The EU proposal for the Artificial Intelligence Act (AIA) defines four risk categories: unacceptable, high, limited, and minimal. However, as these categories statically depend on broad fields of application of AI, the risk magnitude may be wrongly estimated, and the AIA may not be enforced effectively. This problem is particularly challenging when it comes to regulating general-purpose AI (GPAI), which has versatile and often unpredictable applications. Recent amendments to the compromise text, though introducing context-specific assessments, remain insufficient. To address this, (...) we propose applying the risk categories to specific AI scenarios, rather than solely to fields of application, using a risk assessment model that integrates the AIA with the risk approach arising from the Intergovernmental Panel on Climate Change (IPCC) and related literature. This model enables the estimation of the magnitude of AI risk by considering the interaction between (a) risk determinants, (b) individual drivers of determinants, and (c) multiple risk types. We use large language models (LLMs) as an example. (shrink)

In this paper we propose an extension of Defeasible Logic to represent and compute different concepts of defeasible permission. In particular, we discuss some types of explicit permissive norms that work as exceptions to opposite obligations or encode permissive rights. Moreover, we show how strong permissions can be represented both with, and without introducing a new consequence relation for inferring conclusions from explicit permissive norms. Finally, we illustrate how a preference operator applicable to contrary-to-duty obligations can be combined with a (...) new operator representing ordered sequences of strong permissions. The logical system is studied from a computational standpoint and is shown to have linear computational complexity. (shrink)

The EU Artificial Intelligence Act (AIA) defines four risk categories for AI systems: unacceptable, high, limited, and minimal. However, it lacks a clear methodology for the assessment of these risks in concrete situations. Risks are broadly categorized based on the application areas of AI systems and ambiguous risk factors. This paper suggests a methodology for assessing AI risk magnitudes, focusing on the construction of real-world risk scenarios. To this scope, we propose to integrate the AIA with a framework developed by (...) the Intergovernmental Panel on Climate Change (IPCC) reports and related literature. This approach enables a nuanced analysis of AI risk by exploring the interplay between (a) risk determinants, (b) individual drivers of determinants, and (c) multiple risk types. We further refine the proposed methodology by applying a proportionality test to balance the competing values involved in AI risk assessment. Finally, we present three uses of this approach under the AIA: to implement the Regulation, to assess the significance of risks, and to develop internal risk management systems for AI deployers. (shrink)

In this paper we investigate how to represent and reason about legal abrogations and annulments in Defeasible Logic. We examine some options that embed in this setting, and in similar rule-based systems, ideas from belief and base revision. In both cases, our conclusion is negative, which suggests to adopt a different logical model. This model expresses temporal aspects of legal rules, and distinguishes between two main timelines, one internal to a given temporal version of the legal system, and another relative (...) to how the legal system evolves over time. Accordingly, we propose a temporal extension of Defeasible Logic suitable to express this model and to capture abrogation and annulment. We show that the proposed framework overcomes the difficulties discussed in regard to belief and base revision, and is sufficiently flexible to represent many of the subtleties characterizing legal abrogations and annulments. (shrink)

In this paper we present a Gentzen system for reasoning with contrary-to-duty obligations. The intuition behind the system is that a contrary-to-duty is a special kind of normative exception. The logical machinery to formalise this idea is taken from substructural logics and it is based on the definition of a new non-classical connective capturing the notion of reparational obligation. Then the system is tested against well-known contrary-to-duty paradoxes.

In this paper we show that the Hilbert system of agency and ability presented by Dag Elgesem is incomplete with respect to the intended semantics. We argue that completeness result may be easily regained. Finally, we shortly discuss some issues related to the philosophical intuition behind his approach. This is done by examining Elgesem's modal logic of agency and ability using semantics with different flavours.

The first issue of _Artificial Intelligence and Law_ journal was published in 1992. This paper discusses several topics that relate more naturally to groups of papers than a single paper published in the journal: ontologies, reasoning about evidence, the various contributions of Douglas Walton, and the practical application of the techniques of AI and Law.

