Ideality and subideality from a computational point of view

Why should Law need automated proof systems? The answer to this question implies an answer to the following question: Is logic needed in Law? In fact it has been argued that logics are useless for Law (see, for example, Kelsen 1989). We believe that logic, and deontic logics in particular - but also modal logics - have a role to play in Law; for example if one wants to study what the relationships are among the various degrees of adjudication in Italian Law, one should note that they give rise to a transitive, irreflexive and finite structure, which is the frame of the modal logic of provability GL; one of the most important properties of such a logic is that no system, (no court) in this frame, could claim its own correctness without becoming incorrect (Boolos 1993, Smullyan 1988), but the correctness of a lower court can be established by a higher one. This example shows that the study of modal logic can help in finding certain already known properties of legal systems. Moreover, each time we are dealing with the notions of Obligation and Permission, and we are interested in the study of their mutual relationship, we can arrange them into a deontic framework, thus producing a certain kind of deontic logic. Finally a hint for the use of logic in legal reasoning is given, for example in the Italian case, by the law itself; in fact article 192, 1 comma of the "Italian code of criminal procedure" prescribes that the judges state the reasons of their adjudication; moreover several other articles of the same code, state: when evidence is valid, how evidence should be used in order to lead to an adjudication, etc. On this basis the "Italian code of criminal procedure" can be thought of as a deductive system where its articles act as the inference rules, whereas the articles of the "Italian code of criminal law" are the axioms. What does a proof system do? A proof system can work in two ways. The first of them consists of producing admissible steps one after the other according to the inference rules; in this way each step is guaranteed to be correct, but we are not led to the goal we want to prove. The other one consists of verifying whether a conclusion follows from given premises, i.e., if the adjudication follows logically from the evidence, mainly by refuting the negation of the conclusion. The system we propose is based on the logic of ideality and subideality developed by Jones and Porn, and it verifies in the above mentioned logical framework whether a given conclusion follows from given premises. Moreover, due to its basic control structure it can also be used as an analytic direct proof system

Labelled Tableaux for Multi-Modal Logics

In this paper we present a tableau-like proof system for multi-modal logics based on D'Agostino and Mondadori's classical refutation system $KE$. The proposed system, that we call $KEM$, works for the logics $S5A$ and $S5P_{(n)}$ which have been devised by Mayer and van der Hoek for formalizing the notions of actuality and preference. We shall also show how $KEM$ works with the normal modal logics $K45, D45$, and $S5$ which are frequently used as bases for epistemic operators -- knowledge, belief, and we shall briefly sketch how to combine knowledge and belief in a multi-agent setting through $KEM$ modularity

Labelling Ideality and Subideality

In this paper we suggest ways in which logic and law may usefully relate; and we present an analytic proof system dealing with the Jones Porn's deontic logic of Ideality and Subideality, which offers some suggestions about how to embed legal systems in label formalism

KE+: Beyond Refutation

The system KE+, a tableau-like proof system based on D'Agostino-Mondadori KE, is presented in this paper. This system avoids some of the drawbacks of other proof methods. In fact it is completely analytical, it is able to detect whether a formula is either a tautology or a contradiction or only a satisfiable one; in the course of a proof it can detect whether a subformula is a tautology and it uses this fact in the proof of the main formula

Labelled Modal Tableaux

Labelled tableaux are extensions of semantic tableaux with annotations (labels, indices) whose main function is to enrich the modal object language with semantic elements. This paper consists of three parts. In the first part we consider some options for labels: simple constant labels vs labels with free variables, logic depended inference rules vs labels manipulation based on a label algebra. In the second and third part we concentrate on a particular labelled tableaux system called KEM using free variable and a specialised label algebra. Specifically in the second part we show how labelled tableaux (KEM) can account for different types of logics (e.g., non-normal modal logics and conditional logics). In the third and final part we investigate the relative complexity of labelled tableaux systems and we show that the uses of KEM's label algebra can lead to speed up on proofs

An algorithm for the induction Of defeasible logic theories from databases

Defeasible logic is a non-monotonic logic with applications in rule-based domains such as law. To ease the development and improve the accuracy of expert systems based on defeasible logic, it is desirable to automatically induce a theory of the logic from a training set of precedent data. Empirical evidence suggests that minimal theories that describe the training set tend to be more faithful representations of reality. We show via transformation from the hitting set problem that this global minimization problem is intractable, belonging to the class of NP optimisation problems. Given the inherent difficulty of finding the optimal solution, we instead use heuristics and demonstrate that a best-first, greedy, branch and bound algorithm can be used to find good theories in short time. This approach displays significant improvements in both accuracy and theory size as compared to recent work in the area that post-processed the output of an Aprori association rule-mining algorithm, with comparable execution times

A Formal Ontology Reasoning with Individual Optimization: A Realization of the Semantic Web

Answering a query over a group of RDF data pages is a trivial process. However, in the Semantic Web, there is a need for ontology technology. Consequently, OWL, a family of web ontology languages based on description logic, has been proposed for the Semantic Web. Answering a query over the Semantic Web is thus not trivial, but a deductive process. However, the reasoning on OWL with data has an efficiency problem. Thus, we introduce optimization techniques for the inference algorithm. This work demonstrates the techniques for instance checking and instance retrieval problems with respect to ALC description logic which covers certain parts of OWL

Induction of defeasible logic theories in the legal domain

The market for intelligent legal information systems remains relatively untapped and while this might be interpreted as an indication that it is simply impossible to produce a system that satisfies the needs of the legal community, an analysis of previous attempts at producing such systems reveals a common set of deficiencies that in-part explain why there have been no overwhelming successes to date. Defeasible logic, a logic with proven successes at representing legal knowledge, seems to overcome many of these deficiencies and is a promising approach to representing legal knowledge. Unfortunately, an immediate application of technology to the challenges in this domain is an expensive and computationally intractable problem. So, in light of the benefits, we seek to find a practical algorithm that uses heuristics to discover an approximate solution. As an outcome of this work, we have developed an algorithm that integrates defeasible logic into a decision support system by automatically deriving its knowledge from databases of precedents. Experiments with the new algorithm are very promising - delivering results comparable to and exceeding other approaches