. The semantics of a sequent is defined by the satisfaction relation, as shown in Definition 2. The sequent is valid if and only if . Soundness is proved by showing that each sequent rule preserves the validity of the sequent and that the initial sequent is valid. This proof is presented in Section 7.
We note that although we have here fixed some inaccuracies in the presentation of the semantics in de Paiva . (2010), the system presented here is basically the same, excepted that here the propositional rules are presented without nominals. Given that, the soundness of the system proved in de Paiva . (2010) can be still considered valid without further problems. Note also that the proof of soundness provides in Section 7 is regarded the full language of . It considers nominals and assertion on nominals relationship, that is it concerns ABOX and TBOX. The proof of completeness is for the TBOX only. A proof of completeness for ABOX can be done by the method of canonical models. For the purposes of this article, we choose to show the relative completeness proof with the sake of showing a simpler proof concerning TBOX.
6 The completeness of system
We show the relative completeness of regarding the axiomatic presentation of presented in Figure 1. To prove the completeness of it is sufficient to derive in the axioms 1–5 of . It is clear that all substitution instances of IPL theorems can also be proved in using only propositional rules. The MP rule is a derived rule from the using the cut rule. The Nec rule is the p rule in the system with empty. In the first two proofs below do not use nominals for given better intuition of the reader about the use of rules with and without nominals.
Axiom 1: l p r
Axiom 2: l p r
Axiom 3: l r
Axiom 4: r
r l
Axiom 5: r l l r r r
7 Soundness of system
In this section we prove that.
Proposition 1
If is provable in then .
Proof: We prove that each sequent rule preserves the validity of the sequent and that the initial sequents are valid. The definition of a valid sequent () is presented in Definition 2.
The validity of the axioms is trivial. We first observe that any application of the rules r, l,r,l, r,r, l of where the sequents do not have any nominal, neither in nor in , is sound regarded intuitionistic propositional logic kripke semantics, to which the validity definition above collapses whenever there is no nominal in the sequents. Thus, in this proof we concentrate in the case where there are nominals. We first observe that the nominal version of r, the validity of the premises includes
This means that holds in any worlds for the vector of nominals occurring in . This includes the outer nominal in (if any). In this case the semantics of is preserved, since includes . With the sake of a more detailed analysis, we consider the following instance:
r
Consider an structure In this case, for any and any if , , such that, and , we have that , since the premise is valid, by hypothesis. In this case, by the semantics of we have . The conclusion of the rule is valid too.
The argument shown above for the r rule is analogous for the nominal versions of r, l,r,l, r,r, l. Consider the rule r.
r
Since the premise is valid we have that if , , then . This entails that , for . We observe that by the restriction on the rule application, does not occur in , it only occurs in and . The truth of these formulas are subsumed by . The conclusion does not need to consider them any more. The conclusion is valid too. Another way to see its soundness is to prove that if is valid, then so is . This can be show by the following reasoning:
that is the same as:
Using the fact that , we obtain:
The above condition states that is valid.
Consider the rule l:
l
As in the r case, we analyze the simplest validity preservation: if is valid, then so is . The first condition is:
(2) 
Using , eliminating from the term, and, using the fact that is valid, iff, , we obtain
(3) 
Consider the semantics of :
and the following rule:
r
We can see that the premises of the rule entails the conclusion. The premises correspond to the following conditions:
and
Instantiating in both conditions and , this yields , such that , so . Thus, r is sound. The soundness of l is analogous to l.
Finally, it is worth noting that, for each rule, we can derive the soundness of its nonnominal version from the proof of soundness of its nominal version. For instance, the soundness of the nominal version of rule l depends on the diamond conditions F1 and F2. The soundness of its nonnomimal version, is a consequence of the soundness of the nominal version.
The rules below have their soundness proved as a consequence of the following reasonings in firstorder intuitionistic logic that are used for deriving the semantics of the conclusions from the semantics of the premises:
 (p)

