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# Intuitionistic Description Logic for Legal Reasoning

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Presented at EBL 2011 (http://www.cle.unicamp.br/ebl2011/).

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### Intuitionistic Description Logic for Legal Reasoning

1. 1. Intuitionistic Description Logic for Legal Reasoning Intuitionistic Description Logic for Legal Reasoning Edward Hermann Hausler Alexandre Rademaker Valeria de Paiva Departamento de Informática - PUC-Rio - Brasil FGV - Brasil Univ. Birmingham - UK EBL 2011 May
2. 2. Intuitionistic Description Logic for Legal Reasoning Jurisprudence MotivationConsiderations on Legal Ontologies Roles played by a Knowledge Representation artifact/formalism (Davis et alli) It is a set of ontological commitments; It is a fragmentary theory of intelligent reasoning; It is a medium for efﬁcient computation; It is a medium of human expression.
3. 3. Intuitionistic Description Logic for Legal Reasoning Jurisprudence MotivationConsiderations on Legal Ontologies What is an Ontology? A declarative description of a domain. Concretely, an Ontology is a Knowledge Base: A set of Logical Assertions that aims to describe a Domain completely. Consistency is mandatory. Consistency means absence of contradictions. Negation has an essential role.
4. 4. Intuitionistic Description Logic for Legal Reasoning Jurisprudence MotivationConsiderations on Legal Ontologies Main Motivation Solid Jurisprudence + Description Logic
5. 5. Intuitionistic Description Logic for Legal Reasoning Jurisprudence MotivationConsiderations on Legal Ontologies What does the term “Law” mean? What does count as the “unit of law”? Open question, a.k.a. “The individuation problem”. (Raz1972) What is to count as one “complete law”?
6. 6. Intuitionistic Description Logic for Legal Reasoning Jurisprudence MotivationConsiderations on Legal Ontologies Under Legal Positivism: Two main approaches to the “Individuation problem”. 1. Taking the collection of laws as a whole. A law, or general law, is a kind of deontic statement or proposition. 2. Taking into account all individual legal valid statements (ivls or vls for short) as individual laws. An individual law is not a deontic statement, it is not even a proposition. “The law” is the collection of all individual laws.
7. 7. Intuitionistic Description Logic for Legal Reasoning Jurisprudence MotivationConsiderations on Legal Ontologies Formalization of Legal Ontologies following the second approach The ﬁrst-class citizens of any Legal Ontology are vls. Only vls inhabit legal world. Inﬂuence of Kelsen’s characterization of law. There can be concepts on vls and relationships between vls. For example: PILBR , CIVIL, FAMILY , etc, can be concepts. LexDomicilium can be a relationship, a.k.a. a legal connection. £ Facilitates the analysis of structural relationships between laws, viz. Primary and Secondary Rules. Induces natural precedence between laws, e.g. “ Peter is liable” precedes “Peter has a renting contract”.
8. 8. Intuitionistic Description Logic for Legal Reasoning Logical MotivationIntuitionistic versus Classical logic:Which version is more adequate to Law Formalization?? Classical Negation classiﬁes: ¬φ ∨ φ is valid for any φ In BR, 18 is the legal age BR contains all vls in Brazil . “Peter is 17” “Peter is liable”∈ BR iff “Peter is liable”∈ ¬BR Classical negation forces the existence of a liable Peter in some legal system outside Brazil
9. 9. Intuitionistic Description Logic for Legal Reasoning Logical MotivationIntuitionistic versus Classical logic:Which version is more adequate to Law Formalization?? The Intuitionistic Negation |=i ¬A, iff, for all j, if i j then |=j A i ~ 2 Ø |=j A |=k A |=i ¬¬A → A and |=i A ∨ ¬A
10. 10. Intuitionistic Description Logic for Legal Reasoning Logical MotivationIntuitionistic versus Classical logic:Which version is more adequate to Law Formalization?? An Intuitionistically based approach to Law “Peter is liable”∈ BR There is no vls in BR “Peter is liable” ∈ ¬BR means dominating “Peter is liable” neither “Peter is liable”∈ BR nor “Peter is liable”∈ ¬BR
11. 11. Intuitionistic Description Logic for Legal Reasoning Intuitionistic Description LogicsThe logical framework for ontologies formalization iALC and ALC have the same logical language Binary (Roles) and unary (Concepts) predicate symbols, R(x, y ) and C(y ). Prenex Guarded formulas (∀y (R(x, y ) → C(y )), ∃y (R(x, y ) ∧ C(y ))). Essentially propositional (Tboxes), but may involve reasoning on individuals (Aboxes), expressed as “x : C” and xRy . Semantics: Provided by a structure I = (∆I , I , ·I ) closed under reﬁnement, i.e., y ∈ AI and x I y implies x ∈ AI . “¬” and “ ” must be interpreted intuitionistically . It is not First-order Intuitionistic Logic. It is a genuine Hybrid logic.
