Defeasible Semantics for L4

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Paper presented at ProLaLa 2023 Workshop

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Defeasible Semantics for L4
GUIDO GOVERNATORI and MENG WENG WONG, Singapore Management University, Singapore
Reference Format:
Guido Governatori and Meng Weng Wong. 2023. Defeasible Semantics for L4. In ProLaLa 2023. 6 pages.
1 INTRODUCTION
The importance of defeasibility for legal reasoning has been investigated for a long time (see among other [10, 3, 11]).
This notion mostly concerns the issue that textual provisions of (legal) norms typically provide prima facie conditions
for their applicability, but to understand a norm in full, we have to evaluate the norms in the context in which the
norm is used and to see if other norms prevent it either to apply or to be eective. In other words, when evaluating
norms, we must account for possible (prima facie) conicts and exceptions. Indeed, in general, norms rst provide
the basic conditions for their applicability. Then, they give the exceptions and exclusions (and they can go on, with
exceptions/exclusions of the exceptions/exclusions and so on).
The rst issue to address to model legal reasoning is how to model norms. Here, we follow the approach of [12, 4]
and stipulate that a norm is represented by an “IF
· · ·
THEN
. . .
” rule, where the IF part establishes the conditions of
applicability of the norm and the THEN part species the legal eect of the norm. Where the legal eect of the norm is
either that a proposition is taken to hold legally or that a legal requirement (obligation, prohibition, permission) is in
force. Moreover, as we have alluded to, the norms are defeasible; thus, the IF/THEN conditional used to model legal
norms does not correspond to the material implication of classical logic, and it has a non-monotonic nature. Several
approaches have been proposed to reduce or compile the normative IF/THEN conditional. However, in general, as
discussed by [13, 8], they suer from some limitations; for example, the translation to classical propositional logic
requires complete knowledge (for any atomic proposition, we have to determine whether it is true or not), it is not
resilient to contradictions, and changes to the norms might require a complete rewriting of the translation.
In this work, we are going to examine how to provide an eective and constructive non-monotonic interpretation
of (a restricted version of) L4 based on Answer Set Programming (ASP) meta-program. The meta-program gives the
semantics of the underlying L4 constructs as well as a computational framework for them.
2 A QUICK OVERVIEW OF L4
L4 is a Domain Specic Language (DSL) under development in the context of the Centre for Computational Law
1
agship
project Research Programme in Computational Law, supported by the National Research Foundation of Singapore. The
Programme’s ultimate goal is to develop a Domain-Specic Language for expressing laws, contracts, and other rules.
Current investigation of L4 has shown that the language is suciently precise to avoid ambiguities of natural languages
and, at the same time, suciently close to a traditional law text with its characteristic elements such as cross-references,
1https://cclaw.smu.edu.sg/
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
ProLaLa 2023, January 15, 2023, Boston
©2023 Copyright held by the owner/author(s).
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ProLaLa 2023, January 15, 2023, Boston Guido Governatori and Meng Weng Wong
prioritisation of rules and defeasible reasoning. Moreover, once a law has been coded in L4, it can be further processed
for dierent tasks for applications involving some form of legal reasoning.
This paper focuses on one aspect of L4 (and in a restricted form). More specically, we concentrate on the basic
notion of a rule. A basic rule in L4 has the following form.
rule <r> if Preconditions then Conclusion
{restrict: {subject to <s_1>,...,<s_n>} {despite <d_1>,...,<d_m>} }
where
r
, a unique identied, is the label (name) or the rule;
Preconditions
is a (possibly empty) conjunction of
propositions, and, accordingly, we consider it as the set of the propositions;
Conclusion
is a single proposition. The
propositions in a rule can be prexed by one of the following expressions:
MUST
,
MAY
,
SHANT
indicating the deontic
modier (operator) that applies to the proposition. Finally, the keyword
restrict
species what rules are either
stronger (subject to) or weaker despite than the current rule.
3 DEFEASIBILITY
Legal rules can be classied either as constitutive rules (also known as counts as rules) or regulative rules. In turn, a
normative rule can either be a prescriptive rule or a permissive rule. A constitutive rule gives the meaning or denes
a term; regulative rules specify what normative positions (obligations and prohibitions for prescriptive rules and
permissions for permissive rule) and the conditions under which such normative positions hold. Defeasibility applies to
both constitutive rules and regulative rules. Consider the following two real-life examples from Australian regulations
and Acts.
