18 research outputs found
Combining explicit negation and negation by failure via Belnap's logic
AbstractThis paper deals with logic programs containing two kinds of negation: negation as failure and explicit negation. This allows two different forms of reasoning in the presence of incomplete information. Such programs have been introduced by Gelfond and Lifschitz and called extended programs. We provide them with a logical semantics in the style of Kunen, based on Belnap's four-valued logic, and an answer sets' semantics that is shown to be equivalent to that of Gelfond and Lifschitz.The proofs rely on a translation into normal programs, and on a variant of Fitting's extension of logic programming to bilattices
Investigations in Belnap's Logic of Inconsistent and Unknown Information
Nuel Belnap schlug 1977 eine vierwertige Logik vor, die -- im Gegensatz zur klassischen Logik -- die Faehigkeit haben sollte, sowohl mit widerspruechlicher als auch mit fehlender Information umzugehen. Diese Logik hat jedoch den Nachteil, dass sie Saetze der Form 'wenn ..., dann ...' nicht ausdruecken kann. Ausgehend von dieser Beobachtung analysieren wir die beiden nichtklassischen Aspekte, Widerspruechlichkeit und fehlende Information, indem wir eine dreiwertige Logik entwickeln, die mit widerspruechlicher Information umgehen kann und eine Modallogik, die mit fehlender Information umgehen kann. Beide Logiken sind nicht monoton. Wir untersuchen Eigenschaften, wie z.B. Kompaktheit, Entscheidbarkeit, Deduktionstheoreme und Berechnungkomplexitaet dieser Logiken. Es stellt sich heraus, dass die dreiwertige Logik, nicht kompakt und ihre Folgerungsmenge im Allgemeinen nicht rekursiv aufzaehlbar ist. Beschraenkt man sich hingegen auf endliche Formelmengen, so ist die Folgerungsmenge rekursiv entscheidbar, liegt in der Klasse der polynomiellen Zeithierarchie und ist DIFFP-schwer. Wir geben ein auf semantischen Tableaux basierendes, korrektes und vollstaendiges Berechnungsverfahren fuer endliche Praemissenmengen an. Darueberhinaus untersuchen wir Abschwaechungen der Kompaktheitseigenschaft. Die nichtmonotone auf S5-Modellen basierende Modallogik stellt sich als nicht minder komplex heraus. Auch hier untersuchen wir eine sinnvolle Abschwaechung der Kompaktheitseigenschaft. Desweiteren studieren wir den Zusammenhang zu anderen nichtmonotonen Modallogiken wie Moores autoepistemischer Logik (AEL) und McDermotts NML-2. Wir zeigen, dass unsere Logik zwischen AEL und NML-2 liegt. Schliesslich koppeln wir die entworfene Modallogik mit der dreiwertigen Logik. Die dabei enstehende Logik MKT ist eine Erweiterung des nichtmonotonen Fragments von Belnaps Logik. Wir schliessen unsere Betrachtungen mit einem Vergleich von MKT und verschiedenen informationstheoretischen Logiken, wie z.B. Nelsons N und Heytings intuitionistischer Logik ab
Neuere Entwicklungen der deklarativen KI-Programmierung : proceedings
The field of declarative AI programming is briefly characterized. Its recent developments in Germany are reflected by a workshop as part of the scientific congress KI-93 at the Berlin Humboldt University. Three tutorials introduce to the state of the art in deductive databases, the programming language Gödel, and the evolution of knowledge bases. Eleven contributed papers treat knowledge revision/program transformation, types, constraints, and type-constraint combinations
The Liar Paradox: A Consistent and Semantically Closed Solution
This thesis develops a new approach to the formal de nition of a truth predicate that allows a consistent, semantically closed defiition within classical logic. The approach is built on an analysis of structural properties of languages that make Liar Sentences and the paradoxical argument
possible. By focusing on these conditions, standard formal dfinitions of semantics are shown to impose systematic limitations on the definition of formal truth predicates.
The alternative approach to the formal definition of truth is developed by analysing our intuitive procedure for evaluating the truth value of sentences like "P is true". It is observed that the standard procedure breaks down in the case of the Liar Paradox as a side effect of the patterns of naming or reference necessary to the definition of Truth as a predicate. This means there are two ways in which a sentence like "P is true" can be not true, which requires that the T-Schema be modified for such sentences.
