A Deontic Logic Reasoning Infrastructure

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Abstract
A flexible infrastructure for the automation of deontic and normative reasoning is presented. Our motivation is the development, study and provision of legal and moral reasoning competencies in future intelligent machines. Since there is no consensus on the “best” deontic logic formalisms and since the answer may be application specific, a flexible infrastructure is proposed in which candidate logic formalisms can be varied, assessed and compared in experimental ethics application studies. Our work thus links the historically rich research areas of classical higher-order logic, deontic logics, normative reasoning and formal ethics.
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A Deontic Logic Reasoning Infrastructure
Christoph Benzm¨uller, Xavier Parent, and Leendert van der Torre
Computer Science and Communications, University of Luxembourg, Luxembourg
Abstract. A flexible infrastructure for the automation of deontic and
normative reasoning is presented. Our motivation is the development,
study and provision of legal and moral reasoning competencies in future
intelligent machines. Since there is no consensus on the “best” deontic
logic formalisms and since the answer may be application specific, a flexi-
ble infrastructure is proposed in which candidate logic formalisms can be
varied, assessed and compared in experimental ethics application stud-
ies. Our work thus links the historically rich research areas of classical
higher-order logic, deontic logics, normative reasoning and formal ethics.
1 Introduction
If humans and intelligent machines are supposed to peacefully coexist, appropri-
ate forms of machine-control and human-machine-interaction are required. This
motivates the provision of legal and ethical reasoning competencies in intelligent
machines. Bottom-up, learning based approaches (e.g. GenEth [1]) try to acquire
ethical behaviour from dialogs with ethicist or from existing data. Top-down ap-
proaches (e.g. [21]) try to explicitly model selected ethical theories or policies
and to enforce them in intelligent systems; cf. [23, 20] and the references therein.
Our research, which is more in line with the top-down approach, assumes that
suitable declarative, logical reasoning means and competencies are mandatory
in intelligent machines, in particular in the context of legal and moral reason-
ing. Such competencies seem vital not only for guaranteeing sucient degrees of
reliability and accountability, but also for achieving human intuitive interaction
means regarding explainability and transparency of decisions.
This paper therefore addresses the practical development of computational
tools for normative reasoning based on deontic logics. Since ethical and legal
theories/policies as well as suitable deontic logic formalisms are both still under
development [34], we outline a flexible workbench to support empirical studies
with such theories/policies in which the preferred logic formalisms themselves
can still be varied, complemented, assessed and compared. The infrastructure
we propose draws on both recent developments in universal logical reasoning
in classical higher-order logic (HOL) [5] and the coalescence and improvements
in interactive and automated theorem proving (ATP) in HOL, as witnessed by
systems such as Isabelle/HOL [32], LEO-II [11] and Leo-III. We are thus linking
the historically rich research areas of HOL, deontic logics, normative reasoning
and formal ethics. Moreover, since modern HOL theorem provers internally col-
laborate with eective SMT and SAT solvers, a novel bridge is provided from
expressive deontic logics to SMT and SAT technology.
The paper is structured as follows. Section 2 surveys relevant deontic logic
formalisms and discusses recent extensions to model ethical agency. Section 3
introduces norm-based deontic logic, our preferred framework. Sections 4 and
5, the core contribution of this paper, outline our flexible deontic reasoning
infrastructure and sketch a case study in data protection.
2 Traditional Deontic Logic
Deontic logic [39, 25] is the field of logic that is concerned with normative con-
cepts such as obligation, permission, and prohibition. Alternatively, a deontic
logic is a formal system capturing the essential logical features of these concepts.
Typically, a deontic logic uses Op to mean that it is obligatory that p, (or it ought
to be the case that p), and Pp to mean that it is permitted, or permissible, that
p. The term “deontic” is derived from the ancient Greek d´eon, meaning “that
which is binding or proper”. Deontic logic can be used for reasoning about nor-
mative multiagent systems, i.e. about multiagent organisations with normative
systems in which agents can decide whether to follow the explicitly represented
norms, and the normative systems specify how, and to which extent, agents can
modify the norms. Normative multiagent systems need to combine normative
reasoning with agent interaction, and thus raise the challenge to relate the logic
of normative systems to aspects of agency.
