Proof-Theoretic Functional Completeness for the Hybrid Logics of Everywhere and Elsewhere

A hybrid logic is obtained by adding to an ordinary modal logic further expressive power in the form of a second sort of propositional symbols called nominals and by adding so-called satisfaction operators. In this paper we consider hybridized versions of S5 (“the logic of everywhere”) and the modal logic of inequality (“the logic of elsewhere”). We give natural deduction systems for the logics and we prove functional completeness results.     

Modal translations in substructural logics

Substructural logics are logics obtained from a sequent formulation of intuitionistic or classical logic by rejecting some structural rules. The substructural logics considered here are linear logic, relevant logic and BCK logic. It is proved that first-order variants of these logics with an intuitionistic negation can be embedded by modal translations into S40type extensions of these logics with a classical, involutive, negation. Related embeddings via translations like the double-negation translation are also considered. Embeddings into analogues of S4 are obtained with the help of cut climination for sequent formulations of our substructural logics and their modal extensions. These results are proved for systems with equality too.     

Constructing a continuum of predicate extensions of each intermediate propositional logic

Wajsberg and Jankov provided us with methods of constructing a continuum of logics. However, their methods are not suitable for super-intuitionistic and modal predicate logics. The aim of this paper is to present simple ways of modification of their methods appropriate for such logics. We give some concrete applications as generic examples. Among others, we show that there is a continuum of logics (1) between the intuitionistic predicate logic and the logic of constant domains, (2) between a predicate extension ofS4 andS4 with the Barcan formula. Furthermore, we prove that (3) there is a continuum of predicate logics with equality whose equality-free fragment is just the intuitionistic predicate logic.Dedicated to the memory of the late Professor S. MaeharaThis research was supported in part by Grant-in Aid for Encouragement of Young Scientists No. 06740140, Ministry of Education, Science and Culture, Japan.Presented byHiroakira Ono     

Deduction theorems for weak implicational logics

The standard deduction theorem or introduction rule for implication, for classical logic is also valid for intuitionistic logic, but just as with predicate logic, other rules of inference have to be restricted if the theorem is to hold for weaker implicational logics.In this paper we look in detail at special cases of the Gentzen rule for and show that various subsets of these in effect constitute deduction theorems determining all the theorems of many well known as well as not well known implicational logics. In particular systems of rules are given which are equivalent to the relevance logics E,R, T, P-W and P-W-I.     

Normalization and excluded middle. I

The usual rule used to obtain natural deduction formulations of classical logic from intuitionistic logic, namely is stronger then necessary, and will give classical logic when added to minimal logic. A rule which is precisely strong enough to give classical logic from intuitionistic logic, and which is thus exactly equivalent to the law of the excluded middle, is It is a special case of a version of Peirce's law: In this paper it is shown how to normalize logics defined using these last two rules. Part I deals with propositional logics and first order predicate logics. Part II will deal with first order arithmetic and second order logics. This research was supported in part by grants EQ1648, EQ2908, and CE 110 of the program Fonds pour la Formation de Chercheurs et l'aide à la Recherche (F.C.A.R.) of the Quèbec Ministry of Education.     

Preservation of Craig interpolation by the product of matrix logics

The product of matrix logics, possibly with additional interaction axioms, is shown to preserve a slightly relaxed notion of Craig interpolation. The result is established symbolically, capitalizing on the complete axiomatization of the product of matrix logics provided by their meet-combination. Along the way preservation of the metatheorem of deduction is also proved. The computation of the interpolant in the resulting logic is proved to be polynomially reducible to the computation of the interpolants in the two given logics. Illustrations are provided for classical, intuitionistic and modal propositional logics.     

The enduring scandal of deduction

Deductive inference is usually regarded as being “tautological” or “analytical”: the information conveyed by the conclusion is contained in the information conveyed by the premises. This idea, however, clashes with the undecidability of first-order logic and with the (likely) intractability of Boolean logic. In this article, we address the problem both from the semantic and the proof-theoretical point of view. We propose a hierarchy of propositional logics that are all tractable (i.e. decidable in polynomial time), although by means of growing computational resources, and converge towards classical propositional logic. The underlying claim is that this hierarchy can be used to represent increasing levels of “depth” or “informativeness” of Boolean reasoning. Special attention is paid to the most basic logic in this hierarchy, the pure “intelim logic”, which satisfies all the requirements of a natural deduction system (allowing both introduction and elimination rules for each logical operator) while admitting of a feasible (quadratic) decision procedure. We argue that this logic is “analytic” in a particularly strict sense, in that it rules out any use of “virtual information”, which is chiefly responsible for the combinatorial explosion of standard classical systems. As a result, analyticity and tractability are reconciled and growing degrees of computational complexity are associated with the depth at which the use of virtual information is allowed.     

