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This volume is the first ever collection devoted to the field of proof-theoretic semantics. Contributions address topics including the systematics of introduction and elimination rules and proofs of normalization, the categorial characterization of deductions, the relation between Heyting's and Gentzen's approaches to meaning, knowability paradoxes, proof-theoretic foundations of set theory, Dummett's justification of logical laws, Kreisel's theory of constructions, paradoxical reasoning, and the defence of model theory. The field of proof-theoretic semantics has existed for almost 50 years, but the term itself was proposed by Schroeder-Heister in the 1980s. Proof-theoretic semantics explains the meaning of linguistic expressions in general and of logical constants in particular in terms of the notion of proof. This volume emerges from presentations at the Second International Conference on Proof-Theoretic Semantics in Tübingen in 2013, where contributing authors were asked to provide a self-contained description and analysis of a significant research question in this area. The contributions are representative of the field and should be of interest to logicians, philosophers, and mathematicians alike.

Logic --- Philosophy --- Philosophy & Religion --- Philosophy. --- Logic. --- Mathematical logic. --- Mathematical Logic and Foundations. --- Mathematical Logic and Formal Languages. --- 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 --- Psychology --- Science --- Reasoning --- Thought and thinking --- Mental philosophy --- Humanities --- Methodology --- Logic, Symbolic and mathematical. --- Computer science. --- Informatics --- Proof theory --- Semantics - Mathematical models --- Logic, symbolic and mathematical --- Semantics

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This open access book is the first ever collection of Karl Popper's writings on deductive logic. Karl R. Popper (1902-1994) was one of the most influential philosophers of the 20th century. His philosophy of science ("falsificationism") and his social and political philosophy ("open society") have been widely discussed way beyond academic philosophy. What is not so well known is that Popper also produced a considerable work on the foundations of deductive logic, most of it published at the end of the 1940s as articles at scattered places. This little-known work deserves to be known better, as it is highly significant for modern proof-theoretic semantics. This collection assembles Popper's published writings on deductive logic in a single volume, together with all reviews of these papers. It also contains a large amount of unpublished material from the Popper Archives, including Popper's correspondence related to deductive logic and manuscripts that were (almost) finished, but did not reach the publication stage. All of these items are critically edited with additional comments by the editors. A general introduction puts Popper's work into the context of current discussions on the foundations of logic. This book should be of interest to logicians, philosophers, and anybody concerned with Popper's work.

Lògica --- Argumentació --- Dialèctica (Lògica) --- Lògica deductiva --- Filosofia --- Intel·ligència --- Psicologia --- Abstracció --- Alteritat (Filosofia) --- Categories (Filosofia) --- Certesa --- Condicionals (Lògica) --- Definició (Lògica) --- Dilema --- Evidència --- Hipòtesi --- Inconsistència (Lògica) --- Inducció (Lògica) --- Intenció (Lògica) --- Judici lògic --- Lògica deòntica --- Metodologia --- Modalitat (Lògica) --- Nominalisme --- Positivisme lògic --- Probabilitats --- Raó suficient --- Sil·logisme --- Sofismes --- Teoria del coneixement --- Universals (Filosofia) --- Metodologia de la ciència --- Pensament --- Raonament --- Karl R. Popper --- Deductive Logic --- Logical Constants --- Proof-theoretic Semantics --- Classical Logic --- Non-classical Logic --- Inferential Definitions --- Mathematical Logic --- Negation --- Modalities --- History of Logic --- L.E.J. Brouwer --- Paul Bernays --- Rudolf Carnap --- Alonzo Church --- Kalman Joseph Cohen --- Henry George Forder --- Harold Jeffreys --- Stephen Cole Kleene --- Logic. --- Mathematical logic. --- Language and languages --- Mathematical Logic and Foundations. --- Stylistics. --- Style. --- Linguostylistics --- Stylistics --- Literary style --- 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 --- Science --- Reasoning --- Thought and thinking --- Methodology

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Proof theory has long been established as a basic discipline of mathematical logic. It has recently become increasingly relevant to computer science. The - ductive apparatus provided by proof theory has proved useful for metatheoretical purposes as well as for practical applications. Thus it seemed to us most natural to bring researchers together to assess both the role proof theory already plays in computer science and the role it might play in the future. The form of a Dagstuhl seminar is most suitable for purposes like this, as Schloß Dagstuhl provides a very convenient and stimulating environment to - scuss new ideas and developments. To accompany the conference with a proc- dings volume appeared to us equally appropriate. Such a volume not only ?xes basic results of the subject and makes them available to a broader audience, but also signals to the scienti?c community that Proof Theory in Computer Science (PTCS) is a major research branch within the wider ?eld of logic in computer science.

Automatic theorem proving --- Computer Science --- Engineering & Applied Sciences --- Computer science. --- Programming languages (Electronic computers). --- Algorithms. --- Computer logic. --- Mathematical logic. --- Artificial intelligence. --- Computer Science. --- Programming Languages, Compilers, Interpreters. --- Mathematical Logic and Foundations. --- Artificial Intelligence (incl. Robotics). --- Mathematical Logic and Formal Languages. --- Logics and Meanings of Programs. --- Algorithm Analysis and Problem Complexity. --- Logic, Symbolic and mathematical. --- Logic design. --- Computer software. --- Artificial Intelligence. --- Software, Computer --- Computer systems --- Design, Logic --- Design of logic systems --- Digital electronics --- Electronic circuit design --- Logic circuits --- Machine theory --- Switching theory --- AI (Artificial intelligence) --- Artificial thinking --- Electronic brains --- Intellectronics --- Intelligence, Artificial --- Intelligent machines --- Machine intelligence --- Thinking, Artificial --- Bionics --- Cognitive science --- Digital computer simulation --- Electronic data processing --- Logic machines --- Self-organizing systems --- Simulation methods --- Fifth generation computers --- Neural computers --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Mathematics --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Informatics --- Science --- Algorism --- Algebra --- Arithmetic --- Computer science logic --- Logic, Symbolic and mathematical --- Computer languages --- Computer program languages --- Computer programming languages --- Machine language --- Languages, Artificial --- Foundations --- Automatic theorem proving - Congresses

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A model theory that is independent of any concrete logical system allows a general handling of a large variety of logics. This generality can be achieved by applying the theory of institutions that provides a precise mathematical formulation for the intuitive concept of a logical system. Especially in computer science, where the development of a huge number of specification logics is observable, institution-independent model theory simplifies and sometimes even enables a concise model-theoretic analysis of the system. Besides incorporating important methods and concepts from conventional model theory, the proposed top-down methodology allows for a structurally clean understanding of model-theoretic phenomena. As a consequence, results from conventional concrete model theory can be understood more easily, and sometimes even new results are obtained.

Mathematical logic --- Logic --- wiskunde --- logica