In the dissertation epistemic logics with central agent interaction axiom are analysed. The research covers multimodal logics Kn, Tn, K4n and S4n. The aim of the work is finite derivation search sequent calculi for the mentioned logics. A new method to obtain the termination of derivation search is presented in the thesis and this method is applied to the mentioned logics as well as to monomodal logics K4 and S4. / Disertacijoje nagrinėjamos žinių logikos su centrinio agento sąveikos aksioma. Tyrimas apima multimodalines logikas Kn, Tn, K4n ir S4n. Disertacijos tikslas – baigtinės išvedimo paieškos sekvenciniai skaičiavimai minėtoms logikoms. Darbe pristatomas naujas išvedimo paieškos baigtinumą užtikrinantis metodas, kuris yra pritaikomas minėtoms logikoms, o taip pat monomodalinėms logikoms K4 ir S4.

[Abridged abstract] The goal is to give structural proof-theoretical analyses of logical constants, and thereby provide a criterion for what a logical constant is. Another goal is to illustrate the thesis that structural assumptions of logic are basic and that alternative logics (later called substructural logics) differ from each other only in their structural assumptions, and not in their assumptions about logical constants.

Modal logics have found applications in many diferent contexts. For example, epistemic modal logics can be used to reason about security protocols, temporal modal logics can be used to reason about the correctness of distributed systems and propositional dynamic logic can reason about the correctness of programs. However, pure modal logic is expressively weak and cannot represent many interesting secondorder properties that are expressible, for example, in the μ-calculus. Here we investigate the extension of modal logics with propositional quantification modulo bisimulation (bisimulation quantification). We extend existing work on bisimulation quantified modal logic by considering the variety of logics that result by restricting the structures over which they are interpreted. We show this can be a natural extension of modal logic preserving the intuitions of both modal logic and propositional quantification. However, we also find cases where such intuitions are not preserved. We examine cases where the axioms of pure modal logic and propositional quantification are preserved and where bisimulation quantifiers preserve the decidability of modal logic. We translate a number of recent decidability results for monadic second-order logics into the context of bisimulation quantified modal logics, and show how these results can be used to generate a number of interesting bisimulation quantified modal logics.

The modal logic S5 gives us a simple technical tool to analyze some main notions from philosophy (e.g. metaphysical necessity and epistemological concepts such as knowledge and belief). Although S5 can be axiomatized by some simple rules, this logic shows some puzzling properties. For example, an interpolation result holds for the propositional version, but this same result fails when we add first-order quantifiers to this logic. In this dissertation, we study the failure of the Definability and Interpolation Theorems for first-order S5. At the same time, we combine the results of justification logic and we investigate the quantified justification counterpart of S5 (first-order JT45). In this way we explore the relationship between justification logic and modal logic to see if justification logic can contribute to the literature concerning the restoration of the Interpolation Theorem. / A lógica modal S5 nos oferece um ferramental técnico para analizar algumas noções filosóficas centrais (por exemplo, necessidade metafísica e certos conceitos epistemológicos como conhecimento e crença). Apesar de ser axiomatizada por princípios simples, esta lógica apresenta algumas propriedades peculiares. Uma das mais notórias é a seguinte: podemos provar o Teorema da Interpolação para a versão proposicional, mas esse mesmo teorema não pode ser provado quando adicionamos quantificadores de primeira ordem a essa lógica. Nesta dissertação vamos estudar a falha dos Teoremas da Definibilidade e da Interpolação para a versão quantificada de S5. Ao mesmo tempo, vamos combinar os resultados da lógica da justificação e investigar a contraparte da versão quantificada de S5 na lógica da justificação (a lógica chamada JT45 de primeira ordem). Desse modo, vamos explorar a relação entre lógica modal e lógica da justificação para ver se a lógica da justificação pode contribuir para a restauração do Teorema da Interpolação.

