Intuitionistic Description Logic for Legal Reasoning

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

Jurisprudence Motivation

Considerations 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.


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.


Main Motivation

Solid Jurisprudence + Description Logic


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”?


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.


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”.

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”∈ BR iff “Peter is liable”∈ ¬BR

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

Logical Motivation


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

Logical Motivation


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

Intuitionistic Description Logics

The 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.

Intuitionistic Description Logics

Deductive 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

A Case Analysis

Using 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

A Case Analysis

Non-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

A Case Analysis

In 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

Related approaches

Related 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.


Contrary-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

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?


Conclusions

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!!!)


Thanks!

Intuitionistic Description Logic for Legal Reasoning

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Intuitionistic Description Logic for Legal Reasoning