;
 (p)

implies ;
 (pN)

if then for every Kripke model and world , if then .
p  p  pN 
8 Solving Chisholm and other contrarytoduty paradoxes in iALC
The paradoxes discussed in this section are known from the literature as contrarytoduty paradoxes. They are deontic paradoxes under SDL formalization. Usually, there is a primary norm/law/obligation and a secondary norm that comes to effect when the primary obligation is violated. The form of these normative and intuitively coherent situations are in general hard to find a consistent deontic formalization. Because of that they are called paradoxes. A typical example of contrarytoduty paradox appeared in M. (1963):

It ought to be that Jones goes to the assistance of his neighbors.

It ought to be that if Jones does go then he tells them he is coming.

If Jones doesn’t go, then he ought not tell them he is coming.

Jones doesn’t go.

This certainly appears to describe a possible situation. 14 constitute a mutually consistent and logically independent set of sentences.

(1) is a primary obligation, what Jones ought to do unconditionally. (2) is a compatiblewithduty obligation, appearing to say (in the context of 1) what else Jones ought to do on the condition that Jones fulfills his primary obligation. (3) is a contrarytoduty obligation (CTD) appearing to say (in the context of 1) what Jones ought to do conditional on his violating his primary obligation. (4) is a factual claim, which conjoined with (1), implies that Jones violates his primary obligation.
We firstly remember the deontic approch to law and its logic. Differently of ours, it takes laws as propositions. Thus, a norm or law is an obligatory proposition, such as “You must pay your debits” or “It is obligatory to pay the debits”. As a proposition each norm has a truth value. The underlying logic classical. If is a proposition then is a proposition too. intuitively means must be the case ,or It is obligatory that . The paradoxes that we discuss in this work appear just when laws are taken as propositions. They show them up from the most basic deontic logic Standard Deontic Logic (SDL). SDL is a modal logic defined by von Wright19951 Wright (1951) and, according to the modal logic terminology on the names of axioms, it is defined by the following set of axioms. The formulas of SDL include the modality .
 TAUT

all tautologies of the language. This means that if is a propositional tautology then the substitution of for any SDL formula is an SDL tautology too;
 OBK

 OBD

 MP

if and then
 OBNEC

if then
SDL is just the normal modal logic D or KD, with a suggestive notation expressing the intended interpretation. From these, we can prove the principle that obligations cannot conflict, NC of SDL, , see Wright (1951).
The following set of formulas is a straightforward formalization of Chisholm paradox in SDL.
The intuitive meaning of each formula is according the table shown in figure 3 where is “Jones go to the assistance of his neighbours” and is “Jones tells his neighbours he is going”.
Assertion  SDL Formula 

It ought to be that Jones go to  
the assistance of his neighbours.  
It ought to be that if Jones does go then  
he tells them he is going.  
If Jones doesn’t go, then  
he ought not tell them he is going.  
Jones doesn’t go. 
Using the deductive power of SDL we can perform the following derivation of a SDL contradiction.

from (2) by principle OBK we get ,

and then from (1) by MP, we get ;

but by MP alone we get from (3) and (4).

