Normal Logic Research Articles

A new deduction system for deciding validity in modal logic K

A new deduction system for deciding validity for the minimal decidable normal modal logic K is presented in this paper. Modal logics could be very helpful in modeling dynamic and reactive systems such as bio-inspired systems and process algebras. In fact, recently it has been presented the Connectionist Modal Logics which combines the strengths of modal logics and neural networks. Thus, modal logic K is the basis for these approaches. Soundness, completeness, and the fact that the system itself is a decision procedure are proved in this paper. The main advantages of this approach are: first, the system is deterministic, that is, it generates one proof tree for a given formula; second, the system is a validity-checker, hence it generates a proof of a formula (if such exists); third, the language of deduction and the language of a logic coincide. Some of these advantages are compared to other classical approaches.

Ordered Completion for First-Order Logic Programs on Finite Structures

In this paper, we propose a translation from normal first-order logic programs under the answer set semantics to first-order theories on finite structures. Specifically, we introduce ordered completions which are modifications of Clark's completions with some extra predicates added to keep track of the derivation order, and show that on finite structures, classical models of the ordered-completion of a normal logic program correspond exactly to the answer sets (stable models) of the logic program.

Hybrid rules with well-founded semantics

A general framework is proposed for integration of rules and external first-order theories. It is based on the well-founded semantics of normal logic programs and inspired by ideas of Constraint Logic Programming (CLP) and constructive negation for logic programs. Hybrid rules are normal clauses extended with constraints in the bodies; constraints are certain formulae in the language of the external theory. A hybrid program consists of a set of hybrid rules and an external theory. Instances of the framework are obtained by specifying the class of external theories and the class of constraints. An example instance is integration of (non-disjunctive) Datalog with ontologies formalized in description logics. The paper defines a declarative semantics of hybrid programs and a goal-driven formal operational semantics. The latter can be seen as a generalization of SLS-resolution. It provides a basis for hybrid implementations combining Prolog with constraint solvers (such as ontology reasoners). Soundness of the operational semantics is proven. Sufficient conditions for decidability of the declarative semantics and for completeness of the operational semantics are given.

Guarded resolution for Answer Set Programming

AbstractWe investigate a proof system based on a guarded resolution rule and show its adequacy for the stable semantics of normal logic programs. As a consequence, we show that Gelfond–Lifschitz operator can be viewed as a proof-theoretic concept. As an application, we find a propositional theory EP whose models are precisely stable models of programs. We also find a class of propositional theories 𝓒P with the following properties. Propositional models of theories in 𝓒P are precisely stable models of P, and the theories in 𝓒T are of the size linear in the size of P.

La legge della salvezza: Bernardo di Clairvaux e il diritto monastico

Starting from the analysis of Bernard’s of Clairvaux work De praecepto et dispensatione, the article focuses on the ancient and medieval regulae vivendi in order to demonstrate that they represent a specific form of normativity, absolutely different from the classical roman jus and the medieval canon law. I analyze the peculiar way in which Bernard tried to describe the particular normative logic of the Rule of Benedict from the point of view of a western legal history of longue durée. In comparison with the civil law and the ecclesiastical legislation, a “rule of life” (regula vitae) is a special form of law which extends to the complete life of the individual, not a single moment. The juridical institution of the vow has an important place in this special normative logic.

Level Mapping Induced Loop Formulas for Weight Constraint and Aggregate Logic Programs

Level mapping and loop formulas are two different means to justify and characterize answer sets for normal logic programs. Both of them specify conditions under which a supported model is an answer set. Though serving a similar purpose, in the past the two have been studied largely in isolation with each other. In this paper, we study level mapping and loop formulas for weight constraint and aggregate (logic) programs. We show that, for these classes of programs, loop formulas can be devised from level mapping characterizations. First, we formulate a level mapping characterization of stable models and show that it leads to a new formulation of loop formulas for arbitrary weight constraint programs, without using any new atoms. This extends a previous result on loop formulas for weight constraint programs, where weight constraints contain only positive literals. Second, since aggregate programs are closely related to weight constraint programs, we further use level mapping to characterize the underlying answer set semantics based on which we formulate loop formulas for aggregate programs. The main result is that for aggregate programs not involving the inequality comparison operator, the dependency graphs can be built in polynomial time. This compares to the previously known exponential time method.