12. 12. Intuitionistic Description Logic for Legal Reasoning Intuitionistic Description LogicsDeductive Reasoning in iALC ∆⇒ x :A A⇒ B Usual Structural-Rules for Intuitionistic Logic ∈-r ∆⇒ x :B Γ, x : C ⇒ x : C xRy , Γ ⇒ xRy Γ1 ⇒ C Γ2 , D ⇒ δ Γ, C ⇒ D -l -r Γ1 , Γ2 , C D⇒ δ Γ⇒ C D Γ, x : C, x : D ⇒ δ Γ ⇒ x: C Γ ⇒ x: D -l -r Γ, x : (C D) ⇒ δ Γ ⇒ x : (C D) Γ, x : C ⇒ δ Γ, x : D ⇒ δ Γ ⇒ x: C -l 1 -r Γ, x : (C D), ⇒ δ Γ ⇒ x : (C D) Γ, x : ∀R.C, y : C, xRy ⇒ δ Γ, xRy ⇒ y : C ∀-l ∀-r Γ, x : ∀R.C, xRy ⇒ δ Γ ⇒ x : ∀R.C Γ, xRy , y : C ⇒ δ Γ ⇒ xRy Γ ⇒ y: C ∃-l ∃-r Γ, x : ∃R.C ⇒ δ Γ ⇒ x : ∃R.C
13. 13. Intuitionistic Description Logic for Legal Reasoning A Case AnalysisUsing iALC to formalize Conﬂict of Laws in Space A Case Study Peter and Maria signed a renting contract. The subject of the contract is an apartment in Rio de Janeiro. The contract states that any dispute will go to court in Rio de Janeiro. Peter is 17 and Maria is 20. Peter lives in Edinburgh and Maria lives in Rio. Only legally capable individuals have civil obligations: PeterLiable ContractHolds@RioCourt, shortly, pl cmp MariaLiable ContractHolds@RioCourt, shortly, ml cmp Concepts, nominals and their relationships BR is the collection of Brazilian Valid Legal Statements SC is the collection of Scottish Valid Legal Statements PILBR is the collection of Private International Laws in Brazil ABROAD is the collection of VLS outside Brazil LexDomicilium is a legal connection: Legal Connections The pair pl, pl is in LexDomicilium
14. 14. Intuitionistic Description Logic for Legal Reasoning A Case AnalysisNon-Logical Axiom Sequents The sets ∆, of concepts, and Ω, of iALC sequents representing the knowledge about the case ml : BR pl : SC pl cmp ∆= ml cmp pl LexDom pl PILBR ⇒ BR Ω= SC ⇒ ABROAD ∃LexDom.ABROAD ∃Lexk .Lk . . . ⇒ PILBR
15. 15. Intuitionistic Description Logic for Legal Reasoning A Case AnalysisIn Sequent Calculus Ω ∆ ⇒ pl : SC pl : SC ⇒ pl : A ∃LexD.A ⇒ ∃LexD.A Ω cut -R ∆ ⇒ pl : A ∆ ⇒ pl LexD pl ∃LexD.A ⇒ PILBR PILBR ⇒ BR ∃−R cut ∆ ⇒ pl : ∃LexD.A ∃LexD.A ⇒ BR inc − R ∆ ⇒ pl : BR Ω Π ∆ ⇒ pl : BR ml : BR, pl : BR ⇒ cmp : BR cut ∆ ⇒ ml : BR ∆, ml : BR ⇒ cmp : BR cut ∆ ⇒ cmp : BR
16. 16. Intuitionistic Description Logic for Legal Reasoning Related approachesRelated approaches Deontic Logic (since von Wright 1951, Mally 1926). PSPACE-complete: SDL (KD and KDn ), GDL (NSDL and PSDL) and Propositional − BOID Undecidable: LORA, BOID and all FOL versions of KD and its extensions. Limited to deal with contrary-to-duty paradoxes. Defeasible Logic (Sartor 1991). Complexity goes from LinearTime (Propositional) to Undecidability of rule applications. It deals better with contrary-to-duty paradoxes. Is LinearTime regarding a quite restrict logic language. iALC PSPACE-complete, non-FOL based ability to express individuals. Natural precedence can be used detour the contrary-to-duty paradoxes. Seems to expresses same the way that dyadic GDL deals with paradoxes. Defeasible Logic and iALC validate the same VLS, but iALC cannot express the dynamics of a trial.
17. 17. Intuitionistic Description Logic for Legal ReasoningContrary-to-duty paradoxes Consequence of conﬂicting norms It ought to be that Jones go to the assistance of his neighbours. Ob(φ) It ought to be that if Jones does go then he tells them he is coming. Ob(φ → ψ) If Jones doesn’t go, then he ought not tell them he is coming. ¬φ → Ob(¬ψ) Jones doesn’t go. ¬φ
18. 18. Intuitionistic Description Logic for Legal Reasoning Conclusion and Future Work Summary of the Approach Individual Legal Valid Statements are the individuals of the universe. Concepts are Classes of individual laws. Roles (relationships) between individual laws denote kinds of Legal Connections Subsumptions and Negations are intuitionistically interpreted (iALC) Does it avoid the constrary-to-duty paradoxes of the Deontic approach?
19. 19. Intuitionistic Description Logic for Legal Reasoning Conclusion and Future WorkConclusions Seems to be adequate to one jurisprudence theory. Juridic cases can be analyzed in the ABOX. TBOX describes “The Law”. is not always speciﬁed at the level of the TBOX. It seems to scale, but there is no empirical evidence. Is the coherence analysis easier? (PSPACE-complete) (?) Investigate some “hard juridical cases”. (?) Can be the kernel of a tool for helping with a judge’s decision (not a sentence writer!!!)
20. 20. Intuitionistic Description Logic for Legal Reasoning Conclusion and Future Work Thanks!