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Hierarchies of Regulations and Their Logic
Abstract
We study some of the ways in which the imposition of a partial ordering on a code of laws or regulations can serve to overcome logical imperfections in the code itself. In particular, we first show how partial orderings of a code, and derivative orderings of its power set, may be used to confer uniqueness upon otherwise indefinite derogations by ranking remainders; and second, we show how such orderings may be used to resolve contradictions implicit in a code by a process which we shall call delivery. Finally, we investigate the relations between derogation and delivery, showing that although the two processes appear and are generally assumed to be quite different from each other, nevertheless for finite inconsistent codes, the composite process of derogating and then selecting a remainder turns out to be equipowerful with delivery. For consistent codes, where delivery reduces to its underlying consequence operation and so is of no special interest, the correspondence is with a more general process of ‘relative delivery’. Sections 2 and 3, on derogation and the resolution of contradictions respectively, are written so that they may be read in either order. Section 4, on the relations between the two, depends on both. The study is carried out mathematically, and the reader is assumed to be familiar with elementary properties of partial orderings and consequence operations. Throughout, however, attention is also given to the realities of juridical practice.
... This basic principle is encapsulated by certain postulates, such as recovery [12,2], core-retainment, and relevance [17,10], as well as by constructing revision operators that restrict revisions to minimal subsets of the original belief base. 1 However intuitive and plausible minimalism appears to be, it does not always hold true in human belief revision dynamics. Studies in cognitive science show that when people are faced with inconsistencies, they first construct (or seek) explanations to resolve the inconsistencies, which consequently lead them to revise their beliefs in a non-minimal fashion [9,22,26,37]. ...
... Studies in cognitive science show that when people are faced with inconsistencies, they first construct (or seek) explanations to resolve the inconsistencies, which consequently lead them to revise their beliefs in a non-minimal fashion [9,22,26,37]. This is referred to as the explanatory hypothesis: 1 Note that there are various ways to measure minimal change depending on whether one is using a syntactic-based approach or a model-based approach [11]. ...
... The field of belief revision theory has experienced remarkable progress, primarily influenced by the foundational work of Alchourrón, Makinson, and Gärdenfors. Their studies on revisions in legal codes [1], the introduction of rationality postulates for change operators [12], and the development of the AGM model [2] have set the stage for subsequent advancements in the area. It is well known that one of the basic conceptual principles underlying the AGM model, as well as most belief revision frameworks, is the principle of minimalism. ...
In belief revision, agents typically modify their beliefs when they receive some new piece of information that is in conflict with them. The guiding principle behind most belief revision frameworks is that of minimalism, which advocates minimal changes to existing beliefs. However, minimalism may not necessarily capture the nuanced ways in which human agents reevaluate and modify their beliefs. In contrast, the explanatory hypothesis indicates that people are inherently driven to seek explanations for inconsistencies, thereby striving for explanatory coherence rather than minimal changes when revising beliefs. Our contribution in this paper is two-fold. Motivated by the explanatory hypothesis, we first present a novel, yet simple belief revision operator that, given a belief base and an explanation for an explanandum, it revises the belief bases in a manner that preserves the explanandum and is not necessarily minimal. We call this operator explanation-based belief revision. Second, we conduct two human-subject studies to empirically validate our approach and investigate belief revision behavior in real-world scenarios. Our findings support the explanatory hypothesis and provide insights into the strategies people employ when resolving inconsistencies.
... Abgeleitete oder zulässige Regeln (siehe gleich die Erklärung) mögen aber im Prinzip jede endliche Anzahl von Prämissen haben. 6 Die Alternative bestünde darin, bestimmte Formeln als Axiome (bzw. Regeln) herauszugreifen sowie eine Substitutionsregel hinzuzufügen, die aus dieser Formel alle Instanzen des entsprechenden Schemas hervorbringt. ...
... Auch in diesem Satz kommen -auf den ersten Blick -keine Quantoren vor. Seine Formalisierung in der PL ist dennoch naheliegend: (6) Liegt-in(Deutschland, Europa). ...
... Nun könnte es sein, daß wir Länder als Mengen von geographischen Punkten und (6) als die Behauptung einer Mengeninklusion auffassen wollen. D.h. (6) ist dann einfach nur kurz für (7) ∀x(x ∈ Deutschland → x ∈ Europa). ...
... The beginnings of belief change theory can be traced back to the work of Isaac Levi [37], who discussed the fundamental problems concerning this field of research, and to the work of William Harper [32], who proposed a rational way to interrelate belief change operators. However, the main thrust in belief change theory came during the 1980s with Carlos Alchourrón and David Makinson's studies on changes in legal codes [2], and the introduction of rationality postulates for change operators by Peter Gärdenfors [23]. Later, the three authors lay the main building blocks of the AGM model in their seminal work [1]; it is from this work that belief change theory has evolved. ...
... Then, by Observation 1 we have that K⊥ ⊥α = K⊥ ⊥β, and since σ is a function it holds that σ (K⊥ ⊥α) = σ (K⊥ ⊥β) (1). By Definition 6 we have that K σ α = K \ σ (K⊥ ⊥α) and that K σ β = K \ σ (K⊥ ⊥β) (2). Then, from (1) and (2) it follows that K σ α = K σ β. 2 ...
... Also, since κ ⊂ K, then β ∈ K (2). It follows from (1), (2) and our definition of ϕ that β ∈ κ ∩ ϕ(K⊥ ⊥α). Thus, κ ∩ ϕ(K⊥ ⊥α) = ∅. ...