From these two conclusions, by PC, we get , contradicting NC of SDL.
Assertion 14, from Chisholm paradox, leads to inconsistency per SDL. But, 14 do not seem inconsistent at all, the representation cannot be a faithful one. We discuss this in the sequel. For reasons that will become clear, we take Chisholm paradox as stated above in natural language, instead of its SDL version. We use the same letters to denote the propositions/laws as used in the deontic representation of the paradox, for a better comparison.
In first law in the paradox, i.e., the law state in item 1 is a nominal in , and hence it is a Kripke world in our model. The same can be said about item 2. The stateofaffairs, expressed in , is simply the assertions: and . Note that this assertions only state that there are two laws and in the legal universe. Since a Kripke model for intuitionistic logic is a Heyting algebra, and hence it is a lattice too, there must be the meet of these two worlds. This is represented in the model by law , intuitively stating that it is obligatory to do what law and law state. Item 3 of the paradox is a conditional that generally states that if some proposition is truth then some law exists. This is a rather hard expression in judicial terms. Laws exist by promulgation only, they do not have their existence conditioned to anything but their own promulgation. This conditional expression can be raised in a legislative discussion only. But even in this exceptional case, the raising of paradoxes, as the one under discussion, advices that such use should be avoided. What item 3 says, instead, is simply that holds in the world that is the law cited as the consequent of the conditional. Finally, as the model is a lattice, there must be a world that represents the law that it is the conjunctive law related to and , in the same way is related to and . Now, in , it is ensured that holds by the intuitionistic interpretation of the negation. As a result the model depicted in the diagram below is a model for what is known by Chisholm paradox. Thus, it is not a paradox when expressed in in a kelsenian way.

The law , originally

The law , originally

From (3), , we have . If we had the translation would be the same. That is, is .