Traditional (or “standard”) deontic logic (SDL) is a normal propositional
modal logic of type KD, which means that it extends the propositional tautolo-
gies with the axioms K:O(p!q)!(Op !Oq) and D:¬(Op ^O¬p), and it is
closed under the inference rules modus ponens p, p !q/q and generalization or
necessitation p/Op. Prohibition and permission are defined by Fp =O¬pand
Pp =¬O¬p. SDL is an unusually simple and elegant theory. An advantage of
its modal-logical setting is that it can easily be extended with other modalities
such as epistemic or temporal operators and modal accounts of action.
Dyadic deontic logic (DDL) introduces a conditional operator O(p/q), to be
read as “it ought to be the case that p, given q”. Many DDLs have been proposed
to deal with so-called contrary-to-duty reasoning, cf. [18] for an overview on this
area. An example is the DDL proposed by Hansson [28] and ˚
Aqvist [3, 33], and
the one proposed by Carmo and Jones [18, 19].
To enable ethical agency a model of decision needs to be integrated in the
deontic frames. Horty’s STIT logic [29], which combines deontic logic with a
modal logic of action, has been proposed as a starting point. The semantic
condition for the STIT-ought is a utilitarian generalisation of the SDL view that
“it ought be that A” means that Aholds in all deontically optimal worlds.
3 Norm-based Deontic Logic
The term “norm-based” deontic logic has been coined by Hansen [27] to refer
to a family of frameworks analysing the deontic modalities not with reference to
a set of possible worlds (some of them being more ideal than others), but with
reference to a set of explicitly given norms. In such a framework, the central
question is: given some input (e.g. a fact) and a set of explicitly given conditional
norms (a normative system), what norms apply? Thus, the perspective is slightly
dierent from the traditional setting, focusing on inference patterns [35].
We propose to base the AI deontic reasoner on a specific norm-based deontic
logic called input/output (I/O) logic. Initially devised by Makinson [31] and
further developed over the past years by van der Torre and colleagues, I/O logic
has gained increased recognition in the AI community. This is evidenced by the
fact that the framework has its own chapter in the aforementioned Handbook of
Deontic Logic and Normative Systems [34]. I/O logic can be viewed as a rule-
based system. The knowledge base takes the form of a set of rules of the form
(a,b) to be read as “if athen b”. The key feature of I/O logic is that it uses an
operational semantics, based on the notion of detachment, rather than a truth-
functional one in terms of truth-values and possible worlds. On the semantical
side, the meaning of the deontic concepts is given in terms of a set of procedures,
called I/O operations, yielding outputs (e.g., obligations) for inputs (facts). On
the syntactical side, the proof-theory is formulated as a set of inference rules
manipulating pairs of formulas rather than individual formulas. The framework
covers functionalities that are unanimously regarded as characteristic of the legal
domain, and thus required to enable eective legal reasoning:
1. Support for the modelling of constitutive rules, which define concepts or con-
stitute activities that cannot exist without such rules (e.g. legal definitions
such as “property”), and prescriptive rules, which regulate actions by making
them obligatory, permitted, or prohibited.
2. Management of the reification of rules that are objects with properties, such
as jurisdiction, authority, temporal attributes [26].
3. Implementation of defeasibility–see [26, 38]; when the antecedent of a rule is
satisfied by the facts of a case (or via other rules), the conclusion of the rule
presumably holds, but is not necessarily true.
4 Deontic Logic Reasoning Machinery
In a nutshell, a reasoner is a tool that can perform reasoning tasks in a given
application domain. Reasoning thereby refers to the process of deriving or con-
cluding information that is not explicitly encoded in the knowledge base. In our
context, a number of reasoning tasks are particularly relevant. These include:
Compliance checking: Is the current situation compliant with a given regu-
lation (a set of formally represented norms)?
Consistency checking: Is a given regulation consistent? Is such-and-such
norm, as part of a given regulation, consistent with this other set of norms,
stemming from another regulation? Is such-and-such legal interpretation con-
sistent with another one?
