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Starting with simple examples showing the relevance of cutting and pasting logics, the monograph develops a mathematical theory of combining and decomposing logics, ranging from propositional and first-order based logics to higher-order based logics as well as to non-truth functional logics. The theory covers mechanisms for combining semantic structures and deductive systems either of the same or different nature (for instance, two Hilbert calculi or a Hilbert calculus and a tableau calculus). The important issue of preservation of properties is extensively addressed. For instance, sufficient conditions are provided for a combined logic to be sound and complete when the original component logics are known to be sound and complete. The book brings the reader to the front line of current research in the field by showing both recent achievements and directions of future investigations (in particular, multiple open problems). It also provides examples of potential applications in emergent fields like security protocols, quantum computing, networks and argumentation theory, besides discussing more classical applications like software specification, knowledge representation, computational linguistics and modular automated reasoning. This monograph will be of interest to researchers and graduate students in mathematical logic, theory of computation and philosophical logic with no previous knowledge of the subject of combining and decomposing logics, but with a working knowledge of first-order logic. The book will also be relevant for people involved in research projects where logic is used as a tool and the need for working with several logics at the same time is mandatory (for instance, temporal, epistemic and probabilistic logics).

Logic, Symbolic and mathematical. --- Mathematics --- Philosophy. --- Logic of mathematics --- Mathematics, Logic of --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Logic. --- Computer science. --- Mathematical Logic and Foundations. --- Mathematical Logic and Formal Languages. --- Informatics --- Science --- Argumentation --- Deduction (Logic) --- Deductive logic --- Dialectic (Logic) --- Logic, Deductive --- Intellect --- Philosophy --- Psychology --- Reasoning --- Thought and thinking --- Methodology --- Mathematical logic.

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This volume investigates what is beyond the Principle of Non-Contradiction. It features 13 papers on the foundations of reasoning, including logical systems and philosophical considerations. Coverage brings together a cluster of issues centered upon the variety of meanings of consistency, contradiction, and related notions. Most of the papers, but not all, are developed around the subtle distinctions between consistency and non-contradiction, as well as among contradiction, inconsistency, and triviality, and concern one of the above mentioned threads of the broadly understood non-contradiction principle and the related principle of explosion. Some others take a perspective that is not too far away from such themes, but with the freedom to tread new paths. Readers should understand the title of this book in a broad way,because it is not so obvious to deal with notions like contradictions, consistency, inconsistency, and triviality. The papers collected here present groundbreaking ideas related to consistency and inconsistency.

Contradiction. --- Philosophy --- Dialetheism --- Paradox --- Logic. --- Ontology. --- Logic, Symbolic and mathematical. --- Mathematical Logic and Foundations. --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Mathematics --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Being --- Metaphysics --- Necessity (Philosophy) --- Substance (Philosophy) --- Argumentation --- Deduction (Logic) --- Deductive logic --- Dialectic (Logic) --- Logic, Deductive --- Intellect --- Psychology --- Science --- Reasoning --- Thought and thinking --- Methodology --- Mathematical logic. --- Logic, symbolic and mathematical --- Ontology --- Logic

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In the last two decades modal logic has undergone an explosive growth, to thepointthatacompletebibliographyofthisbranchoflogic,supposingthat someone were capable to compile it, would ?ll itself a ponderous volume. What is impressive in the growth of modal logic has not been so much the quick accumulation of results but the richness of its thematic dev- opments. In the 1960s, when Kripke semantics gave new credibility to the logic of modalities? which was already known and appreciated in the Ancient and Medieval times? no one could have foreseen that in a short time modal logic would become a lively source of ideas and methods for analytical philosophers,historians of philosophy,linguists, epistemologists and computer scientists. The aim which oriented the composition of this book was not to write a new manual of modal logic (there are a lot of excellent textbooks on the market, and the expert reader will realize how much we bene?ted from manyofthem)buttoo?ertoeveryreader,evenwithnospeci?cbackground in logic, a conceptually linear path in the labyrinth of the current panorama of modal logic. The notion which in our opinion looked suitable to work as a compass in this enterprise was the notion of multimodality, or, more speci?cally, the basic idea of grounding systems on languages admitting more than one primitive modal operator.

Modality (Logic) --- Semantics (Philosophy) --- Intension (Philosophy) --- Logical semantics --- Semantics (Logic) --- Semeiotics --- Significs --- Syntactics --- Unified science --- Language and languages --- Logic, Symbolic and mathematical --- Logical positivism --- Meaning (Psychology) --- Philosophy, Modern --- Semiotics --- Signs and symbols --- Symbolism --- Analysis (Philosophy) --- Definition (Philosophy) --- Modal logic --- Logic --- Nonclassical mathematical logic --- Bisimulation --- Logic. --- Logic, Symbolic and mathematical. --- Philosophy (General). --- Computer science. --- Mathematical Logic and Foundations. --- Philosophy, general. --- History of Philosophy. --- Mathematical Logic and Formal Languages. --- Informatics --- Science --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Mathematics --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Argumentation --- Deduction (Logic) --- Deductive logic --- Dialectic (Logic) --- Logic, Deductive --- Intellect --- Philosophy --- Psychology --- Reasoning --- Thought and thinking --- Methodology --- Mathematical logic. --- Philosophy. --- Mental philosophy --- Humanities --- Machine theory. --- Formal Languages and Automata Theory. --- History. --- Abstract automata --- Abstract machines --- Automata --- Mathematical machine theory --- Algorithms --- Recursive functions --- Robotics