The problem of knowledge evolution has received considerable attention over the years. Mainly, the study of the dynamics of knowledge has been addressed in the area of Belief Revision, a field emerging as the convergence of the efforts in Philosophy, Logic, and more recently Computer Science, where research efforts usually involve “flat” knowledge bases where there is no additional information about the formulas stored in it. Even when this may be a good fit for particular applications, in many real-world scenarios different information items may be attached to formulas. For instance, when the reliability of the source of the piece of information is attached to it as a measure of some quality (e.g., strength) of the piece of information itself, or when some characteristic informs us on the desirability of the item (e.g., the potential benefit that could be obtained from it). If this type of information is available, we can use it to guide how the belief base is to be modified when new information arrives. In this work, we present a novel approach to the contraction of knowledge bases where formulas have values attached that measure some quality linked to those formulas, exploiting it to define their desirability, and uses such desirability to define which formulas need to be removed to solve conflicts. In this context, we introduce a set of properties for contraction operators by extending classic approaches. We also show how the local treatment of minimal conflicts can induce some counter-intuitive contractions, and we present a way to avoid them by considering optimal resolutions of conflicts using the additional information encoded. We show how the proposed formalization captures any contraction that is optimal under a set of features. Finally, we present a refinement based on the identification of related minimal conflicts that performs contraction in optimal ways without looking into the entire knowledge base. The approach is based on the use of the accrual of beliefs where several formulas collaboratively use their respective values to prevail in the resolution of conflicts.
... Carlos Alchourrón (1931Alchourrón ( -1996 and David Makinson cooperated in studies of changes in legal codes, analysing the logical structure of the derogation procedure in which a norm is removed from a legal code. They tried to find the general principles that any derogation should satisfy, and defined a family of all the possible derogations [5]. The key idea was, given a code A, to create a partial order on the norms of A and induce an order on the set of parts of A. The maximal sets of A that did not imply the norm to be removed were called remainders. ...
... In order to express this more precisely, the following notation is useful: Definition 3.5. [5] Let K be a belief set and p a sentence. The set K ⊥ p ("K remainder p") is the set of sets such that H ∈ K ⊥ p if and only if: ...
... Furthermore, S (K, p) is the set of saturatable sets with respect to p that are subsets of K. As Alchourrón and Makinson proved, K⊥p ⊆ S (K, p) [5]. Partial meet Levi contraction, based on a selection among all the saturatable subsets of K with respect to p, is defined as K − p = ⋂γ(S (K, p)), where γ is a selection function defined in the same way as in the AGM account. ...
It was understood from the beginning that the use of logically closed sets of sentences to represent belief states is not cognitively realistic. In an article published in 1985 Makinson pointed out that “in real life, when we perform a contraction or derogation, we never do it to the theory itself (in the sense of a set of propositions closed under consequence) but rather on some finite or recursive or at least recursively enumerable base for the theory” [238, p. 357]. The use of belief bases rather than (logically closed) belief sets has turned out to increase the expressive power of the belief change framework in important ways.
... Carlos Alchourrón (1931Alchourrón ( -1996 and David Makinson cooperated in studies of changes in legal codes, analysing the logical structure of the derogation procedure in which a norm is removed from a legal code. They tried to find the general principles that any derogation should satisfy, and defined a family of all the possible derogations [5]. The key idea was, given a code A, to create a partial order on the norms of A and induce an order on the set of parts of A. The maximal sets of A that did not imply the norm to be removed were called remainders. ...
... In order to express this more precisely, the following notation is useful: Definition 3.5. [5] Let K be a belief set and p a sentence. The set K ⊥ p ("K remainder p") is the set of sets such that H ∈ K ⊥ p if and only if: ...
... Furthermore, S (K, p) is the set of saturatable sets with respect to p that are subsets of K. As Alchourrón and Makinson proved, K⊥p ⊆ S (K, p) [5]. Partial meet Levi contraction, based on a selection among all the saturatable subsets of K with respect to p, is defined as K − p = ⋂γ(S (K, p)), where γ is a selection function defined in the same way as in the AGM account. ...
An AGM contraction or revision takes us from a belief set to a new belief set. In doing this, it makes use of a selection mechanism such as a selection function or an entrenchment relation. However, it does not provide a new selection mechanism to be used for further changes of the new belief set.
... Carlos Alchourrón (1931Alchourrón ( -1996 and David Makinson cooperated in studies of changes in legal codes, analysing the logical structure of the derogation procedure in which a norm is removed from a legal code. They tried to find the general principles that any derogation should satisfy, and defined a family of all the possible derogations [5]. The key idea was, given a code A, to create a partial order on the norms of A and induce an order on the set of parts of A. The maximal sets of A that did not imply the norm to be removed were called remainders. ...
... In order to express this more precisely, the following notation is useful: 3.8 Partial Meet Operations 21 Definition 3.5. [5] Let K be a belief set and p a sentence. The set K ⊥ p ("K remainder p") is the set of sets such that H ∈ K ⊥ p if and only if: ...
... Furthermore, S (K, p) is the set of saturatable sets with respect to p that are subsets of K. As Alchourrón and Makinson proved, K⊥p ⊆ S (K, p) [5]. Partial meet Levi contraction, based on a selection among all the saturatable subsets of K with respect to p, is defined as K − p = ⋂γ(S (K, p)), where γ is a selection function defined in the same way as in the AGM account. ...
Classical AGM operations model the belief changes of a single agent. They can be extended so that more than one agent is involved. In many situations coherent beliefs based on several sources are needed.
... Carlos Alchourrón (1931Alchourrón ( -1996 and David Makinson cooperated in studies of changes in legal codes, analysing the logical structure of the derogation procedure in which a norm is removed from a legal code. They tried to find the general principles that any derogation should satisfy, and defined a family of all the possible derogations [5]. The key idea was, given a code A, to create a partial order on the norms of A and induce an order on the set of parts of A. The maximal sets of A that did not imply the norm to be removed were called remainders. ...
... In order to express this more precisely, the following notation is useful: Definition 3.5. [5] Let K be a belief set and p a sentence. The set K ⊥ p ("K remainder p") is the set of sets such that H ∈ K ⊥ p if and only if: ...