Entailment checking: Does such-and-such obligation or legal interpretation
follow from a given regulation?
Some of these tasks, e.g. consistency checking, are well supported by model
finders, while others, such as entailment checking, in general require theorem
proving technology. A powerful deontic reasoner should thus ideally provide both
model finding and theorem proving. While this is comparably easy to achieve
for many decidable propositional fragments of deontic logics, it becomes much
less so for their quantified extensions.
The quest for “a single, best deontic logic” is still open, and it eventually
will remain so for a long time to come. While I/O logic is the solution favoured
in our group (see Sec. 3), we also want to stay open-minded regarding alterna-
tive proposals, such as the DDL by Hansson or the one by Carmo and Jones.
Moreover, it is unclear yet whether regulatory texts can always be abstracted
and simplified to the point that pure propositional logic encodings are feasible
and justified, or whether first-order (FO) or even higher-order (HO) encodings
are required instead. This poses a multitude of concrete challenges for a flexible
deontic reasoning infrastructure.
These raised questions motivate empirical studies in which the dierent op-
tions are systematically compared and assessed within well selected application
studies (see Sec. 5). However, for such empirical work to be feasible, implemen-
tations of the dierent deontic candidate logics have to be provided first, both
on the propositional level and ideally also on the FO and HO level. Moreover,
it is reasonable to ensure that these implementations remain maximally compa-
rable regarding the technological foundations they are based on, since this may
improve the fairness and the significance of (conceptual) empirical evaluations.
A meta-logical approach for flexible, deontic logic reasoning. Our currently pre-
ferred solution for the implementation of a most flexible deontic reasoner in the
above sense is to work with a meta-logical approach to universal logic reason-
ing, cf. [5] and the references therein. We subsequently instantiate this approach
in our ongoing project for various deontic candidate logics, including I/O logic
and the other ones as mentioned. The approach, which is based on shallow se-
mantical embeddings (SSE) [10] of these logics in HOL1, has a very pragmatic
motivation, foremost reuse of tools, simplicity and elegance. It utilises HOL as
a unifying meta-logic in which the syntax and semantics of varying other logics
can be explicitly modeled and flexibly combined. O-the-shelf interactive and
automated theorem provers, and model finders can then be employed to reason
about and within the shallowly embedded logics.
Evidence from previous work. Respective experiments with this approach have
e.g. been conducted in metaphysics (cf. [13]). An initial focus thereby has been
on computer-supported assessments of rational arguments, in particular, of mod-
ern, modal logic variants of the ontological argument for the existence of God.
1HOL as addressed here refers to a (simply) typed logic of functions, which has been
proposed by Church [2]. It provides lambda-notation, as an elegant and useful means
to denote unnamed functions, predicates and sets. Types in HOL eliminate paradoxes
and inconsistencies. For more information on HOL see the literature [7].
In the course of these experiments, in which the SSE approach was applied for
automating dierent variants of HO quantified modal logics [10], the HO theo-
rem prover LEO-II even detected an previously unnoticed inconsistency in Kurt
odel’s prominent variant of the ontological argument, while the soundness of
the slightly modified variant by Dana Scott was confirmed and all argument
steps were verified. Further modern variants of the argument have subsequently
been studied with the approach, and theorem provers have even contributed to
the clarification of an unsettled philosophical dispute [12].
Deontic logics already covered in the SSE approach. The following deontic logics
have already been “implemented” by utilising the SSE approach:
SDL: All logics from the modal logic cube, including logic KD, i.e. SDL,
have meanwhile been faithfully implemented in the SSE approach [10]. These
implementations scale for FO and even HO extensions.
the DDL by Carmo and Jones [19]: A semantic embedding of the proposi-
tional fragment of this logic in Isabelle/HOL is already available [6,8], and
most recently the ATP Leo-III has been adapted to accept DDL as input.
I/O logic [31]: The main challenge comes from the fact that the framework
does not have a truth-functional semantics, but an operational one. First
experiments with the semantic embedding of the I/O-operator “out1” (called
simple-minded) in Isabelle/HOL are promising [9]; see also Sect. 5.