The law that represents the infinum of and .
The diagram in figure 4, shows the Kripke model to Chisholm paradox discussed above. Remember that if then .
Only to estimate the range of our approach for solving semantical (contrarytoduty) paradoxes. The free choice permission paradox reported on
Ross (1941)) is derived on the supposition that there is a logic of norms, as for example SDL. The dilemma on the existence of such a logic of normas is also known as Jorgensen dilemma. At the next section we draw the conclusion that is more likely to such a logic than SDL. In fact in the sequel we detail the free choice permission paradox in SDL and we can see that the steps based on the axiom cannot be derived inside .Consider the tautology . We have by necessity SDL rule that:
is derivable in SDL. By the axiom :
is also derivable. On the other hand, by contrapositive on the first sentence, we have
, so, and hence
by necessitation and again. Thus, by contrapositive again, we obtain
Taking as permitted , rather better , we draw:
The free choice permission states that if holds then . Consider that we accept the free choice permission, so to say the formula
As , then by what was discussed above, , then by the free choice permission, we draw that
for any . In summary, in the presence of the free choice permission axiom, we can derive that for every , which should not happen. If we use together with Kelsen jurisprudence, and hence intuitionistic logic, we cannot derive all the steps above, for cannot be used. Anyway, our definition of permission is rather different from what was used in this paragraph, see section 9. We can conclude that many paradoxes that are based on the axiom are not legal paradoxes in a Kelsenian formalization of legal situations in anymore.
9 Conclusion
In this article, we shown how intuitionistic logic and Kelsen’s jurisprudence can be used to express Chisholm paradox faithfully. A key fact in providing a logical model to this paradox is that laws/norms are not taken as propositions. For example, in the explanation above on building the model, if we turn back to deontic expression of laws, we will have that is and is , but we cannot derive that is . is of course the meet () between and , as a meet it is strongly connection to , which is a SDL valid formula. Thus, is the norm , that is an obligation. However, now remembering what norms and are in this particular case, is the meeting that it is . This conclusion, however, does not entail that in can be identified with , since our implication is the intuitionistic implication. This very last aspect of joining Kelsen jurisprudence and also helps to avoid other deontic paradoxes.
Jorgensen’s Dilemma Jorgensen (1937) offers a question, in fact, a dilemma, whether there is, in fact, any deontic logic. The question follows this path: 1) Norms/laws deal with evaluative sentences; 2) Evaluative sentences are not the kind of sentence that can be true or false; 3) Thus, how there is a logic of evaluative sentences? 4) Logic has as goal to define what can be drawn from whatever, and; 5) A sentence follows from a set of sentences on a basis of the relationship between the truth of the sentences in question. Thus, there is no deontic logic. What we have shown in this article, is that deontic logic is possible by considering the logic of norms as a logic on norms, instead. This reading is just what we do in legal ontologies.
We have to touch some aspects that are very wellknown in the deontic approach. One is the deontic concept of permission. This case is modeled by observing that in a society regulated by law, permission is nothing more than an obligation of the State. The State promulgates what is allowed. Concerning prohibitions, the foundation is analogous. However, some subtle and theoretical problems may arise if one wants to recover the definition of forbidden () regarding the very wellknown duality . This discussion will be the subject of another article.
Finally, we would like to comment that professor Farinas del Cerro taught us that the research on logic and AI, mainly the first should be approached by solving partbypart the problem and elegantly putting everything together. Well, we learned the first part of this technique, by following him by reading his articles. We think that in the first part, so to say finding a good foundation on legal ontologies, one that comes from the domain itself, namely Kelsen jurisprudence. Concerning the second part, that is to put everything together in an elegant way, we known that we are far from it. Another thing that we might have learned from prof. Farinas is that in this case, it is a matter of time to have the work in a more mature stage. We hope we reach this stage.
References
 Alchourron Martino (1990) AlchourronMartino1990Alchourron, CE. Martino, A. 1990. Logic without Truth Logic without truth. Ratio Juris3146–67.
 Bellin . (2001) bellinetalBellin, G., de Paiva, V. Ritter, E. 2001. Extended CurryHoward Correspondence for a Basic Constructive Modal Logic Extended curryhoward correspondence for a basic constructive modal logic. Procs of Methods for the Modalities. Procs of methods for the modalities.
 Braüner de Paiva (2006) braunerdepaiva2006Braüner, T. de Paiva, V. 2006. Intuitionistic hybrid logic Intuitionistic hybrid logic. JAL43231–255.
 Buss (1998) buss1998Buss, SR. 1998. Handbook of proof theory Handbook of proof theory. Elsevier.