By modifying the T-Schema, and taking seriously the effects of the patterns of naming/ reference on truth values, the new approach to the definition of truth is developed. Formal truth definitions within classical logic are constructed that provide an explicit and adequate truth definition for their own language, every sentence within the languages has a truth value, and there is no Strengthened Liar Paradox. This approach to solving the Liar Paradox can be easily applied to a very wide range of languages, including natural languages
On the notion of negation in certain non-classical propositional logics
The purpose of this study is to investigate some aspects of how negation functions in certain non-classical propositional logics. These include the intuitionistic system developed by Heyting, the minimal calculus proposed by Johansson, and various intermediate logics between the minimal and the classical systems. Part I contains the new results which can be grouped into two classes: extension-criteria results and infinite chain results. In the first group criteria are given for answering the question: when do formulae added to the axioms of the minimal calculus as extra axioms extend the minimal calculus to various known intermediate logics? One of the results in this group (THEOREM 1 in Chapter II, Section 1) is a generalization of a result of Jankov. In the second group certain intermediate logics are defined which form infinite chains between well-known logical systems. One of the results here (THEOREM 1 in Chapter II, Section 2) is a generalization of a result of McKay. In Part II the new results are discussed from the viewpoint of negation. It is rather difficult, however, to draw definite conclusions which are acceptable to all. For these depend on, and are closely bound up with, certain basic philosophical presuppositions which are neither provable, nor disprovable in a strict sense. Taking an essentially classical position, it is argued that the logics appearing in the defined infinite chains are such that they diverge only in the vicinity of negation, and the notions of negation in them are simply ordered in a sense which is specified during the discussion. In Appendix I a number of conjectures are formulated in connection with the new results.<p
LDS - Labelled Deductive Systems: Volume 1 - Foundations
Traditional logics manipulate formulas. The message of this book is to manipulate pairs; formulas and labels. The labels annotate the formulas. This sounds very simple but it turned out to be a big step, which makes a serious difference, like the difference between using one hand only or allowing for the coordinated use of two hands. Of course the idea has to be made precise, and its advantages and limitations clearly demonstrated. `Precise' means a good mathematical definition and `advantages demonstrated' means case studies and applications in pure logic and in AI. To achieve that we need to address the following: \begin{enumerate} \item Define the notion of {\em LDS}, its proof theory and semantics and relate it to traditional logics. \item Explain what form the traditional concepts of cut elimination, deduction theorem, negation, inconsistency, update, etc.\ take in {\em LDS}. \item Formulate major known logics in {\em LDS}. For example, modal and temporal logics, substructural logics, default, nonmonotonic logics, etc. \item Show new results and solve long-standing problems using {\em LDS}. \item Demonstrate practical applications. \end{enumerate} This is what I am trying to do in this book. Part I of the book is an intuitive presentation of {\em LDS} in the context of traditional current views of monotonic and nonmonotonic logics. It is less oriented towards the pure logician and more towards the practical consumer of logic. It has two tasks, addressed in two chapters. These are: \begin{itemlist}{Chapter 1:} \item [Chapter1:] Formally motivate {\em LDS} by starting from the traditional notion of `What is a logical system' and slowly adding features to it until it becomes essentially an {\em LDS}. \item [Chapter 2:] Intuitively motivate {\em LDS} by showing many examples where labels are used, as well as some case studies of familiar logics (e.g.\ modal logic) formulated as an {\em LDS}. \end{itemlist} The second part of the book presents the formal theory of {\em LDS} for the formal logician. I have tried to avoid the style of definition-lemma-theorem and put in some explanations. What is basically needed here is the formulation of the mathematical machinery capable of doing the following. \begin{itemize} \item Define {\em LDS} algebra, proof theory and semantics. \item Show how an arbitrary (or fairly general) logic, presented traditionally, say as a Hilbert system or as a Gentzen system, can be turned into an {\em LDS} formulation. \item Show how to obtain a traditional formulations (e.g.\ Hilbert) for an arbitrary {\em LDS} presented logic. \item Define and study major logical concepts intrinsic to {\em LDS} formalisms. \item Give detailed study of the {\em LDS} formulation of some major known logics (e.g.