Some relevant I/O logic variants have very recently been studied (see [36]),
and we conjecture that some of these variants are related to certain non-normal
modal logics, e.g. conditional logics with a selection function semantics or similar
logics with a neighbourhood semantics. However, the embedding of such logics
has already been studied in the first authors previous work [4]. It should thus
be possible to benefit from these existing results in the given context.
A most interesting aspect is, that the SSE approach even supports meta-
logical investigations within Isabelle/HOL in which the possible relationships
between I/O and conditional logics we hinted at can be formally assessed. Ex-
amples of meta-logical studies are e.g. mentioned in [10].
Moreover, the SSE approach enables the reuse of existing model finding and
theorem proving technology within the Isabelle/HOL proof assistant [32]. The
automated reasoning systems that are integrated with Isabelle/HOL, respec-
tively that are available via remote calls, include state-of-the-art SMT solvers,
FO and HO theorem provers, and model finders; cf. [15] and the references
therein. This infrastructure, in combination with the SSE approach, meets our
demanding requirements regarding flexibility along dierent axes.
While there is some related work, see e.g. [17, 24, 37, 21], we are not aware
of any other existing deontic logic reasoning approach and associated machinery
that oers the same amount of flexibility and scalability.
Another advantage of the SSE approach, when implemented within powerful
proof assistants such as Isabelle/HOL, is that proof construction (interactive or
automated) can be supported at dierent levels of abstraction. For this note that
proof protocols/objects may generally serve two dierent purposes: (a) they may
provide an independently verifiable explanation in a (typically) well-defined log-
ical calculus, or (b) they may provide an intuitive explanation to the user why
the problem in question has been answered positively or negatively. Many rea-
soning tools, if they are oering proof objects at all, do generate only objects of
type (a). The SSE approach, however, has already demonstrated its capabilities
to provide both types of responses simultaneously in even most challenging logic
settings. For example, a quite powerful, abstract level theorem prover for hyper-
intensional HO modal logic has been provided by Kirchner [30]. He encoded an
abstract level proof calculus for this logic as proof tactics and he demonstrated
how these abstract level proof tactics can again be elegantly automated using
respective tools in Isabelle/HOL. Kirchner then successfully applied his reason-
ing infrastructure to reveal, assess and intuitively communicate a non-trivial
paradox in Zalta’s “Principia-logico Metaphysica” [40].
Drawing on the results and experiences from previous work, the ambition of
our ongoing project is to further extend the already existing implementations
of deontic logics in Isabelle/HOL towards a most powerful, flexible and scalable
deontic logic reasoning infrastructure. A core motivation thereby is to support
empirical studies in various application scenarios, and to assess and compare the
suitability, adequacy and performance of individual deontic logic solutions for
the engineering of moral agents and explainable intelligent systems.
5 Case Study: Data Protection
The General Data Protection Regulation (GDPR, Regulation EU 2016/679) is
a relevant and interesting application scenario for normative reasoning. It is
a regulation by which the European Parliament, the Council of the European
Union and the European Commission intend to strengthen and unify data pro-
tection for all individuals within the European Union. The regulation becomes
enforceable from 25 May 2018. We present two sample norms of the GDPR:
1. Personal data shall be processed lawfully (Art. 5). For example, the data
subject must have given consent to the processing of his or her personal
data for one or more specific purposes (Art. 6/1.a).
2. If the personal data have been processed unlawfully (none of the require-
ments for a lawful processing applies), the controller has the obligation to
erase the personal data in question without delay (Art. 17.d, right to be
forgotten).
When combined with the following a typical CTD-structure is exhibited.
3. It is obligatory e.g. as part of a respective agreement between a customer
and a company) to keep the personal data (as relevant to the agreement)
provided that it is processed lawfully.
4. Some data in the context of such an agreement has been processed unlaw-
fully.
Fig. 1. GDPR example scenario in Isabelle/HOL.
The latter information pieces are not explicit part of the GDPR. Instead
they are to be seen as implicit. 3 comes from another regulation, with which the
GDPR has to co-exists. 4 is a factual information — it is exactly the kind of world
situations the GDPR wants to regulate. This example is given for illustrative
purposes only. It provides a taste of what this knowledge base might look like.