Example 3.1 (Telecommunications Consumer Protections Code (C628:2012). Section 2.1. Denitions).
Complaint means an expression of dissatisfaction made to a Supplier in relation to its Telecommunications
Products or the complaints handling process itself, where a response or Resolution is explicitly or implicitly
expected by the Consumer.
An initial call to a provider to request a service or information or to request support is not necessarily a
Complaint. An initial call to report a fault or service diculty is not a Complaint. However, if a Customer
advises that they want this initial call treated as a Complaint, the Supplier will also treat this initial call as
a Complaint.
Example 3.2 (National Consumer Credit Protection Act 2009 (Act No. 134 of 2009). Section 29).
(1)
A person must not engage in a credit activity if the person does not hold a licence authorising the
person to engage in the credit activity.
(3) For the purposes of subsections (1) and (2), it is a defence if:
(a) the person engages in the credit activity on behalf of another person (the principal); and
(b) the person is:
(i) an employee or director of the principal or of a related body corporate of the principal; or
(ii) a credit representative of the principal; and ...
The semantics/computation for L4 rules we are going to present is based on Defeasible (Deontic) Logic [2, 7]. In
Defeasible Logic, a proposition 𝑝holds (defeasibly) if:
•𝑝is a fact; or
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Defeasible Semantics for L4 ProLaLa 2023, January 15, 2023, Boston
•there is a rule 𝑟such 𝐶𝑜𝑛𝑐 (𝑟)=𝑝, and
–for all 𝑞∈𝑃 𝑟𝑒 (𝑟),𝑞(defeasibly) holds (the rule is applicable), and
–for any rule 𝑠such that 𝐶𝑜𝑛𝑐 (𝑠)=∼𝑝,𝑠is either discarded or defeated
A rule
𝑠
is discarded if there is proposition
𝑞∈𝑃𝑟 𝑒 (𝑠)
such that
𝑞
is refuted, where refuted is the (constructive) failure to
show that it holds. A rule
𝑠
is defeated if there is an applicable rule
𝑡
whose conclusion is the opposite of the conclusion
of 𝑠and 𝑡is stronger than 𝑠.
The logic is sceptical in the sense that if there are two (applicable) rules for opposite conclusions and there is no
means to solve the conict, the logic prevents the conclusion of contradictions. Still, at the same time, it discards the
conclusion of both conclusions. Hence, none of the two opposite conclusions holds. In other words, there is some
ambiguity about which of the two conclusions hold. However, the opposite conclusions can be part of the preconditions
of other rules. Consider, for example, the scenario where there are two equally compelling pieces of evidence, one
supporting the case that a person was legally responsible for A and the second that the person was not responsible for A.
Moreover, if the person was responsible for A, then the person is found guilty. However, according to the presumption
of innocence, a person is assumed to be not guilty. This situation can be represented by
rule <r1> if evidence1 then responsible
rule <r2> if evidence2 then not responsible
rule <r3> if responsible then guilty
{restrict: {despite <r4>}}
rule <r4> if true then not guilty
In this scenario, given the two pieces of evidence, we cannot assert whether
responsible
or
not responsible
holds;
thus, the proposition
responsible
is ambiguous. A statement is ambiguous when there is an argument supporting it
and an argument for its opposite; moreover, there is no way to determine if one of the two arguments is stronger/defeats
the other. However, in this situation, we can go on since we cannot assert that responsible holds, but rule
𝑟4
(encoding
the so-called presumption of innocence) vacuously holds, and we can conclude not guilty.
However, Suppose that, in addition to the conditions stipulated above, if a person was wrongly accused, then the
person is entitled to some compensation. This can be encoded in L4 by the following two rules:
rule <r5> if not guilty then compensation
rule <r6> if true then not compensation
{restrict: {subject to <r5>}}
If we continue our reasoning, rule
𝑟5
is applicable; it defeats
𝑟6
, allowing us to establish that
compensation
holds.