Incompleteness and the Barcan formula

A (normal) system of prepositional modal logic is said to be complete iff it is characterized by a class of (Kripke) frames. When we move to modal predicate logic the question of completeness can again be raised. It is not hard to prove that if a predicate modal logic is complete then it is characterized by the class of all frames for the propositional logic on which it is based. Nor is it hard to prove that if a propositional modal logic is incomplete then so is the predicate logic based on it. But the interesting question is whether a complete propositional modal logic can have an incomplete extension. In 1967 Kripke announced the incompleteness of a predicate extension of S4. The purpose of the present article is to present several such systems. In the first group it is the systemswith the Barcan Formula which are incomplete, while those without are complete. In the second group it is thosewithout the Barcan formula which are incomplete, while those with the Barcan Formula are complete. But all these are based on propositional systems which are characterized by frames satisfying in each case a single first-order sentence.     

The Modal Logic of Bayesian Belief Revision

In Bayesian belief revision a Bayesian agent revises his prior belief by conditionalizing the prior on some evidence using Bayes’ rule. We define a hierarchy of modal logics that capture the logical features of Bayesian belief revision. Elements in the hierarchy are distinguished by the cardinality of the set of elementary propositions on which the agent’s prior is defined. Inclusions among the modal logics in the hierarchy are determined. By linking the modal logics in the hierarchy to the strongest modal companion of Medvedev’s logic of finite problems it is shown that the modal logic of belief revision determined by probabilities on a finite set of elementary propositions is not finitely axiomatizable.

     

Predicate Logics on Display

The paper provides a uniform Gentzen-style proof-theoretic framework for various subsystems of classical predicate logic. In particular, predicate logics obtained by adopting van Behthem's modal perspective on first-order logic are considered. The Gentzen systems for these logics augment Belnap's display logic by introduction rules for the existential and the universal quantifier. These rules for x and x are analogous to the display introduction rules for the modal operators and and do not themselves allow the Barcan formula or its converse to be derived. En route from the minimal modal predicate logic to full first-order logic, axiomatic extensions are captured by purely structural sequent rules.     

Identity in modal logic theorem proving

THINKER is an automated natural deduction first-order theorem proving program. This paper reports on how it was adapted so as to prove theorems in modal logic. The method employed is an indirect semantic method, obtained by considering the semantic conditions involved in being a valid argument in these modal logics. The method is extended from propositional modal logic to predicate modal logic, and issues concerning the domain of quantification and existence in a world's domain are discussed. Finally, we look at the very interesting issues involved with adding identity to the theorem prover in the realm of modal predicate logic. Various alternatives are discussed.     

First-order Expressivity for S5-models: Modal vs. Two-sorted Languages

Standard models for model predicate logic consist of a Kripke frame whose worlds come equipped with relational structures. Both modal and two-sorted predicate logic are natural languages for speaking about such models. In this paper we compare their expressivity. We determine a fragment of the two-sorted language for which the modal language is expressively complete on S5-models. Decidable criteria for modal definability are presented.     

Game logic and its applications I

This paper provides a logic framework for investigations of game theoretical problems. We adopt an infinitary extension of classical predicate logic as the base logic of the framework. The reason for an infinitary extension is to express the common knowledge concept explicitly. Depending upon the choice of axioms on the knowledge operators, there is a hierarchy of logics. The limit case is an infinitary predicate extension of modal propositional logic KD4, and is of special interest in applications. In Part I, we develop the basic framework, and show some applications: an epistemic axiomatization of Nash equilibrium and formal undecidability on the playability of a game. To show the formal undecidability, we use a term existence theorem, which will be proved in Part II.The authors thank Hiroakira Ono for helpful discussions and encouragements from the early stage of this research project, and Philippe Mongin, Mitio Takano and a referee of this journal for comments on earlier versions of this paper. The first and second authors are partially supported, respectively, by Tokyo Center of Economic Research and Grant-in-Aids for Scientific Research 04640215, Ministry of Education, Science and Culture.Presented by H. Ono     