In a recent technical report [8] we illustrate the practical challenge of such a
CTD scenario. Namely, when the above norms are encoded in SDL, an inconsis-
tency follows, meaning that everything is implied in the given context, including
arbitrarily weird and unethical conclusions such as the obligation to “kill the
boss”. In the same report we demonstrate that our SSE based implementation
of Carmo and Jones’s DDL is in contrast not suering from this eect. No in-
consistency follows when the above scenario is modelled in DDL. The obligation
to erase the data, however, is implied as intended.
We now analyse the above CTD scenario in the context of I/O logic. This is
done is Fig. 1, which first presents, in lines 4-13, an SSE based implementation
of I/O logic in Isabelle/HOL.2A possible world semantics is employed in this
embedding to adequately address an extensionality issue we have revealed in our
previous work. This issue, and its solution, is discussed in more detail in a techni-
cal report [9]. The prescriptive rules of the GDPR scenario are then modelled in
lines 19-25, where the set of given Norms is defined as {(>,process data lawfully),
(¬process data lawfully,erase data),(process data lawfully,¬erase data))}.The
given Situation,inwhichwehave¬process data lawfully, is defined in line 27.
Subsequently, three dierent queries are answered by the reasoning tools in-
tegrated with Isabelle/HOL. The first query asks whether the data should be
erased in the given context. The ATPs integrated with Isabelle/HOL via the
Sledgehammer tool [15] respond quickly: the SMT solver CVC4 [22] and the
first-order prover Spass [16] return a proof within a few milliseconds. For queries
2 and 3 the ATPs fail (not shown here), but now the countermodel finder Nitpick
[14] responds and presents counterarguments to both queries. That is, we receive
the intended negative answers to queries 2 and 3 when the GDPR example is
modelled in our preferred I/O logic. It is worth mentioning that I/O logic (and
also DDL) have never been automated before.
6 Conclusion
The deontic logic reasoning infrastructure we have presented supports empirical
studies on legal and ethical theories/policies in which the particular deontic logic
formalisms itself can be varied, assessed and compared in context. We believe
that this infrastructure can fruitfully support the development of much needed
logic based approaches towards ethical agency. The solution we have presented
supports a wide range of specific deontic logic variants, and it also scales for their
first-order and higher-order extensions. In fact, our infrastructure already now
2The semantical embedding of out1as presented here is technically still an approxi-
mative solution. For a complete embedding, xneeds to be defined as a consequence
of an arbitrary number of facts (instead of just i,jand k) in lines 13 and 14.
implements a wider range of deontic and related logics than any other competitor
systems we are aware of.
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    Principia Logico-Metaphysica proposes a foundational logical theory for metaphysics, mathematics, and the sciences. It contains a canonical development of Abstract Object Theory [AOT], a metaphysical theory (inspired by ideas of Ernst Mally, formalized by Zalta) that differentiates between ordinary and abstract objects. This article reports on recent work in which AOT has been successfully represented and partly automated in the proof assistant system Isabelle/HOL. Initial experiments within this framework reveal a crucial but overlooked fact: a deeply-rooted and known paradox is reintroduced in AOT when the logic of complex terms is simply adjoined to AOT's specially-formulated comprehension principle for relations. This result constitutes a new and important paradox, given how much expressive and analytic power is contributed by having the two kinds of complex terms in the system. Its discovery is the highlight of our joint project and provides strong evidence for a new kind of scientific practice in philosophy, namely, computational metaphysics. Our results were made technically possible by a suitable adaptation of Benzm\"uller's metalogical approach to universal reasoning by semantically embedding theories in classical higher-order logic. This approach enables the fruitful reuse of state-of-the-art higher-order proof assistants, such as Isabelle/HOL, for mechanizing and experimentally exploring challenging logics and theories such as AOT. Our results also provide a fresh perspective on the question of whether relational type theory or functional type theory better serves as a foundation for logic and metaphysics.