In this dissertation we prove the Completeness, Soundness and Compactness of the Monadic Hybrid Calculus MHC and we prove its expressive equivalence to the Monadic Predicate Calculus MPC. The Monadic Hybrid Calculus MHC is a new system that is based on the (propositional) modal logic S5. It is “Hybrid” in the sense that it includes quantifier free MPC and therefore, unlike S5, allows free individual constants. The main innovation in this system is the elimination of bound variables. In MHC, upper case letters denote properties and lower case letters denote individuals. Universal quantification is represented by square brackets, [], and existential quantification is represented by angled brackets, 〈〉. Thus, All Athenians are Greek and mortal is formalized as [A](G∧M), Some mortal Greeks are Athenians as 〈M∧G〉A, and Socrates is mortal and Athenian as s(M∧A). We give the formal syntax and the formal semantics of [MHC] and give Beth-style Tableau Rules (Inference Rules). In these rules, if [P]Q is on the right then we select a new constant [v] and we add [vP] on left, vQ on the right, and we cancel the formula. If [P]Q is on the left then we select a pre-used constant p and split the tree. We add pP on the right of one branch and pQ on the left of the other branch. We treat 〈P〉Q similarly. Our Completeness proof uses induction on formulas down a path in the proof tree. Our Soundness proof uses induction up a path. To prove that MPC is logically equivalent to the Monadic Predicate Calculus, we present algorithms that transform formulas back and forth between these two systems. Compactness follows immediately. Finally, we examine the pragmatic usage of the Monadic Hybrid Calculus and we compare it with the Monadic Predicate Calculus using natural language examples. We also examine the novel notions of the Hybrid Predicate Calculus along with their pragmatic implications. / Graduate / 0800 / 0984

The fusion Ll ? Lr of two normal modal logics formulated in languages with disjoint sets of modal operators is the smallest normal modal logic containing Ll [ Lr. This paper proves that decidability, interpolation, uniform interpolation, and Halld?encompleteness are preserved under forming fusions of normal polyadic polymodal logics. Those problems remained open in [Fine & Schurz [3]] and [Kracht & Wolter [10]]. The paper defines the fusion `l ? `r of two classical modal consequence relations and proves that decidability transfers also in this case. Finally, these results are used to prove a general decidability result for modal logics based on superintuitionistic logics.

This paper investigates (modal) extensions of Heyting–Brouwer logic, i.e., the logic which results when the dual of implication (alias coimplication) is added to the language of intuitionistic logic. We first develop matrix as well as Kripke style semantics for those logics. Then, by extending the Gödel-embedding of intuitionistic logic into S4 , it is shown that all (modal) extensions of Heyting–Brouwer logic can be embedded into tense logics (with additional modal operators). An extension of the Blok–Esakia-Theorem is proved for this embedding.

This paper is a comparative study of the propositional intuitionistic (non-modal) and classical modal languages interpreted in the standard way on transitive frames. It shows that, when talking about these frames rather than conventional quasi-orders, the intuitionistic language displays some unusual features: its expressive power becomes weaker than that of the modal language, the induced consequence relation does not have a deduction theorem and is not protoalgebraic. Nevertheless, the paper develops a manageable model theory for this consequence and its extensions which also reveals some unexpected phenomena. The balance between the intuitionistic and modal languages is restored by adding to the former one more implication.

The paper considers the standard concept description language ALC augmented with various kinds of modal operators which can be applied to concepts and axioms. The main aim is to develop methods of proving decidability of the satisfiability problem for this language and apply them to description logics with most important temporal and epistemic operators, thereby obtaining satisfiability checking algorithms for these logics. We deal with the possible world semantics under the constant domain assumption and show that the expanding and varying domain assumptions are reducible to it. Models with both finite and arbitrary constant domains are investigated. We begin by considering description logics with only one modal operator and then prove a general transfer theorem which makes it possible to lift the obtained results to many systems of polymodal description logic.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20159号 / 理博第4244号 / 新制||理||1610(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 照井 一成, 教授 岡本 久, 教授 長谷川 真人 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM