Intuitionistic Description Logic for Legal Reasoning

Intuitionistic Description Logic for Legal Reasoning

Intuitionistic Description Logic for LegalReasoning

Edward Hermann Hausler Alexandre RademakerValeria de Paiva

Departamento de Informática - PUC-Rio - BrasilFGV - Brasil

Univ. Birmingham - UK

EBL 2011 May

Intuitionistic Description Logic for Legal Reasoning

Jurisprudence Motivation

Considerations on Legal Ontologies

Roles played by a Knowledge Representation artifact/formalism(Davis et alli)

I It is a set of ontological commitments;I It is a fragmentary theory of intelligent reasoning;I It is a medium for efficient computation;I It is a medium of human expression.

Intuitionistic Description Logic for Legal Reasoning

Jurisprudence Motivation

Considerations on Legal Ontologies

What is an Ontology?

I A declarative description of a domain.I Concretely, an Ontology is a Knowledge Base: A set of

Logical Assertions that aims to describe a Domaincompletely.

I Consistency is mandatory.I Consistency means absence of contradictions.I Negation has an essential role.

Intuitionistic Description Logic for Legal Reasoning

Jurisprudence Motivation

Considerations on Legal Ontologies

Main Motivation

Solid Jurisprudence + Description Logic

Intuitionistic Description Logic for Legal Reasoning

Jurisprudence Motivation

Considerations on Legal Ontologies

What does the term “Law” mean?

I What does count as the “unit of law”? Open question,a.k.a. “The individuation problem”.

I (Raz1972) What is to count as one “complete law”?

Intuitionistic Description Logic for Legal Reasoning

Jurisprudence Motivation

Considerations 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 generallaw, 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 isnot a deontic statement, it is not even a proposition. “Thelaw” is the collection of all individual laws.

Intuitionistic Description Logic for Legal Reasoning

Jurisprudence Motivation

Considerations on Legal Ontologies

Formalization of Legal Ontologies following the secondapproach

I The first-class citizens of any Legal Ontology are vls. Onlyvls inhabit legal world. Influence of Kelsen’scharacterization of law.

I There can be concepts on vls and relationships betweenvls. For example: PILBR, CIVIL, FAMILY , etc, can beconcepts. LexDomicilium can be a relationship, a.k.a. alegal connection.

I Facilitates the analysis of structural relationshipsbetween laws, viz. Primary and Secondary Rules. Inducesnatural precedence between laws, e.g. “ Peter is liable”precedes “Peter has a renting contract”.

Intuitionistic Description Logic for Legal Reasoning

Logical Motivation

Intuitionistic 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”6∈ BR iff “Peter is liable”∈ ¬BR

Classical negation forces the existence of a liablePeter in some legal system outside Brazil

Intuitionistic Description Logic for Legal Reasoning

Logical Motivation

Intuitionistic versus Classical logic:

Which version is more adequate to Law Formalization??

The Intuitionistic Negation|=i ¬A, iff, for all j , if i j then 6|=j A

i

~~

|=j A

6|=k A

6|=i ¬¬A→ A and 6|=i A ∨ ¬A

Intuitionistic Description Logic for Legal Reasoning

Logical Motivation

Intuitionistic versus Classical logic:

Which version is more adequate to Law Formalization??

An Intuitionistically based approach to Law

“Peter is liable”6∈ BR

“Peter is liable” ∈ ¬BR means There is no vls in BRdominating “Peter is liable”

neither “Peter is liable”6∈ BR nor “Peter is liable”∈ ¬BR

Intuitionistic Description Logic for Legal Reasoning

Intuitionistic Description Logics

The logical framework for ontologies formalizationiALC and ALC have the same logical language

I Binary (Roles) and unary (Concepts) predicate symbols,R(x , y) and C(y).

I Prenex Guarded formulas (∀y(R(x , y)→ C(y)),∃y(R(x , y) ∧ C(y))).

I Essentially propositional (Tboxes), but may involvereasoning on individuals (Aboxes), expressed as “x : C”and xRy .

I Semantics: Provided by a structure I = (∆I ,I , ·I) closedunder refinement, i.e., y ∈ AI and x I y implies x ∈ AI .“¬” and “v” must be interpreted intuitionistically .

I It is not First-order Intuitionistic Logic. It is a genuineHybrid logic.