... Furthermore, S (K, p) is the set of saturatable sets with respect to p that are subsets of K. As Alchourrón and Makinson proved, K⊥p ⊆ S (K, p) [5]. Partial meet Levi contraction, based on a selection among all the saturatable subsets of K with respect to p, is defined as K − p = ⋂γ(S (K, p)), where γ is a selection function defined in the same way as in the AGM account. ...
This is a brief introduction to the theory of belief change. It provides an example of a belief change problem, and lists some of the major issues that are investigated in this research area.
... Carlos Alchourrón (1931Alchourrón ( -1996 and David Makinson cooperated in studies of changes in legal codes, analysing the logical structure of the derogation procedure in which a norm is removed from a legal code. They tried to find the general principles that any derogation should satisfy, and defined a family of all the possible derogations [5]. The key idea was, given a code A, to create a partial order on the norms of A and induce an order on the set of parts of A. The maximal sets of A that did not imply the norm to be removed were called remainders. ...
... In order to express this more precisely, the following notation is useful: 3.8 Partial Meet Operations 21 Definition 3.5. [5] Let K be a belief set and p a sentence. The set K ⊥ p ("K remainder p") is the set of sets such that H ∈ K ⊥ p if and only if: ...
... Furthermore, S (K, p) is the set of saturatable sets with respect to p that are subsets of K. As Alchourrón and Makinson proved, K⊥p ⊆ S (K, p) [5]. Partial meet Levi contraction, based on a selection among all the saturatable subsets of K with respect to p, is defined as K − p = ⋂γ(S (K, p)), where γ is a selection function defined in the same way as in the AGM account. ...
The purpose of this chapter is to introduce the AGM account of belief change, originally developed by Alchourrón, Gärdenfors and Makinson [4]. In Sections 3.1–3.3 we introduce the formal apparatus of belief sets and in Section 3.4 the operations of change. In Sections 3.5–3.6 we introduce the axioms of the AGM model. In Section 3.7 the relations between contraction and revision are specified and in Section 3.8 we introduce the basic construtive method of the AGM model, partial meet contraction and revision functions.
... The design of logics for exceptions has been of interest for several years [5,106,134], and is related to the broader area of non-monotonic reasoning [97,113,117,126]. ...
... A bloodbank is not prevented from using a different kind of test for source plasma donations. (6) can be represented by first producing (5), and then inferring that (5) and (6) imply the following: ...
... Identifying a compelling semantics for says is an important open problem in access control logics (see [1]), and we do not address it in this work. 5 We begin by defining models (Kripke structures): ...
We consider the problem of checking whether an organization conforms to a body of regulation. Conformance is studied in a runtime verification setting. The regulation is translated to a logic, from which we synthesize monitors. The monitors are evaluated as the state of an organization evolves over time, raising an alarm if a violation is detected. An important challenge to this approach comes from the fact that regulations are commonly expressed in natural language. The translation to logic is difficult. Our goal is to assist in this translation by: (a) the design of logics that let us formalize regulation one sentence at a time, and (b) the use of natural language processing as an aid in the sentential translation.
There are many features that are needed in a logic, to accommodate a sentential translation of regulation. We study two features, motivated by a case study. First, statements in regulation refer to others for conditions or exceptions. Second, sentences in regulation convey legal concepts, e.g., obligation and permission. Obligations and permissions can be nested to convey concepts, such as, rights. We motivate and design a logic to accomodate these two features of regulatory texts. The common theme is the importance of the notion of {\em saying} in such constructs.
We begin by extending linear temporal logic to allow statements to refer to others. Inter-sentential references are expressed via the use of a predicate, called "says", whose interpretation is determined by inferences from laws. The "says" predicate offers a unified analysis of various kinds of inter-sentential references, e.g., priorities of exceptions over rules, and references to definitions or list items.
We then augment the logic with obligation and permission, by considering problems in access control and conformance. Saying and permission are combined using an axiom that permits a principal to speak on behalf of another. The combination yields benefits to both applications. For access control, we overcome the problematic interactions between hand-off and classical reasoning. For conformance, we obtain a characterization of legal power by nesting saying with obligation and permission. A useful fragment of the logic has a polynomial time decision procedure.
Finally, we turn to the use of natural language processing to translate a sentence to logic. We study one component of the translation in a supervised learning setting. Linguistic theories have argued for a level of logical form as a prelude to translating a sentence into logic. Logical form encodes a resolution of scope ambiguties. We define a restricted kind of logical form, called abstract syntax trees (ASTs), based on the logic developed. Guidelines for annotating ASTs are formulated, using a case study of the Food and Drug Administration's Code of Federal Regulations.
We describe experiments on a modest-sized corpus, of about 200 sentences, annotated with ASTs. The main step in computing ASTs is the ordering or ranking of operators. We adapt a learning model for ranking to order operators. Features are designed by studying subproblems, such as, disambiguating between de re and de dicto interpretations. We obtain an F-score of 90.6% on the set of pairwise ordering decisions.
... It seems vain to attempt an exhaustive review of defeasible reasoning. Before the backdrop of foundational law theory [16], there are sometimes diverging proposals for integrating defeasibility, sometimes opting for non-monotonic logics [15], sometimes taking a more classical stance [2]. Defeasible rule-based reasoning in the context of argumentation theory is discussed in [10,3]. ...
... The only fact is is_legal(a), and there are 2 modifiers despite(1, 2), strong_subject_to (3,2). There are no minimal inconsistent sets. ...
The paper studies defeasible reasoning in rule-based systems, in particular about legal norms and contracts. We identify rule modifiers that specify how rules interact and how they can be overridden. We then define rule transformations that eliminate these modifiers, leading in the end to a translation of rules to formulas. For reasoning with and about rules, we contrast two approaches, one in a classical logic with SMT solvers as proof engines, one in a non-monotonic logic with Answer Set Programming solvers.
