Labelled Natural Deduction for Substructural Logics

In this paper a uniform methodology to perform Natural Deduction over the family of linear, relevance and intuitionistic logics is proposed. The methodology follows the Labelled Deductive Systems (LDS) discipline, where the deductive process manipulates declarative units { formulas labelled according to a labelling algebra. In the system de-scribed here, labels are either ground terms or variables of a given labelling language and inference rules manipulate formulas and labels simultaneously, generating (whenever necessary) constraints on the labels used in the rules. A set of natural deduction style inference rules is given, and the notion of a derivation is dened which associates a la-belled natural deduction style \structural derivation " with a set of generated constraints. Algorithmic procedures, based on a technique called resource abduction, are dened to solve the constraints generated within a derivation, and their termination conditions dis-cussed. A natural deduction derivation is correct with respect to a given substructural logic, if, under the condition that the algorithmic procedures terminate, the associated set of constraints is satised with respect to the underlying labelling algebra. This is shown by proving that the natural deduction system is sound and complete with respect to the LKE tableaux system [DG94].

Labelled natural deduction for substructural logics

In this paper a uniform methodology to perform Natural Deduction over the family of linear, relevance and intuitionistic logics is proposed. The methodology follows the Labelled Deductive Systems (LDS) discipline, where the deductive process manipulates declarative units - formulas labelled according to a labelling algebra. In the system described here, labels are either ground terms or variables of a given labelling language and inference rules manipulate formulas and labels simultaneously, generating (whenever necessary) constraints on the labels used in the rules. A set of natural deduction style inference rules is given, and the notion of a derivation is defined which associates a labelled natural deduction style "structural derivation" with a set of generated constraints. Algorithmic procedures, based on a technique called resource abduction, are defined to solve the constraints generated within a derivation, and their termination conditions discussed. A natural deduction derivation is correct with respect to a given substructural logic, if, under the condition that the algorithmic procedures terminate, the associated set of constraints is satisfied with respect to the underlying labelling algebra. This is shown by proving that the natural deduction system is sound and complete with respect to the LKE tableaux system

Logics for modelling collective attitudes

We introduce a number of logics to reason about collective propositional attitudes that are defined by means of the majority rule. It is well known that majoritarian aggregation is subject to irrationality, as the results in social choice theory and judgment aggregation show. The proposed logics for modelling collective attitudes are based on a substructural propositional logic that allows for circumventing inconsistent outcomes. Individual and collective propositional attitudes, such as beliefs, desires, obligations, are then modelled by means of minimal modalities to ensure a number of basic principles. In this way, a viable consistent modelling of collective attitudes is obtained

Refutation Systems : An Overview and Some Applications to Philosophical Logics

Refutation systems are systems of formal, syntactic derivations, designed to derive the non-valid formulas or logical consequences of a given logic. Here we provide an overview with comprehensive references on the historical development of the theory of refutation systems and discuss some of their applications to philosophical logics

Modal tableaux for nonmonotonic reasoning

The tableau-like proof system KEM has been proven to be able to cope with a wide variety of (normal) modal logics. KEM is based on D'Agostino and Mondadori's (1994) classical proof system KE, a combination of tableau and natural deduction inference rules which allows for a restricted ("analytic") Use of the cut rule. The key feature of KEM, besides its being based neither on resolution nor on standard sequent/tableau inference techniques, is that it generates models and checks them using a label scheme to bookkeep "world" paths. This formalism can be extended to handle various system of multimodal logic devised for dealing with nonmonotonic reasoning, by relying in particular on Meyer and van der Hoek's (1992) logic for actuality and preference. In this paper we shall be concerned with developing a similar extension this time by relying on Schwind and Siegel's (1993,1994) system H, another multimodal logic devised for dealing with nonmonotonic inference

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

Goal-directed proof theory

This report is the draft of a book about goal directed proof theoretical formulations of non-classical logics. It evolved from a response to the existence of two camps in the applied logic (computer science/artificial intelligence) community. There are those members who believe that the new non-classical logics are the most important ones for applications and that classical logic itself is now no longer the main workhorse of applied logic, and there are those who maintain that classical logic is the only logic worth considering and that within classical logic the Horn clause fragment is the most important one. The book presents a uniform Prolog-like formulation of the landscape of classical and non-classical logics, done in such away that the distinctions and movements from one logic to another seem simple and natural; and within it classical logic becomes just one among many. This should please the non-classical logic camp. It will also please the classical logic camp since the goal directed formulation makes it all look like an algorithmic extension of Logic Programming. The approach also seems to provide very good compuational complexity bounds across its landscape

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

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