However, the two pieces of evidence were equally reliable; it does not sound right that the person was wrongly accused.
Indeed, it was ambiguous whether the accused was responsible or not. This scenario illustrates that we have to account
for two forms of defeasibility: ambiguity blocking and ambiguity propagation. Governatori [5] argues that these two
forms of defeasibility account for dierent (legal) proof standards. However, Defeasible Logic can accommodate the
two variants.
The semantics we gave above is for the ambiguity blocking case. For the ambiguity propagating case, a few changes
are needed [1]. First, a conclusion is supported if it is a fact or there is a rule such that all the preconditions are supported
and the rule s not weaker than an applicable rule for the opposite. Second, rules attaching a conclusion are discarded if
they are not supported (instead of applicable). These changes simplify attacking a conclusion, and it is easy to verify
that both guilty and ‘not guilty are supported, and we prevent the conclusion of compensation.
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ProLaLa 2023, January 15, 2023, Boston Guido Governatori and Meng Weng Wong
4 DEFEASIBLE ENCODING OF L4 IS ANSWER SET PROGRAMMING
In this section, we give a meta-program in Answer Set Programming to encode the reasoning mechanism we presented
in the previous section to model defeasibility with both ambiguity blocking and ambiguity propagation. The ASP meta-
program clauses to capture ambiguity blocking and ambiguity propagation are based on Defeasible Logic variants and
meta-program given in [1]. The deontic extension is based on the Defeasible Deontic Logic of [7] and the meta-program
techniques of [9]. In addition to the meta-program clauses to model and compute the logic aspects, we discuss how to
encode an L4 theory in the meta-program to compute the extension of the theory.
The rst step for encoding Defeasible Deontic Logic in ASP is to provide the predicates and the clauses dening the
language and the basic notions of the logic.
negation(non(X),X) :- atom(X).
negation(X,non(X)) :- atom(X).
The predicate
negation/2
takes two arguments (meant to correspond to the positive and negative literals for an atomic
proposition, i.e.,
atom/1
in the encoding parlance), and it establishes that
non(X)
and
X
are the negation of each other,
and thus they cannot be true at the same time. It is worth noting that we use a meta-encoding; thus, we do not use the
negations (classical and negation as failure) of ASP to represent the negation of literals in Defeasible Deontic Logic.
conflict(X,Y) :- strongConflict(X,Y).
conflict(Y,X) :- strongConflict(X,Y).
We need the following clauses to model the computation for ambiguity blocking aspect of defeasibility.
defeasible(X) :- fact(X).
defeasible(X) :- opposes(X,X1), not fact(X1), rule(R,X), applicable(R), not overruled(R,X).
overruled(R,X) :- opposes(X,X1), rule(R,X), rule(R1,X1), applicable(R1), not defeated(R1,X1).
defeated(R,X) :- opposes(X,X2), rule(R2,X2), superior(R2,R), applicable(R2).
The predicate
opposes/2
establishes that two propositions
X
and
Y
cannot hold simultaneously. The predicate
applicable/1
takes as its argument a rule in L4, and its truth is determined by the ASP encoding of the L4 rule (we discuss the full
procedure of how to encode an L4 rule below).
For ambiguity propagation, we need the following clauses (notice that the clauses below oer an alternative version
of the defeasible predicate to the denition given above).
support(X) :- fact(X).
support(X) :- rule(R,X), supported(R), not beaten(R,X).
beaten(R,X) :- rule(R,X), opposes(X,X1), fact(X1).
beaten(R,X) :- rule(R,X), opposes(X,X1), rule(R1,X1), supported(R1), superior(R1,R).
defeasible(X) :- fact(X).
defeasible(X) :- opposes(X,X1), not fact(X1), rule(R,X), applicable(R), not overruled(R,X).
overruled(R,X) :- opposes(X,X1), rule(R,X), rule(R1,X1), supported(R1), not defeated(R1,X1).
defeated(R,X) :- opposes(X,X2), rule(R2,X2), superior(R2,R), applicable(R2).
support(X) :- defeasible(X).
supported(X) :- applicable(X).