... With this article, we intend to provide further mathematical foundations of selective revision. 2 The rest of the paper is organized as follows: In Section 2 we introduce the notations and recall the main background concepts that will be needed throughout this article. In Section 3 we present a formal definition of selective base revision and introduce some desirable properties that transformation functions should satisfy. ...
... In this example, the new information is considered to be "a dinosaur broke my youngest daughter's grandmother's vase in the living room" and not "my youngest daughter just told me that a dinosaur broke her grandmother's vase in the living room" (if that were the case, then the new information should be fully incorporated).2 It may also serve to complement the philosophical foundations of this kind of operation that might come in the future.Content courtesy of Springer Nature, terms of use apply. ...
Belief Revision addresses the problem of rationally incorporating pieces of new information into an agent’s belief state. In the AGM paradigm, the most used framework in Belief Revision, primacy is given to the new information, which is fully incorporated into the agent’s belief state. However, in real situations, one may want to reject the new information or only accept a part of it. A constructive model called Selective Revision was proposed to meet this need but, as in the AGM framework, focused on belief sets (sets closed under logical consequence). In this paper we adapt the selective revision operators, that were proposed for belief sets, to the belief base context, obtaining a model in which an agent’s epistemic state is represented by a belief base and that allows the acceptance of only part of the new information. We present several representation theorems for selective base revision operators based on different base revision operators.
... First, a person might be held responsible with reference to a certain normative source (set of norms) and not responsible with reference to another. The ultimate decision on whether a person or a group of people is to be blamed for something often depends on a hierarchy of normative sources [1]. Different norms can disagree, giving rise to normative conflicts; in this case, a norm can be derogated due to its incompatibility with a more important one. ...
... For a more elaborated formulation of a hierarchy among normative sources, see[1]. ...
The present article is devoted to a logical treatment of some fundamental concepts involved in responsibility attribution. We specify a theoretical framework based on a language of temporal deontic logic with agent-relative operators for deliberate causal contribution. The framework is endowed with a procedure to solve normative conflicts which arise from the assessment of different normative sources. We provide a characterization result for a basic system within this framework and illustrate how the concepts formalized can be put at work in the analysis of examples of legal reasoning.
... En segundo lugar, porque toda ordenación tiene efectos derogatorios, como ya mostraron Alchourrón y Makinson. Véase Alchourron, Carlos;Makinson, David (1981). "Hierarchies of Regulation and Th eir Logic". ...
... En segundo lugar, porque toda ordenación tiene efectos derogatorios, como ya mostraron Alchourrón y Makinson. Véase Alchourron, Carlos;Makinson, David (1981). "Hierarchies of Regulation and Th eir Logic". ...
The present work constitutes an attempt of reformulation, in iusrealistic terms, of the theory of the dogmatic concepts. It is structured into five sections. The first is about trying to delimit the scope of dogmatic concepts, distinguishing them from other types of concepts present in legal language. Second, we analyze and criticize the reconstruction of Alf Ross of dogmatic concepts, to thirdly address how they are formulated. The fourth section deals with the functions frequently performed by those within the science of law. It concludes with an analysis of the problems encountered in the use and formulation of these concepts.
... The similarity was first extended to conditional versions of those logics by Lewis (1973). Interestingly, AGM belief revision theory has an equally important normative origin in Alchourrón and Makinson (1981), although the main bulk of subsequent theorizing was devoted to formal epistemology. As mentioned, the argumentation theoretic approach treats all arguments on a par, anyway. ...
... Even if that order is largely implicit, it operates in the background and controls the consistency of all such changes. This has indeed been the original intention of Alchourrón and Makinson (1981). ...
The paper is motivated by the need of accounting for the practical syllogism as a piece of defeasible reasoning. To meet the need, the paper first refers to ranking theory as an account of defeasible descriptive reasoning. It then argues that two kinds of ought need to be distinguished, purely normative and fact-regarding obligations (in analogy to intrinsic and extrinsic utilities). It continues arguing that both kinds of ought can be iteratively revised and should hence be represented by ranking functions, too, just as iteratively revisable beliefs. Its central proposal will then be that the fact-regarding normative ranking function must be conceived as the sum of a purely normative ranking function and an epistemic ranking function (as suggested in qualitative decision theory). The distinctions defends this proposal with a comparative discussion of some critical examples and some other distinctions made in the literature. It gives a more rigorous justification of this proposal. Finally, it starts developing the logic of purely normative and of fact-regarding normative defeasible reasoning, points to the difficulties of completing the logic of the fact-regarding side, but reaches the initial aim of accounting for the defeasible nature of the practical syllogism.
... Let a, p and x respectively denote the propositions that you go to the party; you drink; and you drive. In terms of a hierarchical normative systems [1], these norms are respectively represented as (a, p) 1 , (p, ¬x) 3 and (a, x) 2 . These three norms are visualized in Figure 1(a). ...
... An argument is represented as a path of a directed graph starting from a node in the context. In this simple example, there are four arguments A 0 , A 1 In this example, the last link principle thus gives the same result as the Reduction approach, and weakest link gives the same result as the Greedy approach. Liao et al. [35] show that this is not a coincidence, but it holds for all totally ordered normative systems. ...
... Various solutions have been offered to overcome this problem when formalizing legal texts. Among them, Alchourrón and Makinson (1981) have proposed partially ordering the precedence of different legal sources, a concept which can be captured by relatively weak logics. ...
There is an inherent tension between knowledge representation and reasoning. For an optimal representation and validation, an expressive language should be used. For an optimal automated reasoning, a simple one is preferred. Which language should we choose for our legal knowledge representation if our goal is to apply automated legal reasoning? In this paper, we investigate the properties and requirements of each of these two applications. We suggest that by using Legal Linguistic Templates, one can solve the above tension in some practical situations.