SLGAD Resolution for Inference on Logic Programs with Annotated Disjunctions

Logic Programs with Annotated Disjunctions (LPADs) allow to express probabilistic information in logic programming. The semantics of an LPAD is given in terms of the well-founded models of the normal logic programs obtained by selecting one disjunct from each ground LPAD clause. Inference on LPADs can be performed using either the system Ailog2, that was developed for the Independent Choice Logic, or SLDNFAD, an algorithm based on SLDNF. However, both of these algorithms run the risk of going into infinite loops and of performing redundant computations. In order to avoid these problems, we present SLGAD resolution that computes the (conditional) probability of a ground query from a range-restricted LPAD and is based on SLG resolution for normal logic programs. As SLG, it uses tabling to avoid some infinite loops and to avoid redundant computations. The performances of SLGAD are evaluated on classical benchmarks for normal logic programs under the well-founded semantics, namely a 2-person game and the ancestor relation, and on games of dice. SLGAD is compared with Ailog2 and SLDNFAD on the problems in which they do not go into infinite loops, namely those that are described by a modularly acyclic program. The results show that SLGAD is sometimes slower than Ailog2 and SLDNFAD but, if the program requires the repeated computations of the same goals, as for the dice games, then SLGAD is faster than both.

Fusion of sequent modal logic systems labelled with truth values

Fusion is a well-known form of combining normal modal logics endowed with a Hilbert calculi and a Kripke semantics. Herein, fusion is studied over logic systems using sequent calculi labelled with truth values and with a semantics based on a two-sorted algebra allowing, in particular, the representation of general Kripke structures. A wide variety of logics, including non-classical logics like, for instance, modal logics and intuitionistic logic can be presented by logic systems of this kind. A categorical approach of fusion is defined in the context of these logic systems. Preservation of soundness and completeness by fusion is studied. Soundness is preserved without further requirements, completeness is preserved under mild assumptions.

Building Peace between Global and Local Politics: The Cosmopolitical Ethics of Liberal Peacebuilding

What are the ethical implications of the shortcomings of liberal peacebuilding operations? This article addresses the problem by investigating the normative premises of liberal peacebuilding and its critics. Three ideal types of peacebuilding are singled out to illuminate the normative logic of current ‘revisionist’ proposals. These are: ‘re-liberal peacebuilding’ which prescribes a more coercive approach; ‘social peacebuilding’ that emphasizes local agency and the promotion of socioeconomic rights; and ‘multicultural peacebuilding’ that roots peace in indigenous norms and institutions. These alternatives are assessed with regard to their ability to promote both the autonomy and the basic needs of the affected population. It is concluded that only ‘social peacebuilding’ passes this test. It exemplifies a model of global governance where a cosmopolitan human rights agenda is consistent with the communitarian defence of political autonomy and cultural diversity.

Institutionalizing social entrepreneurship in regulatory space: Reporting and disclosure by community interest companies

In 2005, the British parliament passed legislation to make available the first new legal form of incorporation in over a century: the Community Interest Company (CIC). This initiative represented an important element within a larger set of public policy measures that aimed to create a more enabling environment for the accelerated growth of social entrepreneurship and, specifically, social enterprises. In an exploratory study, this paper presents an analysis of the regulatory space within which the reporting and disclosure practices for CICs were negotiated. Three elements within the regulatory space are identified as having explanatory value: regulatory boundaries that set and limit the terms of negotiation around regulatory practice; the key actors that engage in a process of negotiation around the establishment of actual practice; the range of debate and conflicting ideas that inform regulatory negotiation and legitimating consensus. The analysis suggests that a normative logic of light touch regulation was of particular importance within the wider UK policy context from within which CICs emerged and that the CIC Regulator acts as a mediator of disclosure information across multiple user constituencies. Empirically, this paper draws upon a sample of 80 published CIC annual reports to consider two aspects of CIC reporting: the quantity of information provided and the type of data presented. These data demonstrate the limitations and challenges of current CIC regulatory disclosure practices for key users of reporting information, particularly in terms of perceptions of organizational legitimacy. Conclusions are drawn concerning these limitations, particularly in terms of their implications for public policy. In terms of new research, this paper makes two important contributions. First, it develops theory in terms of (social) reporting and public policy with respect to the regulatory mechanisms that relate the two. This has yet to be explored in social entrepreneurship research. Second, this paper includes a preliminary examination of the reporting practices of CICs in their policy context, including an analysis of a sample of the publicly available CIC annual reports that have been filed to date. This data set has yet to be the subject of any other academic research.

