Heinrich Wansing

Displaying modal logic

Preface. 1. Introduction. 2. Sequents Generalized. 3. Display Logic. 4. Properly Displayable Logics, Displayable Logics and Strong Cut-Elimination. 5. A Proof-Theoretic Proof of Functional Completeness for Many Modal and Tense Logics. 6. Modal Tableaux Based on Residuation. 7. Strong Cut-Elimination and Labelled Modal Tableaux. 8. Tarskian Structured Consequence Relations and Functional Completeness. 9. Constructive Negation and the Modal Logic of Consistency. 10. Displaying as Temporalizing. 11. Translation of Hypersequents into Display Sequents. 12. Predicate Logics on Display. 13. Appendix. Bibliography. Index.

Proof theory of modal logic

Preface. Part I: Standard Proof Systems. 1. A Contraction-free Sequent Calculus for S4 J. Hudelmaier. 2. Transfer of Sequent Calculus Strategies to Resolution for S4 G. Mints, et al. 3. A Linear Approach to Modal Proof Theory H. Schellinx. 4. Refutations and Proofs in S4 T. Skura. Part II: Extended Formalisms. 5. Relational Proof Systems for Modal Logics E. Orlowska. 6. The Display Problem N. Belnap. 7. Power and Weakness of the Modal Display Calculus M. Kracht. 8. A Proof-Theoretic Proof of Functional Completeness for Many Modal and Tense Logics H. Wansing. 9. On the Completeness of Classical Modal Display Logic R. Gore. 10. Modal Sequents C. Cerrato. 11. Modal Functional Completeness K. Dosen, Z. Petric. 12. A Computational Interpretation of Modal Proofs S. Martini, A. Masini. 13. Gabbay-Style Calculi S. Mikulas. Part III: Translation-Based Proof Systems. 14. Translating Graded Modalities into Predicate Logic H.J. Ohlbach, et al. 15. From Classical to Normal Modal Logics O. Gasquet, A. Herzig.

What is Negation

Preface. Part I: Models, Relevance and Impossibility. Negation: Two Points of View A. Avron. A Comparative Study of Various Model-Theoretic Treatments of Negation: A History of Formal Negation J.M. Dunn. Negation in Relevant Logics (How I Stopped Worrying and Learned to Love the Routley Star) G. Restall. Negation in the Light of Modal Logic K. Dosen. Part II: Paraconsistency, Partiality and Logic Programming. Negation and Contradiction D. Gabbay, A. Hunter. What not? A Defence of Dialetheic Theory of Negation G. Priest. Partial Logics with Two Kinds of Negation as a Foundation for Knowledge-Based Reasoning H. Herre, et al. From Here to There: Stable Negation in Logic Programming D. Pearce. Part III: Absurdity, Falsity and Refutability. Antirealism and Falsity M. Hand. Negation, Absurdity and Contrariety N. Tennant. Negation as Falsity: A Reply to Tennant H. Wansing. Part IV: Negations, Natural Language and The Liar. Models for Non-Boolean Negations in Natural Languages Based on Aspect Analysis M. La Palme Reyes, et al. Negation, Denial and Language Change in Philosophical Logic J. Tappenden. What is That Item Designated Negation? R. Sylvan. Index.

Sequent Systems for Modal Logics

This chapter surveys the application of various kinds of sequent systems to modal and temporal logic, also called tense logic. The starting point are ordinary Gentzen sequents and their limitations both technically and philosophically. The rest of the chapter is devoted to generalizations of the ordinary notion of sequent. These considerations are restricted to formalisms that do not make explicit use of semantic parameters like possible worlds or truth values, thereby excluding, for instance, Gabbay’s labelled deductive systems, indexed tableau calculi, and Kanger-style proof systems from being dealt with. Readers interested in these types of proof systems are referred to [Gabbay, 1996], [Gore, 1999] and [Pliuskeviene, 1998]. Also Orlowska’s [1988; 1996] Rasiowa-Sikorski-style relational proof systems for normal modal logics will not be considered in the present chapter. In relational proof systems the logical object language is associated with a language of relational terms. These terms may contain subterms representing the accessibility relation in possible-worlds models, so that semantic information is available at the same level as syntactic information. The derivation rules in relational proof systems manipulate finite sequences of relational formulas constructed from relational terms and relational operations. An overview of ordinary sequent systems for non-classical logics is given in [Ono, 1998], and for a general background on proof theory the reader may consult [Troelstra and Schwichtenberg, 2000].