... Nonetheless, it can be argued, that some of the first steps were given in Isaac Levi's work in [34], where the fundamental problems concerning this field of research were discussed, and in the work of William Harper's in [26], where a rational way to interrelate belief change operators were proposed. Later, essential developments in the field came from the work of Carlos Alchourrón and David Makinson [2], and also by Peter Gärdenfors [21]. These works eventually came together with the three authors laying the main building blocks of the AGM model in their seminal work [1], setting the foundations from where belief change theory eventually evolved. ...
Merging operators represent a significant tool to extract a consistent and informative view from a set of agents. The consideration of practical scenarios where someagents can be more credible than others has contributed to substantially increase the interest in developing systems working with trust models. In this context, we propose an approach to the problem of merging knowledge in a multiagent scenario where every agent assigns to other agents a value reflecting its perception on how credible each agent is. The focus of this paper is the introduction of an operator for merging Datalog± ontologies considering agents’ credibility. We present a procedure to enhance a conflict resolution strategy by exploiting the credibility attached to a set of formulas; the approach is based on accrual functions that calculate the value of formulas according to the credibility of the agents that inform them. We show how our new operator can obtain the best-valued knowledge base among consistent bases available, according to the credibilities attached to the sources.
... Belief revision (Gärdenfors and Rott 1994, Williams and Rott 2001, Rott 2001, among many others) is concerned with belief change fired by incoming information. It emerged as a proper research field when philosophical traditions dealing with both the requirements of rational belief change and the mechanisms by which scientific theories develop (Harper 1977) converged with computerscience oriented approaches on database updates (Doyle 1979, de Kleer 1986 and deontic studies focused on derogations in legal codes (Alchourrón and Makinson 1981). The seminal Alchourrón et al. (1985), the AGM approach, is considered the birth of the field; since then, many of its concerns and ideas have been proved relevant in philosophy, computer science, learning theory and other fields. ...
Belief revision is concerned with belief change fired by incoming information. Despite the variety of frameworks representing it, most revision policies share one crucial feature: incoming information outweighs current information and hence, in case of conflict, incoming information will prevail. However, if one is interested in representing the way actual humans revise their beliefs, one might not always want for the agent to blindly believe everything they are told. This manuscript presents a semantic approach to non-prioritized belief revision. It uses plausibility models for depicting an agent’s beliefs, and model operations for displaying the way beliefs change. The first proposal, semantically-based screened revision, compares the current model with the one the revision would yield, accepting or rejecting the incoming information depending on whether the ‘differences’ between these models go beyond a given threshold. The second proposal, semantically-based gradual revision, turns the binary decision of acceptance or rejection into a more general setting in which a revision always occurs, with the threshold used rather to choose ‘the right revision’ for the given input and model.
... It follows from compacteness and Zorn's lemma (cf. (Alchourrón and Makinson 1981, Proof of Observation 2.2)) that, given a set of sentences A and a sentence α, every subset D of A that does not imply α is contained in some remainder of A by α. This property is known as upper bound property: ...
In this paper, we propose and axiomatically characterize residual contractions, a new kind of contraction operators for belief bases. We establish that the class of partial meet contractions is a strict subclass of the class of residual contractions. We identify an extra condition that may be added to the definition of residual contractions, which is such that the class of residual contractions that satisfy it coincides with the class of partial meet contractions. We investigate the interrelations in the sense of (strict) inclusion among the class of residual contractions and other classes of well known contraction operators for belief bases.
... (7) La séptima tesis -como se recordará-consiste en sostener que, interpretando holísticamente las disposiciones jurídicas, los jueces pueden llegar a decidir cualquier controversia, 39 Alchourrón & Makinson, 1981. de manera consistente y moralmente correcta. ...
La expresión “neoconstitucionalismo” es sumamente ambigua: la cultura contemporánea la usa para denotar cosas tan heterogéneas como una forma de estado, una política de activismo judicial, una reconstrucción teórica de los sistemas jurídicos contemporáneos, una metodología jurídica y una adhesión ideológica al derecho. Este trabajo se centra en los aspectos de política jurisprudencial, de teoría del derecho y de metodología jurídica, enfatizando dos posibles perspectivas del neoconstitucionalismo: una negativa que delinea dicha multifacética concepción del derecho como un conjunto de oposiciones a las tesis defendidas por las concepciones dominantes, y otra positiva que pretende haber identificado una serie de “nuevos descubrimientos” en los sistemas jurídicos contemporáneos. En esta versión positiva se examinan ocho tesis: (1) el derecho de los estados constitucionales es una combinación de reglas y principios, (2) reglas y principios son aptos para constituir un conjunto sistematico de normas, (3) las constituciones contemporáneas incorporan valores morales, de manera que habrían creado una conexión relevante entre derecho y moral, (4) tal incorporación implica un deber de interpretar las disposiciones constitucionales mediante una lectura moral, (5) toda norma jurídica es derrotable, (6) algunos estándares jurídicos requieren de un razonamiento ponderativo o de balance, opuesto al tradicional razonamiento subsutivo, (7) interpretando holísticamente las disposiciones jurídicas, los jueces pueden llegar a decidir cualquier controversia, y, (8) las normas jurídicas no pueden tener cualquier contenido, porque su contenido es determinado en última instancia por juicios morales. El trabajo termina ilustrando aciertos y equivocaciones de cada versión del neoconstitucionalismo.
... Alchourrón and Makinson were the first to logically study the changes of a legal code [2,3,1]. The addition of a new norm n causes an enlargement of the code, consisting of the new norm plus all the regulations that can be derived from n. Alchourrón and Makinson distinguish two other types of change. ...
This paper presents a belief revision operator for legal systems that considers time intervals. This model relates techniques about 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.