A New Fuzzy Set Theory Satisfying All Classical Set Formulas

A new fuzzy set theory, C-fuzzy set theory, is introduced in this paper. It is a particular case of the classical set theory and satisfies all formulas of the classical set theory. To add a limitation to C-fuzzy set system, in which all fuzzy sets must be “non-uniform inclusive” to each other, then it forms a family of sub-systems, the Z-fuzzy set family. It can be proved that the Z0-fuzzy set system, one of Z-fuzzy set systems, is equivalent to Zadeh’s fuzzy set system. Analysis shows that 1) Zadeh’s fuzzy set system defines the relations A = B and A ⊆ B between two fuzzy sets A and B as “∀u ∈ U, (μA ∈ (u) = μB(u))” and “∀u ∈ U, (μA(u) ≤ μB(u))” respectively is inappropriate, because it makes all fuzzy sets be “non-uniformly inclusive”; 2) it is also inappropriate to define two fuzzy sets’ union and intersection operations as the max and min of their grades of membership, because this prevents fuzzy set’s ability to correctly reflect different kinds of fuzzy phenomenon in the natural world. Then it has to work around the problem by invent unnatural functions that are hard to understand, such as augmenting max and min for union and intersection to min{a + b, 1} and max{a + b – 1, 0}, but these functions are incorrect on inclusive case. If both pairs of definitions are used together, not only are they unnatural, but also they are still unable to cover all possible set relationships in the natural world; and 3) it is incorrect to define the set complement as 1 – μA(u), because it can be proved that set complement cannot exist in Zadeh’s fuzzy set, and it causes confusion in logic and thinking. And it is seriously mistaken to believe that logics of fuzzy sets necessarily go against classical and normal thinking, logic, and conception. The C-fuzzy set theory proposed in this paper overcomes all of the above errors and shortcomings, and more reasonably reflects fuzzy phenomenon in the natural world. It satisfies all relations, formulas, and operations of the classical set theory. It is consistent with normal, natural, and classical thinking, logic, and concepts.

Syntactic transformation rules under P-Stable semantics: theory and implementation

We study the simplification of normal logic programs under the P-Stable semantics, with respect to the notions of equivalence, using many of the transformati...

Political representation and the normative logic of two-level games

This paper develops the normative logic of two-level games linking international negotiation and domestic acceptability. The kernel of the logic is to be found in the claim that normatively governed relations involve agents simultaneously asking whether the expectations that they have under an agreement are reasonable given the expectations of others under that agreement. This normative logic mirrors the empirical logic that Putnam (1988) identified in his seminal account.The normative logic is derived from a consideration of relevant concepts of representation, and in particular the concepts of authorization in international negotiation and accountability in domestic ratification. Rawls’ (1996) distinction between the reasonable and the rational is then deployed to state normative conditions of domestic acceptability as well as the obligations of fairness that states owe to one another. Two implications for democratic theory are drawn.

Placing the Normative Logics of Accountability in “Thick” Perspective

This article provides a critical reflection on the heavily normative nature of current accountability debates. In particular, it explores three streams of normative discourse on nonprofit accountability: improving board governance, improving performance-based reporting, and demonstrating progress toward mission. The article describes how these logics are associated with three basic accountability regimes: coercive, technocratic, and adaptive. It then discusses how a focus on these normative logics and regimes, although important, can mask the realities of social structure and the relations of power that underlie them. The article proposes a more empirical approach to framing accountability—“thick description”—that might enable scholars better understand how social regimes of accountability actually operate in different contexts and in which the instruments of accountability are at least as likely to reproduce relationships of inequality as they are to overturn them.

Integrative economic ethics: foundations of a civilized market economy

List of figures Preface Translator's note Introduction: orientation in economic-ethical thinking Part I. Fundamental Concepts of Modern Ethics and the Approach of Integrative Economic Ethics: 1. The phenomenon of human morality: the normative logic of interpersonal relations 2. The moral point of view: philosophical developmental lines of rational ethics 3. Morality and economic rationality: integrative economic ethics as the rational ethics of economic activity Part II. Reflections on the Foundations of Economic Ethics I: A Critique of Economism: 4. 'Inherent necessity' of competition? A critique of economic determinism 5. 'Morality' of the market? A critique of economic reductionism Part III. Reflections on the Foundations of Economic Ethics II: Rational Economic Activity and the Lifeworld: 6. The question of meaning: economic activity and the good life 7. The question of legitimation: economic activity and the just social life Part IV. A Topology of Economic Ethics: The 'Sites' of Morality in Economic Life: 8. Economic citizens' ethics 9. Regulatory ethics 10. Corporate ethics Bibliography Indexes.