Some Useful 16-Valued Logics: How a Computer Network Should Think

In Belnap’s useful 4-valued logic, the set 2={T,F} of classical truth values is generalized to the set 4=℘(2)={∅,{T},{F},{T,F}}. In the present paper, we argue in favor of extending this process to the set 16=℘(4) (and beyond). It turns out that this generalization is well-motivated and leads from the bilattice FOUR2 with an information and a truth-and-falsity ordering to another algebraic structure, namely the trilattice SIXTEEN3 with an information ordering together with a truth ordering and a (distinct) falsity ordering. Interestingly, the logics generated separately by the algebraic operations under the truth order and under the falsity order in SIXTEEN3 coincide with the logic of FOUR2, namely first degree entailment. This observation may be taken as a further indication of the significance of first degree entailment. In the present setting, however, it becomes rather natural to consider also logical systems in the language obtained by combining the vocabulary of the logic of the truth order and the falsity order. We semantically define the logics of the two orderings in the extended language and in both cases axiomatize a certain fragment comprising three unary operations: a negation, an involution, and their combination. We also suggest two other definitions of logics in the full language, including a bi-consequence system. In other words, in addition to presenting first degree entailment as a useful 16-valued logic, we define further useful 16-valued logics for reasoning about truth and (non-)falsity. We expect these logics to be an interesting and useful instrument in information processing, especially when we deal with a net of hierarchically interconnected computers. We also briefly discuss Arieli’s and Avron’s notion of a logical bilattice and state a number of open problems for future research.

A general possible worlds framework for reasoning about knowledge and belief

In this paper non-normal worlds semantics is presented as a basic, general, and unifying approach to epistemic logic. The semantical framework of non-normal worlds is compared to the model theories of several logics for knowledge and belief that were recently developed in Artificial Intelligence (AI). It is shown that every model for implicit and explicit belief (Levesque), for awareness, general awareness, and local reasoning (Fagin and Halpern), and for awareness and principles (van der Hoek and Meyer) induces a non-normal worlds model validating precisely the same formulas (of the language in question).

Constructive negation, implication, and co-implication

In this paper, a family of paraconsistent propositional logics with constructive negation, constructive implication, and constructive co-implication is introduced. Although some fragments of these logics are known from the literature and although these logics emerge quite naturally, it seems that none of them has been considered so far. A relational possible worlds semantics as well as sound and complete display sequent calculi for the logics under consideration are presented.

Inconsistency-tolerant Description Logic: Motivation and Basic Systems

In this paper, three systems of constructive and inconsistency-tolerant description logic are motivated and defined semantically. Moreover, sound and complete tableau calculi for these systems are presented.

Inconsistency-tolerant description logic. Part II: A tableau algorithm for CALC C

Abstract In the first part of this paper, we motivated and defined three systems of constructive and inconsistency-tolerant description logic. The variety of arising systems is conditioned by the variety of approaches to defining modalities in the constructive setting. We also presented sound and complete tableau calculi for the logics under consideration. Whereas these calculi were not meant to give rise to tableau algorithms, in the present second part of the paper, after providing some motivation and recalling the main definitions, we adapt methods developed by R. Dyckhoff and by I. Horrocks and U. Sattler in order to define a tableau algorithm for our basic four-valued constructive description logic CALC C . Notice that among the three logics defined in the first part of the paper, CALC C is the only logic which lends itself to applications, because for the other logics it is unknown whether they are elementarily decidable. The presented algorithm for CALC C is the first example of an elementary decision procedure for a constructive description logic.

The Idea of a Proof-Theoretic Semantics and the Meaning of the Logical Operations

This is a purely conceptual paper. It aims at presenting and putting into perspective the idea of a proof-theoretic semantics of the logical operations. The first section briefly surveys various semantic paradigms, and Section 2 focuses on one particular paradigm, namely the proof-theoretic semantics of the logical operations.