... OTIVATED by the need to consider real situations that do not fall into the strict rules of classical logic, parallel studies have been created and have resulted in other logics that are alternatives to the classical one [1]. Non-classical logics investigate, among other things, the situations excluded from the classical logic which are, for example, the existing logical values other than the "True" and the "False", allowing better adaptation of concepts such as; uncertainties, ambiguities, paradoxes and inconsistencies [2], [3]. ...
In a modern society the factor corresponding to the increase in the level of quality in industrial production demand new techniques of control and machinery automation. In this context, this work presents the implementation of a Paraconsistent-Fuzzy Digital PID controller. The controller is based on the treatment of inconsistencies both in the Paraconsistent Logics and in the Fuzzy Logic. Paraconsistent analysis is performed on the signals applied to the system inputs using concepts from the Paraconsistent Annotated Logic with annotation of two values (LPA2v). The signals resulting from the paraconsistent analysis are two values defined as Dc-Degree of Certainty and Dct-Degree of Contradiction, which receive a treatment according to the Fuzzy Logic theory, and the resulting output of the logic actions is a single value called the crisp value, which is used to control dynamic system. Through an example, it was demonstrated the application of the proposed model. Initially, the Paraconsistent-Fuzzy Digital PID controller was built and tested in an isolated MATLAB environment and then compared to the equivalent Digital PID function of this software for standard step excitation. After this step, a level control plant was modeled to execute the controller function on a physical model, making the tests closer to the actual. For this, the control parameters (proportional, integral and derivative) were determined for the configuration of the conventional Digital PID controller and of the Paraconsistent-Fuzzy Digital PID, and the control meshes in MATLAB were assembled with the respective transfer function of the plant. Finally, the results of the comparison of the level process control between the Paraconsistent-Fuzzy Digital PID controller and the conventional Digital PID controller were presented.
... OTIVATED by the need to consider real situations that do not fall into the strict rules of classical logic, parallel studies have been created and have resulted in other logics that are alternatives to the classical one [1]. Non-classical logics investigate, among other things, the situations excluded from the classical logic which are, for example, the existing logical values other than the "True" and the "False", allowing better adaptation of concepts such as; uncertainties, ambiguities, paradoxes and inconsistencies [2], [3]. Classical logic considers only two possible states "true" or "false". ...
In a modern society, the factor corresponding to the increase in the level of quality in industrial production demand new techniques of control and machinery automation. This work presents the implementation of a Paraconsistent-Fuzzy Digital PID controller simulation in Matlab. The controller is based on the treatment of non-trivial inconsistencies both in the Paraconsistent Logics and in the Fuzzy Logic. The paraconsistent analysis is performed on the signals applied to the system inputs using concepts from the Paraconsistent Logic Annotated with 2 values-LPA2v. The signals resulting from the paraconsistent analysis are two values defined as Gc-Degree of Certainty and Gi-Degree of inconsistency, which receives a treatment according to the Fuzzy Logic theory, and the resulting output of the two logic actions is a single value called the crisp value, which is used to control dynamic system. Initially, the Paraconsistent-Fuzzy Digital PID controller was built and tested in an isolated Matlab environment and then compared to the equivalent Fuzzy PID function of this software for standard excitations such as step and pulse. After this step, a level control plant was modeled to execute the controller function on a physical model, making the tests closer to the actual ones. For this, the control parameters (proportional, integral and derivative) were calculated for the configuration of the fuzzy controller conventional and of the Paraconsistent-Fuzzy PID Digital controller, and the control meshes in Matlab were assembled with the respective transfer function of the plant. Finally, the results of the comparison of the level process control between the Paraconsistent-Fuzzy PID Digital controller and the conventional Fuzzy PID controller were presented.
... Since the work of Alchourrón and Makinson[1]on hierarchies of regulations and their logic, in which a partial ordering on a code of laws or regulations is used to overcome logical imperfections in the code itself, reasoning with prioritized norms has been a central challenge in deontic logic[13,4,12]. The goal of this paper is to study the open issue of reasoning with priorities over norms through the lens of argumentation theory[10]. ...
To resolve conflicts among norms,various nonmonotonic formalisms can be used to perform prioritized normative reasoning. Meanwhile, formal argumentation provides a way to represent nonmonotonic logics. In this paper, we propose a representation of prioritized normative reasoning by argumentation. Using hierarchical abstract normative systems, we define three kinds of prioritized normative reasoning approaches, called Greedy, Reduction, and Optimization. Then, after formulating an argumentation theory for a hierarchical abstract normative system, we show that for a totally ordered hierarchical abstract normative system, Greedy and Reduction can be represented in argumentation by applying the weakest link and the last link principles respectively, and Optimization can be represented by introducing additional defeats capturing implicit conflicts between arguments.
... 37 Though, as I suggested, quite a large number of the theories had bigger ambitions since their creators often wanted to constitute a 'universal' deontic logic capturing logical properties of "ought sentences" all at once in both their descriptive and prescriptive interpretations. 38 Alchourrón and Makinson (1981), Meyer (1988), Brown (2004), Lindström and Segerberg (2006), or Stolpe (2010) can be seen as contributions to this area. 39 This doesn't hold if we consider complex systems with hierarchized prescriptions. ...
Philosophers like G.H. von Wright and D. Makinson have pointed to serious challenges regarding the foundations of deontic logic. In this paper, I suggest that to deal successfully with these challenges a reconsideration of the research program of the discipline is useful. Some problems that have troubled this particular field of logical study for decades may disappear or appear more tractable if we view them from the perspective of a language game introduced by D. Lewis involving three characters: the Master, the Slave and the Kibitzer. The adoption of this perspective opens a natural approach to a new layout of the domain of deontic studies. I propose dividing deontic logic into six sub-areas which are distinguished (i) by their focus on the different idioms typical of the individual players, (ii) by conceiving the language game as either being static or as dynamic and (iii) by the aims of the logical inquiry. What kind of insights the proposed perspective provides is illustrated by an analysis of the so-called Ross paradox—a problem that has troubled deontic logic since its origins and, though it was many times pronounced solved, still keeps coming back ‘alive and kicking’.