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    The quest for a most general framework supporting universal reasoning is very prominently represented in the works of Leibniz. He envisioned a scientia generalis founded on a characteristica universalis, that is, a most universal formal language in which all knowledge about the world and the sciences can be encoded. A quick study of the survey literature on logical formalisms suggests that quite the opposite to Leibniz’ dream has become reality. Instead of a characteristica universalis, we are today facing a very rich and heterogenous zoo of different logical systems, and instead of converging towards a single superior logic, this logic zoo is further expanding, eventually even at accelerated pace. As a consequence, the unified vision of Leibniz seems farther away than ever before. However, there are also some promising initiatives to counteract these diverging developments. Attempts at unifying approaches to logic include categorial logic algebraic logic and coalgebraic logic.
  • Conference Paper
    As intelligent systems are increasingly making decisions that directly affect society, perhaps the most important upcoming research direction in AI is to rethink the ethical implications of their actions. Means are needed to integrate moral, societal and legal values with technological developments in AI, both during the design process as well as part of the deliberation algorithms employed by these systems. In this paper, we describe leading ethics theories and propose alternative ways to ensure ethical behavior by artificial systems. Given that ethics are dependent on the socio-cultural context and are often only implicit in deliberation processes, methodologies are needed to elicit the values held by designers and stakeholders, and to make these explicit leading to better understanding and trust on artificial autonomous systems.
  • Article
    Full-text available
    As intelligent systems are increasingly making decisions that directly affect society, perhaps the most important upcoming research direction in AI is to rethink the ethical implications of their actions. Means are needed to integrate moral, societal and legal values with technological developments in AI, both during the design process as well as part of the deliberation algorithms employed by these systems. In this paper, we describe leading ethics theories and propose alternative ways to ensure ethical behavior by artificial systems. Given that ethics are dependent on the socio-cultural context and are often only implicit in deliberation processes, methodologies are needed to elicit the values held by designers and stakeholders, and to make these explicit leading to better understanding and trust on artificial autonomous systems.
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    A universal reasoning approach based on shallow semantical embeddings of higher-order modal logics into classical higher-order logic is exemplarily employed to analyze several modern variants of the ontological argument on the computer. Several novel findings are reported which contribute to the clarification of a long-standing dispute between Anderson and Hájek. The technology employed in this work, which to some degree realizes Leibniz’s dream of a characteristica universalis and a calculus ratiocinator for solving philosophical controversies, is ready to be fruitfully adopted in larger scale by philosophers.
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    This paper discusses the discovery of the inconsistency in Gödel's ontological argument as a success story for artificial intelligence. Despite the popularity of the argument since the appearance of Gödel's manuscript in the early 1970's, the inconsistency of the axioms used in the argument remained unnoticed until 2013, when it was detected automatically by the higher-order theorem prover Leo-II. Understanding and verifying the refutation generated by the prover turned out to be a time-consuming task. Its completion, as reported here, required the reconstruction of the refutation in the Isabelle proof assistant, and it also led to a novel and more efficient way of automating higher-order modal logic S5 with a universal accessibility relation. Furthermore, the development of an improved syntactical hiding for the utilized logic embedding technique allows the refutation to be presented in a human-friendly way, suitable for non-experts in the technicalities of higher-order theorem proving. This brings us a step closer to wider adoption of logic-based artificial intelligence tools by philosophers .
  • Chapter
    How is deontic logic possible on a positivistic philosophy of norms? D. Makinson (1999) considered this question the ‘fundamental problem of deontic logic’, and called to reconstruct deontic logic as a logic of reasoning about norms. A solution is to use a semantics which defines the truth of monadic and dyadic deontic sentences with respect to an explicitly modelled set of norms. Here, I explore how such a norm-based semantics can be adapted to include not just mandatory but also permissive norms that possibly conflict with the first, and describe how this may affect the validity of well-known theorems. All studied proposals are based on a definition of consistency from the later theory of G. H. von Wright. The approach may shed new light on the problem of ‘free choice permission,’ while D. Lewis’s ‘problem of permission’ persists. Finally, I question a persistent belief about permissions: that unlike obligations they must be considered one by one, and not collectively.