Intuitionistic Description Logic for Legal Reasoning

Intuitionistic Description Logics

Deductive Reasoning in iALC

Usual Structural-Rules for Intuitionistic Logic∆⇒ x : A A⇒ B ∈-r

∆⇒ x : B

Γ, x : C ⇒ x : C xRy, Γ⇒ xRy

Γ1 ⇒ C Γ2,D ⇒ δv-l

Γ1, Γ2,C v D ⇒ δ

Γ,C ⇒ Dv-r

Γ⇒ C v D

Γ, x : C, x : D ⇒ δu-l

Γ, x : (C u D)⇒ δ

Γ⇒ x : C Γ⇒ x : D u-rΓ⇒ x : (C u D)

Γ, x : C ⇒ δ Γ, x : D ⇒ δt-l

Γ, x : (C t D), ⇒ δ

Γ⇒ x : C t1-rΓ⇒ x : (C t D)

Γ, x : ∀R.C, y : C, xRy ⇒ δ∀-l

Γ, x : ∀R.C, xRy ⇒ δ

Γ, xRy ⇒ y : C∀-r

Γ ⇒ x : ∀R.C

Γ, xRy, y : C ⇒ δ∃-l

Γ, x : ∃R.C ⇒ δ

Γ⇒ xRy Γ⇒ y : C∃-r

Γ⇒ x : ∃R.C

Intuitionistic Description Logic for Legal Reasoning

A Case Analysis

Using iALC to formalize Conflict of Laws in Space

A Case StudyPeter and Maria signed a renting contract. The subject of the contract

is an apartment in Rio de Janeiro. The contract states that any dispute willgo to court in Rio de Janeiro. Peter is 17 and Maria is 20. Peter lives inEdinburgh and Maria lives in Rio.

Only legally capable individuals have civil obligations:PeterLiable ContractHolds@RioCourt , shortly, pl cmpMariaLiable ContractHolds@RioCourt , shortly, ml cmpConcepts, nominals and their relationshipsBR is the collection of Brazilian Valid Legal StatementsSC is the collection of Scottish Valid Legal StatementsPILBR is the collection of Private International Laws in BrazilABROAD is the collection of VLS outside BrazilLexDomicilium is a legal connection:

Legal Connections The pair 〈pl, pl〉 is in LexDomicilium

Intuitionistic Description Logic for Legal Reasoning

A Case Analysis

Non-Logical Axiom Sequents

The sets ∆, of concepts, and Ω, of iALC sequents representing theknowledge about the case

∆ =ml : BR pl : SC pl cmp

ml cmp pl LexDom pl

Ω =PILBR ⇒ BR

SC⇒ ABROAD∃LexDom.ABROAD t ∃Lexk.Lk t . . .⇒ PILBR

Intuitionistic Description Logic for Legal Reasoning

A Case Analysis

In Sequent Calculus

∆⇒ pl : SCΩ

pl : SC⇒ pl : Acut

∆⇒ pl : A ∆⇒ pl LexD pl∃ − R

∆⇒ pl : ∃LexD.A

∃LexD.A⇒ ∃LexD.At-R

∃LexD.A⇒ PILBR

Ω

PILBR ⇒ BRcut

∃LexD.A⇒ BRinc − R

∆⇒ pl : BR

∆⇒ ml : BR

Π

∆⇒ pl : BR

Ω

ml : BR, pl : BR⇒ cmp : BRcut

∆,ml : BR⇒ cmp : BRcut

∆⇒ cmp : BR

Intuitionistic Description Logic for Legal Reasoning

Related approaches

Related approachesI 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 allFOL versions of KD and its extensions. Limited to deal withcontrary-to-duty paradoxes.

I Defeasible Logic (Sartor 1991). Complexity goes fromLinearTime (Propositional) to Undecidability of rule applications.It deals better with contrary-to-duty paradoxes. Is LinearTimeregarding a quite restrict logic language.

I iALC PSPACE-complete, non-FOL based ability to expressindividuals. Natural precedence can be used detour thecontrary-to-duty paradoxes. Seems to expresses same the waythat dyadic GDL deals with paradoxes.

I Defeasible Logic and iALC validate the same VLS, but iALCcannot express the dynamics of a trial.

Intuitionistic Description Logic for Legal Reasoning

Contrary-to-duty paradoxes

Consequence of conflicting norms

It ought to be that Jones go tothe assistance of his neighbours. Ob(φ)

It ought to be that if Jones does go thenhe 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

Intuitionistic Description Logic for Legal Reasoning

Conclusion and Future Work

Summary of the Approach

I Individual Legal Valid Statements are the individuals of the universe.I Concepts are Classes of individual laws.I Roles (relationships) between individual laws denote kinds of Legal

ConnectionsI Subsumptions and Negations are intuitionistically interpreted (iALC)I Does it avoid the constrary-to-duty paradoxes of the Deontic approach?

Intuitionistic Description Logic for Legal Reasoning

Conclusion and Future Work

Conclusions

I Seems to be adequate to one jurisprudence theory.I Juridic cases can be analyzed in the ABOX.I TBOX describes “The Law”.I is not always specified at the level of the TBOX.I It seems to scale, but there is no empirical evidence.I Is the coherence analysis easier? (PSPACE-complete)I (?) Investigate some “hard juridical cases”.I (?) Can be the kernel of a tool for helping with a judge’s decision (not a

sentence writer!!!)

Intuitionistic Description Logic for Legal Reasoning

Conclusion and Future Work

Thanks!