... Definition 10 (Alchourrón and Makinson 1981;Hansson 1993) Let K and A be subsets of L. Then K ⊥A is the set of all subsets K of K such that A ∩ Cn(K ) = ∅ and that there is no set K such that K ⊂ K ⊆ K and A ∩ Cn(K ) = ∅. ...
In this paper we study AGM contraction and revision of rules using input/output logical theories. We replace propositional formulas in the AGM framework of theory change by pairs of propositional formulas, representing the rule based character of theories, and we replace the classical consequence operator Cn by an input/output logic. The results in this paper suggest that, in general, results from belief base dynamics can be transferred to rule base dynamics, but that a similar transfer of AGM theory change to rule change is much more problematic. First, we generalise belief base contraction to rule base contraction, and show that two representation results of Hansson still hold for rule base contraction. Second, we show that the six so-called basic postulates of AGM contraction are consistent only for some input/output logics, but not for others. In particular, we show that the notorious recovery postulate can be satisfied only by basic output, but not by simple-minded output. Third, we show how AGM rule revision can be defined in terms of AGM rule contraction using the Levi identity. We highlight various topics for further research.
... Naturalmente, el código mismo no será modificado, para hacerlo consistente, eliminando las contradicciones, operación que transformaría su naturaleza misma, aunque en ello consiste buena parte de la labor argumentativa de los jueces. Este tipo de transformación o cambio de un código ha sido propuesto por Makinson y Alchourrón (1981), como un modelo posible para explicar características del razonamiento jurídico llevado a cabo por jueces y abogados. En realidad, cuando se procura hacer consistente a un código contradictorio no permanecemos propiamente dicho en el código original, sino que lo reemplazamos por otro, dado que el conjunto que lo constituye tendrá una composición diferente a la del conjunto original. ...
Este artículo profundiza en ámbitos relacionales entre lógicas no clásicas, prácticas jurídicas y educación del derecho. A través de una revisión de la literatura científica, se abordan en particular las lógicas intuicionistas, difusas y paraconsistente, examinando sus desafíos y contribuciones al área jurídica. A pesar de los obstáculos, como la resistencia al cambio, se constatan avances significativos en la adopción de estos enfoques. El artículo concluye exponiendo las diversas perspectivas empleadas y sosteniendo una transformación en la educación del derecho, tanto en el currículo oficial como en el oculto de la formación del abogado, con el objetivo de abordar de manera más integral los problemas complejos, contradictorios e inciertos que caracterizan al mundo actual.
El trabajo estudia los conflictos constitucionales como una especie de teoría moral deontológica pluralista moderada, por lo que se centra en los principios constitucionales, la proporcionalidad y la función de los tribunales constitucionales. A raíz de un análisis de las estructuras morales deontológicas (cuantitativas y no cuantitativas) y de un repaso por los conceptos de derogación, laguna normativa y jerarquía normativa, se analiza el teorema Alchourrón-Makinson. Se concluye, en base al teorema, que la proporcionalidad constitucional es una modalidad de cambio normativo. Según esto, los tribunales constitucionales actuarían como órganos de revisión constitucional que derogan algunas consecuencias normativas.
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.
This article argues that Robert Alexy's influential theory of balancing is affected by a contradiction that makes it unfeasible as an instrument by which to explain some aspects of law and legal reasoning it aims to clarify. In particular, I will show that one of the premises of Alexy's theory of balancing is incompatible with its conclusion. Alexy's theory is based upon a sharp distinction between rules and principles. However, as my analysis will demonstrate, its conclusion implies that it is impossible to distinguish between rules and principles. This is because the so‐called weight formula and the law of colliding principles (i.e., the two main notions used by Alexy to explain balancing) cancel out any difference between these two types of norms.
Two of the most well-known belief contraction operators are partial meet contractions (PMCs) and kernel contractions (KCs). In this paper we propose two new classes of contraction operators, namely the class of generalized partial meet contractions (GPMC) and the class of generalized kernel contractions (GKC), which strictly contain the classes of PMCs and of KCs, respectively. We identify some extra conditions that can be added to the definitions of GPMCs and of GKCs, which give rise to some interesting subclasses of those classes of functions, namely the classes of extensional and of uniform GPMCs/GKCs. In the context of contractions on belief sets the classes of partial meet contractions, uniform GPMCs and extensional GPMCs are all identical. Nevertheless, when considered as operations on belief bases, the class of uniform GPMCs coincides with the class of partial meet contractions, but the extensional GPMCs constitute a new kind of belief base contraction functions whose characterizing postulate of irrelevance of syntax is extensionality—the same postulate of irrelevance of syntax which occurs in the classical axiomatic characterization of partial meet contractions for belief sets—rather than the postulate of uniformity—which is the irrelevance of syntax postulate used in the axiomatic characterization for partial meet contractions on belief bases. Analogous results are obtained regarding the classes of extensional and of uniform GKCs. We present the interrelations in the sense of inclusion among all the new classes of operators presented in this paper and several well known classes of PMCs and of KCs.
This paper aims at comparing and relating belief revision and argumentation as approaches to model reasoning processes. Referring to some prominent literature references in both fields, we will discuss their (implicit or explicit) assumptions on the modeled processes and hence commonalities and differences in the forms of reasoning they are suitable to deal with. The intended contribution is on one hand assessing the (not fully explored yet) relationships between two lively research fields in the broad area of defeasible reasoning and on the other hand pointing out open issues and potential directions for future research.