 Clément (2008) clement08Clément, I. 2008. Proof Theoretical Foundations for Constructive Description Logic Proof theoretical foundations for constructive description logic . MontrelSchool of Computer Science. McGill University.
 de Paiva (2006) depaiva2003de Paiva, V. 2006. Constructive Description Logics: what, why and how Constructive description logics: what, why and how. Context Representation and Reasoning. Context representation and reasoning.
 de Paiva . (2010) PHR:2010de Paiva, V., Hausler, EH. Rademaker, A. 2010. Constructive Description Logic: HybridStyle Constructive description logic: Hybridstyle. Proceedings of Hybrid Logic and Applications (HyLo 2010). Proceedings of hybrid logic and applications (HyLo 2010).
 FischerServi (1984) Fisher1984FischerServi, G. 1984. Axiomatizations for some Intuitionistic Modal Logics Axiomatizations for some intuitionistic modal logics. Rendiconti del Seminario Matematico Università e Politecnico di Torino42.
 Girard . (1989) girard1989Girard, J., Taylor, P. Lafont, Y. 1989. Proofs and Types Proofs and types. Cambridge University Press.
 Haeusler, de Paiva Rademaker (2010) HPR:2010bHaeusler, EH., de Paiva, V. Rademaker, A. 2010. Intuitionistic Logic and Legal Ontologies Intuitionistic logic and legal ontologies. Proc. JURIX 2010 Proc. jurix 2010 ( 155–158). IOS Press.
 Haeusler, de Paiva Rademaker (2010) HPR:2010abHaeusler, EH., de Paiva, V. Rademaker, A. 2010. Using Intuitionistic Logic as a basis for Legal Ontologies Using intuitionistic logic as a basis for legal ontologies. Proceedings of the 4th Workshop on Legal Ontologies and Artificial Intelligence Techniques Proceedings of the 4th workshop on legal ontologies and artificial intelligence techniques ( 69–76). Fiesole, Florence, ItalyEuropean University Institute.
 Haeusler . (2011) HPR:2011Haeusler, EH., de Paiva, V. Rademaker, A. 2011. Intuitionistic Description Logic and Legal Reasoning Intuitionistic description logic and legal reasoning. Proceedings of International Workshop Data, Logic and Inconsistency with DEXA 2011. Proceedings of international workshop data, logic and inconsistency with dexa 2011.
 Haeusler, Paiva Rademaker (2010) HPR:2010aHaeusler, EH., Paiva, VD. Rademaker, A. 2010. Using Intuitionistic Logic as a Basis for Legal Ontologies Using intuitionistic logic as a basis for legal ontologies. Informatica e DirittoXIX12289–298.
 Hughes (1971) GrahamHughes1971Hughes, G. 1971. Validity and the Basic Norm Validity and the basic norm. California Law Review593695–715.
 Jorgensen (1937) Jorgensen1937Jorgensen, J. 1937. Imperatives and Logic Imperatives and logic. Erkenntnis7288–296.
 Kelsen (1946) KelsenGTLSKelsen, H. 1946. General Theory of Law and State General theory of law and state. Harvard University Press. Translation from german by Wedberg, A. and Kraus, W.H.
 Kelsen (1967) KelsenRRIIKelsen, H. 1967. Pure theory of law Pure theory of law [Book]. Berkeley : University of California Press. Translated from Reine Rechtslehre, second edition, 1960
 Kelsen (1991) Kelsen1991Kelsen, H. 1991. General theory of norms General theory of norms. USAOxford Univ. Press.
 M. (1963) Chisholm1963M., CR. 1963. ContrarytoDuty Imperatives and Deontic Logic Contrarytoduty imperatives and deontic logic. Analysis2433–36.
 M.E. Coniglio (2009) ConiglioNewtonM.E. Coniglio, NP. 2009. A Paraconsistent Approach to Chisholm’s Paradox A paraconsistent approach to chisholm’s paradox. Principia133299–326.
 Mendler Scheele (2010) mendlerscheeleMendler, M. Scheele, S. 2010. Towards constructive DL for abstraction and refinement Towards constructive DL for abstraction and refinement. JAR443207–243. Proc. 21st International DL Workshop
 Negri Von Plato (2008) negri2008Negri, S. Von Plato, J. 2008. Structural proof theory Structural proof theory. Cambridge University Press.
 Plotkin Stirling (1986) plotkin1986Plotkin, G. Stirling, C. 1986. A framework for intuitionistic modal logics: extended abstract A framework for intuitionistic modal logics: extended abstract. Proceedings of the 1986 conference on Theoretical aspects of reasoning about knowledge Proceedings of the 1986 conference on theoretical aspects of reasoning about knowledge ( 399–406). San Francisco, CA, USAMorgan Kaufmann Publishers Inc. http://dl.acm.org/citation.cfm?id=1029786.1029823
 Rademaker (2012) Rademaker2012Rademaker, A. 2012. A Proof Theory for Description Logics A proof theory for description logics. Springer.
 Ranalter (2010) ranalter09Ranalter, K. 2010. Embedding Constructive into Intuitionistic Embedding constructive into intuitionistic . Electr. Notes Theor. Comput. Sci.262205–219.
 Raz (1972) Raz1972Raz, J. 1972. Legal Principles and the Limits of Law Legal principles and the limits of law. Yale Law Journal81823–854.
 Ross (1941) Ross1941Ross, A. 1941. Imperatives and Logic Imperatives and logic. Theoria, 7.
 Simpson (1993) simpson95Simpson, A. 1993. The Proof Theory and Semantics of Intuitionistic Modal Logic The proof theory and semantics of intuitionistic modal logic . University of Edinburgh.
 Takeuti (2013) takeuti2013Takeuti, G. 2013. Proof theory Proof theory. Courier Dover Publications.

Valente (1995)
Valente1995Valente, A.
1995.
Legal knowledge engineering: A modelling approach Legal knowledge engineering: A modelling approach.
IOS Press. Amsterdam  Wright (1951) vonWright1951Wright, GV. 1951. Deontic Logic Deontic logic. Mind601–15.
Comments
There are no comments yet.