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Defeasible Semantics for L4 ProLaLa 2023, January 15, 2023, Boston
Finally, to model the Defeasible Deontic Logic proposed in [7], we can use the following clauses (due to space
reasons, we refer the readers to [7] for the description of the logic and its motivation. However, the predicate names are
self-describing, and the clauses provide an alternative description of the logic).
deonticRule(R,X) :- prescriptiveRule(R,X).
deonticRule(R,X) :- permissiveRule(R,X).
obligation(X) :- prescriptiveRule(R,X), applicable(R), not deonticOverruled(R,X).
obligation(X) :- constitutiveRule(R,X), obligationApplicable(R), not deonticOverruled(R,X).
permission(X) :- permissiveRule(R,X), applicable(R), not deonticOverruled(R,X).
permission(X) :- constitutiveRule(R,X), permissionApplicable(R), not deonticOverruled(R,X).
deonticOverruled(R,X) :- prescriptiveRule(R,X), deonticRule(R1,X1), opposes(X,X1),
applicable(R1), not deonticDefeated(R1,X1).
deonticOverruled(R,X) :- permissiveRule(R,X), prescriptiveRule(R1,X1), opposes(X,X1),
applicable(R1), not deonticDefeated(R1,X1).
deonticDefeated(R,X) :- opposes(X,X1), prescriptiveRule(R,X), deonticRule(R1,X1),
applicable(R1), superior(R1,R).
deonticDefeated(R,X) :- opposes(X,X1), permissiveRule(R,X), prescriptiveRule(R1,X1),
applicable(R1), superior(R1,R).
The process of encoding an L4 rule in the meta-program has the following step. First, we rewrite each
SHANT p
as
MUST not p
. Then we group all propositions in
Preconditions
in three groups (the groups can be empty): the rst
group is the set of propositions that do not occurs in the scope of a deontic operator (
MUST
,
MAY
). The second and third
groups are, respectively, the sets of propositions in the scope of ‘MUST and MAY. Thus a rule 𝑟has the generic form
rule <r>
if a_1 && ... && a_n && MUST o_1 && ... && MUST o_m && MAY p_1 && ... && MAY p_k
then [MUST|MAY] c
{restrict: {subject to <s_1>...<s_l>} {despite <d_1>...<d_w>}}
A rule 𝑟is encoded as
prescriptiveRule(r,c). % if Conc(r) = MUST c
permissiveRule(r,c). % if Conc(r) = MAY c
constitutiveRule(r,c). % otherwise
applicable(r) :-
defeasible(a_1), ... , defeasible(a_n), % for each a_i not in the scope of MUST|MAY
obligation(o_1), ... , obligation(o_m), % for each o_i in the scope of MUST
permission(p_1), ... , permission(p_k). % for each p_i in the scope of MAY
nally, we add
superior(r,d). % for each d_i in restrict {despite <d_i>}
superior(s,r). % for each r_i in restrict {subject to <s_i>}
As we discussed, ambiguity propagation and ambiguity blocking intuition could be seen as dierent legal proof
standards. A decision in a legal proceeding can use conclusions with dierent proof standards. The meta-programming
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ProLaLa 2023, January 15, 2023, Boston Guido Governatori and Meng Weng Wong
approach presented in this paper allows us to accommodate them. For example, instead of using
defeasible/1
, we can re-
place it with
defeasible/2
where the rst argument is the type of defeasible conclusion (ambiguity blocking or ambigu-
ity propagation) and the second is the proposition. In addition, the encoding of a rule can specify what type of defeasibility
is required for a precondition in given rules. Thus we can have
defeasible(propagation,a_1), defeasible(blocking,a_2)
in one rule and
defeasible(blocking,a_1), defeasible(_,a_2)
in another rule. [6] proved that this combination
is sound and complete and is a conservative extension of the individual variants. This shows that the defeasible deontic
logic meta-programming encoding of L4 rules oers an ecient, exible and powerful environment for modelling legal
rules and a feasible and viable Rules as Code framework.
ACKNOWLEDGMENTS
This research/project is supported by the National Research Foundation, Singapore under its Industry Alignment Fund
– Pre-positioning (IAF-PP) Funding Initiative. Any opinions, ndings and conclusions or recommendations expressed
in this material are those of the author(s) and do not reect the views of National Research Foundation, Singapore.
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