基于广义谢弗竖的分析性模态公理系统

沿着安德森等人开创的方向,我们将分析性公理系统从经典逻辑推向模态逻辑,所定义的广义谢弗竖混合了模态词和广义析舍。在这篇论文中,我们给出常见的正规模态逻辑的分析性公理系统及其强完全性定理和插值定理,并讨论演绎关系的性质：单调性和切割性。     

Algorithmic Proof Methods and Cut Elimination for Implicational Logics Part I: Modal Implication

In this work we develop goal-directed deduction methods for the implicational fragment of several modal logics. We give sound and complete procedures for strict implication of K, T, K4, S4, K5, K45, KB, KTB, S5, G and for some intuitionistic variants. In order to achieve a uniform and concise presentation, we first develop our methods in the framework of Labelled Deductive Systems [Gabbay 96]. The proof systems we present are strongly analytical and satisfy a basic property of cut admissibility. We then show that for most of the systems under consideration the labelling mechanism can be avoided by choosing an appropriate way of structuring theories. One peculiar feature of our proof systems is the use of restart rules which allow to re-ask the original goal of a deduction. In case of K, K4, S4 and G, we can eliminate such a rule, without loosing completeness. In all the other cases, by dropping such a rule, we get an intuitionistic variant of each system. The present results are part of a larger project of a goal directed proof theory for non-classical logics; the purpose of this project is to show that most implicational logics stem from slight variations of a unique deduction method, and from different ways of structuring theories. Moreover, the proof systems we present follow the logic programming style of deduction and seem promising for proof search [Gabbay and Reyle 84, Miller et al. 91].     

Gentzen-Type Methods for Bilattice Negation

A general Gentzen-style framework for handling both bilattice (or strong) negation and usual negation is introduced based on the characterization of negation by a modal-like operator. This framework is regarded as an extension, generalization or re- finement of not only bilattice logics and logics with strong negation, but also traditional logics including classical logic LK, classical modal logic S4 and classical linear logic CL. Cut-elimination theorems are proved for a variety of proposed sequent calculi including CLS (a conservative extension of CL) and CLS_cw (a conservative extension of some bilattice logics, LK and S4). Completeness theorems are given for these calculi with respect to phase semantics, for SLK (a conservative extension and fragment of LK and CLS_cw, respectively) with respect to a classical-like semantics, and for SS4 (a conservative extension and fragment of S4 and CLScw, respectively) with respect to a Kripke-type semantics. The proposed framework allows for an embedding of the proposed calculi into LK, S4 and CL.     

Vagueness-Adaptive Logic: A Pragmatical Approach to Sorites Paradoxes

Kripke-completeness of every classical modal logic with Sahlqvist formulas is one of the basic general results on completeness of classical modal logics. This paper shows a Sahlqvist theorem for modal logic over the relevant logic Bin terms of Routley-Meyer semantics. It is shown that usual Sahlqvist theorem for classical modal logics can be obtained as a special case of our theorem.     

An analytic tableau calculus for a temporalised belief logic

A tableau is a refutation-based decision procedure for a related logic, and is among the most popular proof procedures for modal logics. In this paper, we present a labelled tableau calculus for a temporalised belief logic called TML⁺, which is obtained by adding a linear-time temporal logic onto a belief logic by the temporalisation method of Finger and Gabbay. We first establish the soundness and the completeness of the labelled tableau calculus based on the soundness and completeness results of its constituent logics. We then sketch a resolution-type proof procedure that complements the tableau calculus and also propose a model checking algorithm for TML⁺ based on the recent results for model checking procedures for temporalised logics. TML⁺ is suitable for formalising trust and agent beliefs and reasoning about their evolution for agent-based systems. Based on the logic TML⁺, the proposed labelled tableau calculus could be used for analysis, design and verification of agent-based systems operating in dynamic environments.     

A general tableau method for propositional interval temporal logics: Theory and implementation

In this paper, we focus our attention on tableau methods for propositional interval temporal logics. These logics provide a natural framework for representing and reasoning about temporal properties in several areas of computer science. However, while various tableau methods have been developed for linear and branching time point-based temporal logics, not much work has been done on tableau methods for interval-based ones. We develop a general tableau method for Venema's CDT logic interpreted over partial orders (BCDT⁺ for short). It combines features of the classical tableau method for first-order logic with those of explicit tableau methods for modal logics with constraint label management, and it can be easily tailored to most propositional interval temporal logics proposed in the literature. We prove its soundness and completeness, and we show how it has been implemented.