Sound and Complete SLD-Resolution for Bilattice-Based Annotated Logic Programs

We introduce the class of normal bilattice-based annotated first-order logic programs (BAPs) and develop declarative and operational semantics for them. In particular, SLD-resolution for these programs is defined and its soundness and completeness established.

What groups do, can do, and know they can do: an analysis in normal modal logics

We investigate a series of logics that allow to reason about agents' actions, abilities, and their knowledge about actions and abilities. These logics include Pauly's Coalition Logic CL, Alternating-time Temporal Logic ATL, the logic of ‘seeing-to-it-that' (STIT), and epistemic extensions thereof. While complete axiomatizations of CL and ATL exist, only the fragment of the STIT language without temporal operators and without groups has been axiomatized by Xu (called Ldm). We start by recalling a simplification of the Ldm that has been proposed in previous work, together with an alternative semantics in terms of standard Kripke models. We extend that semantics to groups via a principle of superadditivity, and give a sound and complete axiomatization that we call Ldm G. We then add a temporal ‘next' operator to Ldm G, and again give a sound and complete axiomatization. We show that Ldm G subsumes coalition logic CL. Finally, we extend these logics with standard S5 knowledge operators. This enables us to express that agents see to something under uncertainty about the present state or uncertainty about which action is being taken. We focus on the epistemic extension of X-Ldm G, noted E-X-Ldm G. In accordance with established terminology in the planning community, we call this extension of X-Ldm G the conformant X-Ldm G. The conformant X-Ldm G enables us to express that agents are able to perform a uniform strategy. We conclude that in that respect, our epistemic extension of X-Ldm G is better suited than epistemic extensions of ATL.

Nonmonotonic abductive inductive learning

Inductive Logic Programming (ILP) is concerned with the task of generalising sets of positive and negative examples with respect to background knowledge expressed as logic programs. Negation as Failure (NAF) is a key feature of logic programming which provides a means for nonmonotonic commonsense reasoning under incomplete information. But, so far, most ILP research has been aimed at Horn programs which exclude NAF, and has failed to exploit the full potential of normal programs that allow NAF. By contrast, Abductive Logic Programming (ALP), a related task concerned with explaining observations with respect to a prior theory, has been well studied and applied in the context of normal logic programs. This paper shows how ALP can be used to provide a semantics and proof procedure for nonmonotonic ILP that utilises practical methods of language and search bias to reduce the search space. This is done by lifting an existing method called Hybrid Abductive Inductive Learning (HAIL) from Horn clauses to normal logic programs. To demonstrate its potential benefits, the resulting system, called XHAIL, is applied to a process modelling case study involving a nonmonotonic temporal Event Calculus (EC).

Extended ASP Tableaux and rule redundancy in normal logic programs

AbstractWe introduce an extended tableau calculus for answer set programming (ASP). The proof system is based on the ASP tableaux defined in the work by Gebser and Schaub (Tableau calculi for answer set programming. In Proceedings of the 22nd International Conference on Logic Programming (ICLP 2006), S. Etalle and M. Truszczynski, Eds. Lecture Notes in Computer Science, vol. 4079. Springer, 11–25) with an added extension rule. We investigate the power of Extended ASP Tableaux both theoretically and empirically. We study the relationship of Extended ASP Tableaux with the Extended Resolution proof system defined by Tseitin for sets of clauses, and separate Extended ASP Tableaux from ASP Tableaux by giving a polynomial-length proof for a family of normal logic programs {Φn} for which ASP Tableaux has exponential-length minimal proofs with respect to n. Additionally, Extended ASP Tableaux imply interesting insight into the effect of program simplification on the lengths of proofs in ASP. Closely related to Extended ASP Tableaux, we empirically investigate the effect of redundant rules on the efficiency of ASP solving.

An algebraic generalization of Kripke structures

AbstractThe Kripke semantics of classical propositional normal modal logic is made algebraic via an embedding of Kripke structures into the larger class of pointed stably supported quantales. This algebraic semantics subsumes the traditional algebraic semantics based on lattices with unary operators, and it suggests natural interpretations of modal logic, of possible interest in the applications, in structures that arise in geometry and analysis, such as foliated manifolds and operator algebras, via topological groupoids and inverse semigroups. We study completeness properties of the quantale based semantics for the systems K, T, K4, S4 and S5, in particular obtaining an axiomatization for S5 which does not use negation or the modal necessity operator. As additional examples we describe intuitionistic propositional modal logic, the logic of programs PDL and the ramified temporal logic CTL.