This paper proposes an approach to investigate norm-governed learning agents which combines a logic-based formalism with an equation-based counterpart. This dual formalism enables us to describe the reasoning of such agents and their interactions using argumentation, and, at the same time, to capture systemic features using equations. The approach is applied to norm emergence and internalisation in systems of learning agents. The logical formalism is rooted into a probabilistic defeasible logic instantiating Dung’s argumentation framework. Rules of this logic are attached (...) with probabilities to describe the agents’ minds and behaviours as well as uncertain environments. Then, the equation-based model for reinforcement learning, defined over this probability distribution, allows agents to adapt to their environment and self-organise. (shrink)

In this paper we show how defeasible logic could formally account for the non-monotonic properties involved in motivational attitudes like intention and obligation. Usually, normal modal operators are used to represent such attitudes wherein classical logical consequence and the rule of necessitation comes into play, i.e., ⊢A/⊢ □A, that is from ⊢A derive ⊢ □A. This means that such formalisms are affected by the Logical Omniscience problem. We show that policy-based intentions exhibit non-monotonic behaviour which could be captured through a (...) non-monotonic system like defeasible logic. To this end we outline a defeasible logic of intention that specifies how modalities can be introduced and manipulated in a non-monotonic setting without giving rise to the problem of logical omniscience. In a similar way we show how to add deontic modalities defeasibly and how to integrate them with other motivational attitudes like beliefs and goals. Finally we show that the basic aspect of the BOID architecture is captured by this extended framework. (shrink)

This paper presents a belief revision operator that considers time intervals for modelling norm change in the law. This approach relates techniques from belief revision formalisms and time intervals with temporalised rules for legal systems. Our goal is to formalise a temporalised belief base and corresponding timed derivation, together with a proper revision operator. This operator may remove rules when needed or adapt intervals of time when contradictory norms are added in the system. For the operator, both constructive definition and (...) an axiomatic characterisation by representation theorems are given. (shrink)

This handbook offers a deep analysis of the main forms of legal reasoning and argumentation from both a logical-philosophical and legal perspective. These forms are covered in an exhaustive and critical fashion, and the handbook accordingly divides in three parts: the first one introduces and discusses the basic concepts of practical reasoning. The second one discusses the main general forms of reasoning and argumentation relevant for legal discourse. The third one looks at their application in law as well as at (...) the different areas of legal reasoning. The handbook’s division in three parts reflects its conceptual architecture, since legal reasoning and argumentation are considered in relation to the more general types of reasoning. (shrink)

This paper offers a semantic study in multi-relational semantics of quantified N-Monotonic modal logics with varying domains with and without the identity symbol. We identify conditions on frames to characterise Barcan and Ghilardi schemata and present some related completeness results. The characterisation of Barcan schemata in multi-relational frames with varying domains shows the independence of BF and CBF from well-known propositional modal schemata, an independence that does not hold with constant domains. This fact was firstly suggested for classical modal systems (...) by Stolpe, 557–575, 2003), but unfortunately that work used only models and not frames. (shrink)

A number of significant contributions in the last four decades show that non-normal modal logics can be fruitfully employed in several applied fields. Well-known domains are epistemic logic, deontic logic, and systems capturing different aspects of action and agency such as the modal logic of agency, concurrent propositional dynamic logic, game logic, and coalition logic. Semantics for such logics are traditionally based on neighbourhood models. However, other model-theoretic semantics can be used for this purpose. Here, we systematically study multi-relational structures, (...) whose investigation is still relatively underdeveloped. After a brief introduction to two different versions of multi-relational semantics – which we call strong and weak multi-relational semantics – we proceed to study several modal schemata. Special attention is paid to the schemata CON and D. Finally we offer completeness proofs for several systems using both strong and weak semantic tools: The proofs thus cover both classical system.. (shrink)

This paper provides a computational framework, based on defeasible logic, to capture some aspects of institutional agency. Our background is Kanger-Lindahl-Pörn account of organised interaction, which describes this interaction within a multi-modal logical setting. This work focuses in particular on the notions of counts-as link and on those of attempt and of personal and direct action to realise states of affairs. We show how standard defeasible logic (DL) can be extended to represent these concepts: the resulting system preserves some basic (...) properties commonly attributed to them. In addition, the framework enjoys nice computational properties, as it turns out that the extension of any theory can be computed in time linear to the size of the theory itself. (shrink)

The present collection represents an attempt to bring together several contributions to the ongoing debate pertaining to supervenience of the normative in law and morals and strives to be the first work that addresses the topic comprehensively. It addresses the controversies surrounding the idea of normative supervenience and the philosophical conceptions they generated, deserve a recapitulation, as well as a new impulse for further development. Recently, there has been renewed interest in the concepts of normativity and supervenience. The research on (...) normativity – a term introduced to the philosophical jargon by Edmund Husserl almost one hundred years ago – gained impetus in the 1990s through the works of such philosophers as Robert Audi, Christine Korsgaard, Robert Brandom, Paul Boghossian or Joseph Raz. The problem of the nature and sources of normativity has been investigated not only in morals and in relation to language, but also in other domains, e.g. in law or in the c ontext of the theories of rationality. Supervenience, understood as a special kind of relation between properties and weaker than entailment, has become analytic philosophers’ favorite formal tool since 1980s. It features in the theories pertaining to mental properties, but also in aesthetics or the law. In recent years, the ‘marriage’ of normativity and supervenience has become an object of many philosophical theories as well as heated debates. It seems that the conceptual apparatus of the supervenience theory makes it possible to state precisely some claims pertaining to normativity, as well as illuminate the problems surrounding it. (shrink)

