From Many-Valued Consequence to Many-Valued Connectives

Given a consequence relation in many-valued logic, what connectives can be defined? For instance, does there always exist a conditional operator internalizing the consequence relation, and which form should it take? In this paper, we pose this question in a multi-premise multi-conclusion setting for the class of so-called intersective mixed consequence relations, which extends the class of Tarskian relations. Using computer-aided methods, we answer extensively for 3-valued and 4-valued logics, focusing not only on conditional operators, but on what we call Gentzen-regular connectives (including negation, conjunction, and disjunction). For arbitrary N-valued logics, we state necessary and sufficient conditions for the existence of such connectives in a multi-premise multi-conclusion setting. The results show that mixed consequence relations admit all classical connectives, and among them pure consequence relations are those that admit no other Gentzen-regular connectives. Conditionals can also be found for a broader class of intersective mixed consequence relations, but with the exclusion of order-theoretic consequence relations.Comment: Updated version [corrections of an incorrect claim in first version; two bib entries added

Секвенційні числення композиційно-номінотивних логік квазіарних предикатів

Побудовано секвенційні числення першопорядкових композиційно-номінативних логік часткових однозначних, тотальних неодно-значних та часткових неоднозначних квазіарних предикатів кванторного рівня. Такі числення запропоновано для загального випад-ку логік квазіарних предикатів, для логік однозначних еквітонних та логік тотальних антитонних предикатів. Для побудованих чис-лень доведено теореми коректності та повноти.We construct sequent calculi for first-order composition-nominative logics of partial single-valued, total multiple-valued and partial multi-ple-valued quasi-ary predicates of quantifier level. The defined calculi are proposed for a general case of logics of quasi-ary predicates, for logics of single-valued equitone predicates and for logics of total multiple-valued antytone predicates. For the introduced calculi soundness and completeness theorems are proved

Композиційно-номітативні логіки часткових та неоднозначних предикатів

Досліджено чисті першопорядкові логіки часткових однозначних, тотальних неоднозначних та часткових неоднозначних предикатів. Розглянуто низку розширень цих логік за допомогою узагальнених реномінацій та спеціальних предикатів-індикаторів наявності значення для змінних. Описано мови та семантичні моделі таких логік, досліджено їх семантичні властивості, для різних класів цих логік запропоновано числення секвенційного типу.Исследованы чистые первопорядковые логики частичных однозначных, тотальных неоднозначных и частичных неоднозначных предикатов. Рассмотрен ряд расширений этих логик с помощью обобщенных реноминаций и специальных предикатов-индикаторов наличия значения для переменных. Описаны языки и семантические модели таких логик, для разных классов этих логик предложены исчисления секвенциального типа.We study pure fisrt-order logics of partial single-valued, total multi-valued and partial multi-valued predicates. Various extensions of the introduced logics with generalized renominations and special variable definedness predicates are considered. For such logics, we define languages and semantic models, investigate their semantic properties and specify sequent calculi

Evidence-Based Beliefs in Many-Valued Modal Logics

Rational agents, humans or otherwise, build their beliefs from evidence – a process which we call consolidation. But how should this process be carried out? In this thesis, we study a multi-agent logic of evidence and the question how agents should form beliefs in this logic. The main contributions of this thesis are twofold. First, we present and study a many-valued modal logic, and show how it can be suitable for modelling multi-agent scenarios where each agent has access to some evidence, which in turn can be processed into beliefs. This is a technical and practical contribution to many-valued modal logics. Second, we open new paths for research in the field of evidence logics: we show a new approach based on many-valued logics, we highlight the concept of consolidations and the importance of looking at their dynamic nature, and build a methodology based on rationality postulates to evaluate them

Viewpoint Development of Stochastic Hybrid Systems

Nowadays, due to the explosive spreading of networked and highly distributed systems, mastering system complexity becomes a critical issue. Two development and verification paradigms have become more popular: viewpoints and randomisation. The viewpoints offer large freedom and introduce concurrency and compositionality in the development process. Randomisation is now a traditional method for reducing complexity (comparing with deterministic models) and it offers finer analytical analysis tools (quantification over non-determinism, multi-valued logics, etc). In this paper, we propose a combination of these two paradigms introducing a viewpoint methodology for systems with stochastic behaviours