Belief revision is a hallmark of knowledge representation, logic, and philosophy. However, despite the extensive research in the area, we believe a fresh take on belief revision is needed. To that end, it is our conviction that believing a piece of information depends on trust in information sources that conveyed said piece and that trust in information sources is affected by changes in beliefs. Trust is also an impresa of philosophy and all time favorite of psychology and multi-agent systems. Hence, many approaches were developed for trust representation, yet, in isolation from belief revision. While admittedly crucial to a realistic treatment of belief revision, trust revision, to our dismay, did not receive the same level of attention. In this paper, we argue that a formal treatment for the joint interdependent revision of belief and trust is called for. Moreover, we propose a new framework called information revision that captures the joint revision of belief and trust. Further, we provide postulates that govern such process of revision. Finally, we provide a class of operators called relevant change propagation operators and provide their representation theorem.
AGM’s belief revision is one of the main paradigms in the study of belief change operations. Despite its popularity and importance to the area, it is well recognized that AGM’s work relies on a strong idealisation of the agent’s capabilities and on the nature of beliefs themselves. Particularly, it is well-recognized in the area of Epistemology that Belief and Knowledge are hyperintensional notions, but to our knowledge only a few works have explicitly considered how hyperintensionality affects belief change. In this work, we investigate belief change operations based on Berto’s topic-sensitive framework and provide three different pseudo-contraction operations to account for the hyperintensional behaviour of beliefs. Our work highlights the connection of a foundational hyperintensional theory of belief with the results of AGM Belief Change. Also we propose and characterise different possible contraction-like operations in this framework.
Conflicts between legal norms are common in reality. In many legislations, legal conflicts between norms are resolved
by applying ordered principles. This work presents a formalization of the conflict resolution mechanism and introduces
action legal logic (ALL) to reason about the normative consequences of possibly conflicting legal systems. The semantics of
ALL is explicitly based on legal systems consisting of norms and ordered principles. Legal systems specify the legal status
of transitions in transition systems and the language of ALL describes the legal status of paths in transition systems. The
formalization is used to study abstract revisions of legal systems. The expressivity of ALL is studied and its completeness is
proved.
Updating a knowledge base to remove an unwanted consequence is a challenging task. Some of the original sentences must be either deleted or weakened in such a way that the sentence to be removed is no longer entailed by the resulting set. On the other hand, it is desirable that the existing knowledge be preserved as much as possible, minimising the loss of information. Several approaches to this problem can be found in the literature. In particular, when the knowledge is represented by an ontology, two different families of frameworks have been developed in the literature in the past decades with numerous ideas in common but with little interaction between the communities: applications of AGM-like Belief Change and justification-based Ontology Repair. In this paper, we investigate the relationship between pseudo-contraction operations and gentle repairs. Both aim to avoid the complete deletion of sentences when replacing them with weaker versions is enough to prevent the entailment of the unwanted formula. We show the correspondence between concepts on both sides and investigate under which conditions they are equivalent. Furthermore, we propose a unified notation for the two approaches, which might contribute to the integration of the two areas.
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.
All formal models of belief change involve choices between different ways to accommodate new information. However, the models differ in their loci of choice, i.e. in what formal entities the choice mechanism is applied to. Four models of belief change with different loci of choice are investigated in terms of how they satisfy a set of important properties of belief contraction and revision. It is concluded that the locus of epistemic choice has a large impact on the properties of the resulting belief change operation.
The one that is considered the standard model in the area of logic of belief change was proposed by Alchourrón, Gärdenfors and Makinson in 1985 and is, nowadays, known as the AGM model.
Two of the main shortcomings pointed out to the AGM model of belief change are the (impractical) use of belief sets to represent belief states and the (unrealistic) acceptance of any new information. In this paper we study a kind of operators—known as shielded base contractions—which address both those issues. Indeed, on the one hand, these operators are defined on belief bases (rather than belief sets) and, on the other hand, they are constructed with the underlying idea that not all new informations are accepted.
We propose twenty different classes of shielded base contractions and obtain axiomatic characterizations for each of them. Additionally we thoroughly investigate the interrelations (in the sense of inclusion) among all those classes. More precisely, we analyse whether each of those classes is or is not (strictly) contained in each of the remaining ones.
Normative systems accommodate temporary norms of several types, which can also be modified in different, and codified ways. In this paper we address the problem of modifying temporary norms that are represented by means of the combination of two known formalisms in the current literature. The framework evolves from a known one, which provides a system of norms at two distinct layers, and represents changes at the two layers as means to provide room for the codified change types. This results in four novel operators that anticipate and extend norms in two different combined ways, by preserving or not the effects of the norms in the period of time generated by the temporal modifications. We study these new operators and show how they relate to the operators of annulment and abrogation analysed elsewhere.
This paper proposes three subtle revision policies that are not propositionally successful (after a single application the agent might not believe the given propositional formula), but nevertheless are not propositionally idempotent (further applications might affect the agent’s epistemic state). It also compares them with two well-known revision policies, arguing that the subtle ones might provide a more faithful representation of humans’ real-life revision processes.
The paper is intended as a first, tentative, contribution to the clarification of the place that negation has in prescriptive discourse. In particular, the paper analyzes the ways in which rules may be said to be negated and the meanings they assume when they are so regarded. In so doing, the differences between external and internal negation of conditional rules are examined. The paper also deals with the effects of inconsistency between conditional rules, understood as the conjunction of a conditional rule and its corresponding conditional denial. The main result of the paper is that both rules negation and rules inconsistency are unclear concepts, casting their shadows over the very concept of rule.
As Paul V. Spade remarks “[t]here are many puzzles for historians of medieval logic” and “[o]ne of them concerns the peculiar form of disputation described in treatises de obligationibus.” (Spade 1992, 171.) In the present paper, we claim that the theory of obligationes as presented by Walter Burley in the section de positione of his Treatise on obligations can be seen as a theory of belief change.
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