In this paper we investigate the role of performative contradictions in legal discourse. First of all we identify the argumentative roles of performative contradictions and two possible interpretations of them. With this done, we show that one use of performative contradictions can be fruitfully applied in analysing normative speech acts implementing norm enactment, namely, those speech acts that are designed to produce new legal norms. We conclude the paper by showing that our analysis provides strong support for Robert Alexy's claim-to-correctness (...) thesis, according to which speech acts of the norm-enacting kind raise a claim to correctness. (shrink)

This handbook offers a deep analysis of the main forms of legal reasoning and argumentation from both a logical-philosophical and legal perspective. These forms are covered in an exhaustive and critical fashion, and the handbook accordingly divides in three parts: the first one introduces and discusses the basic concepts of practical reasoning. The second one discusses the main general forms of reasoning and argumentation relevant for legal discourse. The third one looks at their application in law as well as at (...) the different areas of legal reasoning. The handbook’s division in three parts reflects its conceptual architecture, since legal reasoning and argumentation are considered in relation to the more general types of reasoning. (shrink)

As lawmakers produce norms, the underlying normative system is affected showing the intrinsic dynamism of law. Through undertaken actions of legal change, the normative system is continuously modified. In a usual legislative practice, the time for an enacted legal provision to be in force may differ from that of its inclusion to the legal system, or from that in which it produces legal effects. Even more, some provisions can produce effects retroactively in time. In this article we study a simulation (...) of such process through the formalisation of a temporalised logical framework upon which a novel belief revision model tackles the dynamic nature of law. Represented through intervals, the temporalisation of sentences allows differentiating the temporal parameters of norms. In addition, a proposed revision operator allows assessing change to the legal system by including a new temporalised literal while preserving the time-based consistency. This can be achieved either by pushing out conflictive pieces of pre-existing norms or through the modification of intervals in which such norms can be either in force, or produce effects. Finally, the construction of the temporalised revision operator is axiomatically characterised and its rational behavior proved through a corresponding representation theorem. (shrink)

In his work, Jules Coleman has held that the rule of recognition, if conceived of as a shared cooperative activity, should be the gateway through which to incorporate moral constraints on the content of law. This analysis, however, leaves unanswered two important questions. For one thing, we do not know when or even why morality becomes a criterion of legality. And, for another thing, we still do not know what conception of morality it is that we are dealing with. In (...) this article, we will attempt to clarify in greater depth what relations there are between the social practice of law and morality. We will thus see how the cooperative nature of social practices imbues law with a moral force, and how this makes it possible to establish a "weak" connection between law and morality: To see this, we will need to single out some basic features of cooperative social practices, thus setting out a suitable framework for the view just mentioned. (shrink)

This handbook offers a deep analysis of the main forms of legal reasoning and argumentation from both a logical-philosophical and legal perspective. These forms are covered in an exhaustive and critical fashion, and the handbook accordingly divides in three parts: the first one introduces and discusses the basic concepts of practical reasoning. The second one discusses the main general forms of reasoning and argumentation relevant for legal discourse. The third one looks at their application in law as well as at (...) the different areas of legal reasoning. The handbook’s division in three parts reflects its conceptual architecture, since legal reasoning and argumentation are considered in relation to the more general types of reasoning. (shrink)

In this paper we analyze the notion of collective trust within a multi-modal setting. We argue that collective trust is a scalable concept and therefore definable in qualitatively distinct levels or strengths. We show possible connections between different forms of group trust and the emergence of obligations within groups of agents. In particular, the notion of collective trust appears to be strong enough to entail forms of delegation that may have a deontic connotation: the trust a group puts in an (...) agent’s performance of an action yields to the result that it becomes obligatory in the interest of the group that the agent performs this action. (shrink)

This chapter offers a concise and elementary introduction to fundamental concepts in deductive and deontic reasoning.

In this paper we provide a formal analysis of the idea of normative co-ordination. We argue that this idea is based on the assumption that agents can achieve flexible co-ordination by conferring normative positions to other agents. These positions include duties, permissions, and powers. In particular, we explain the idea of declarative power, which consists in the capacity of the power-holder of creating normative positions, involving other agents, simply by proclaiming such positions. In addition, we account also for the concepts (...) of representation, namely the representatives capacity of acting in the name of his principal, and of mandate, which is the mandatees duty to act as the mandator has requested. Finally, we show how the framework can be applied to represent the contract-net protocol. Some brief remarks on future research and applications conclude this contribution. (shrink)