\ modal logics, resource logics) and demonstrate its advantages. \item Translate {\em LDS} into classical logic (reduce the `new' to the `old'), and explain {\em LDS} in the context of classical logic (two sorted logic, metalevel aspects, etc). \end{itemize} \begin{itemlist}{Chapter 1:} \item [Chapter 3:] Give fairly general definitions of some basic concepts of {\em LDS} theory, mainly to cater for the needs of the practical consumer of logic who may wish to apply it, with a detailed study of the metabox system. The presentation of Chapter 3 is a bit tricky. It may be too formal for the intuitive reader, but not sufficiently clear and elegant for the mathematical logician. I would be very grateful for comments from the readers for the next draft. \item [Chapter 4:] Presents the basic notions of algebraic {\em LDS}. The reader may wonder how come we introduce algebraic {\em LDS} in chapter 3 and then again in chapter 4. Our aim in chapter 3 is to give a general definition and formal machinery for the applied consumer of logic. Chapter 4 on the other hand studies {\em LDS} as formal logics. It turns out that to formulate an arbitrary logic as an {\em LDS} one needs some specific labelling algebras and these need to be studied in detail (chapter 4). For general applications it is more convenient to have general labelling algebras and possibly mathematically redundant formulations (chapter 3). In a sense chapter 4 continues the topic of the second section of chapter 3. \item [Chapter 5:] Present the full theory of {\em LDS} where labels can be databases from possibly another {\em LDS}. It also presents Fibred Semantics for {\em LDS}. \item [Chapter 6:] Presents a theory of quantifers for {\em LDS}. The material for this chapter is still under research. \item [Chapter 7:] Studies structured consequence relations. These are logical system swhere the structure is not described through labels but through some geometry like lists, multisets, trees, etc. Thus the label of a wff is implicit, given by the place of in the structure. \item [Chapter 8:] Deals with metalevel features of {\em LDS} and its translation into two sorted classical logic. \end{itemlist} Parts 3 and 4 of the book deals in detail with some specific families of logics. Chapters 9--11 essentailly deal with substructural logics and their variants. \begin{itemlist}{Chapter10:} \item [Chapter 9:] Studies resource and substructural logics in general. \item [Chapter 10:] Develops detailed proof theory for some systems as well as studying particular features such as negation. \item [Chapter 11:] Deals with many valued logics. \item [Chapter 12:] Studies the Curry Howard formula as type view and how it compres with labelling. \item [Chapter 13:] Deals with modal and temporal logics. \end{itemlist} Part 5 of the book deals with {\em LDS} metatheory. \begin{itemlist}{Chapter15:} \item [Chapter 14:] Deals with labelled tableaux. \item [Chapter 15:] Deals with combining logics. \item [Chapter 16:] Deals with abduction. \end{itemlist
Paradoxes and the foundations of semantics and metaphysics
Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Linguistics and Philosophy, 2000.Includes bibliographical references (p. 175-180).Numerous philosophical problems, otherwise quite different in character, are of the following form. Certain claims which seem not only obviously true, but even constitutive of the meanings of the expressions employed, can be shown to lead to absurdity when taken together (perhaps in conjunction with contingent facts about the world). All such problems can justly be called paradoxes. The paradoxes I examine are the liar paradox, the sorites paradox, and the personal identity paradox posed by the fission problem. I argue that, in all of the cases examined, the claims that jointly lead to absurdity really are constitutive of the meanings of the expressions employed, in the following ways. First, semantic competence with the expressions involves being disposed to accept these claims. Second, the claims are reference-determining, in that the semantic values of the expressions employed are constrained by the condition that these claims should come out true, or as nearly true as possible. If a claim or principle is constitutive of meaning in both of these ways, I call it meaning-constitutive. When the meaning-constitutive principles for some expressions of a language are inconsistent, I call the language inconsistent. This is a stipulative definition; but it accords well with what other theorists who have talked about languages being inconsistent, for example Alfred Tarski, have had in mind. In chapters one and two, I argue that our language is inconsistent. In chapter three, I relate my theses to Frank Jackson's and David Lewis's views on how reference is determined. Another problem posed by the liar paradox concerns important theses in the philosophy of language. The liar reasoning shows that under certain conditions, a natural language cannot contain a predicate satisfying the T-schema. But many theses in the philosophy of language presuppose that truth satisfies the T-schema. I resolve this conflict in a Tarskian way: by saying that truth then is expressed only in an essentially richer metalanguage. However, I argue that taking this route means having to embrace the existence of absolutely inexpressible properties - and even to embrace the conclusion that some properties of which we appear to have concepts are absolutely inexpressible. All this is dealt with in chapter four. In the fifth chapter I show that my arguments of the previous chapters have (dis)solved the liar paradox. And finally, in the sixth chapter, I discuss the philosophical significance of truth and logic, and argue that these questions are significant only if understood in a new way. In this last chapter I also discuss the implications of the liar paradox for metaphysics; more specifically, its implications for the issue of how metaphysical claims are justified.by Matti Eklund.Ph.D