Couverture du livre "Introduction à la théorie de la démonstration Élimination des coupures, normalisation et preuves de cohérence" par P. Mancosu, S. Galvan, et R. Zach

Introduction à la théorie de la démonstration: Élimination des coupures, normalisation et preuves de cohérence

Mancosu, Paolo, Sergio Galvan, and Richard Zach. 2022. Introduction à la théorie de la démonstration: Élimination des coupures, normalisation et preuves de cohérence. Paris: Vrin. Traduction française de An Introduction to Proof Theory. Cet ouvrage offre une introduction accessible à la théorie de la démonstration : il donne les détails des preuves et comporte de nombreux … Continue reading Introduction à la théorie de la démonstration: Élimination des coupures, normalisation et preuves de cohérence

An epimorphism between Fine and Ferguson’s matrices for Angell’s AC

Zach, Richard. 2022. “An Epimorphism Between Fine and Ferguson’s Matrices for Angell’s AC.” Logic and Logical Philosophy, Forthcoming, 1–19. https://doi.org/10.12775/LLP.2022.025. Angell’s logic of analytic containment AC has been shown to be characterized by a 9-valued matrix NC by Ferguson, and by a 16-valued matrix by Fine. It is shown that the former is the image … Continue reading An epimorphism between Fine and Ferguson’s matrices for Angell’s AC

Epsilon theorems in intermediate logics

Baaz, Matthias, and Richard Zach. 2022. Epsilon theorems in intermediate logics. The Journal of Symbolic Logic 87(2), pp. 682–720. DOI: 10.1017/jsl.2021.103. Open access. Any intermediate propositional logic (i.e., a logic including intuitionistic logic and contained in classical logic) can be extended to a calculus with epsilon- and tau-operators and critical formulas. For classical logic, this … Continue reading Epsilon theorems in intermediate logics

Cut-free completeness for modular hypersequent calculi for modal logics K, T, and D

We investigate a recent proposal for modal hypersequent calculi. The interpretation of relational hypersequents incorporates an accessibility relation along the hypersequent. These systems give the same interpretation of hypersequents as Lellman's linear nested sequents, but were developed independently by Restall for S5 and extended to other normal modal logics by Parisi. The resulting systems obey Došen's principle: the modal rules are the same across different modal logics. Different modal systems only differ in the presence or absence of external structural rules. With the exception of S5, the systems are modular in the sense that different structural rules capture different properties of the accessibility relation. We provide the first direct semantical cut-free completeness proofs for K, T, and D, and show how this method fails in the case of B and S4.

Cut elimination and normalization for generalized single and multi-conclusion sequent and natural deduction calculi

Zach, Richard. 2021. “Cut Elimination and Normalization for Generalized Single and Multi-Conclusion Sequent and Natural Deduction Calculi.” The Review of Symbolic Logic 14 (3): 645–86. doi:10.1017/S1755020320000015. Any set of truth-functional connectives has sequent calculus rules that can be generated systematically from the truth tables of the connectives. Such a sequent calculus gives rise to a … Continue reading Cut elimination and normalization for generalized single and multi-conclusion sequent and natural deduction calculi

An Introduction to Proof Theory: Normalization, Cut-elimination, and Consistency Proofs

Paolo Mancosu, Sergio Galvan, and Richard Zach. An Introduction to Proof Theory: Normalization, Cut-elimination, and Consistency Proofs. Oxford: Oxford University Press, 2021. DOI: 10.1093/oso/9780192895936.001.0001 Published in the UK on August 17, 2021. Available in hardcover and paperback. An Introduction to Proof Theory provides an accessible introduction to the theory of proofs, with details worked out … Continue reading An Introduction to Proof Theory: Normalization, Cut-elimination, and Consistency Proofs

Learning Outcomes and Grade Specifications in a Formal Logic Course

Zach, Richard. 2021. “Learning Outcomes and Grade Specifications in a Formal Logic Course.” Poster presentation presented at the Mastery Grading University Conference 2021, Online, June 11. https://drive.google.com/file/d/1q6rAyXfrz2k79NtaHoMREQ9IDagnbvLN/view. Formal logic is a staple of undergraduate philosophy departments. Courses are typically run as straight lectures, with problem sets and exams. I will describe a design for such … Continue reading Learning Outcomes and Grade Specifications in a Formal Logic Course

The significance of the Curry-Howard isomorphism

Zach, Richard. 2019. “The Significance of the Curry-Howard Isomorphism.” In Philosophy of Logic and Mathematics. Proceedings of the 41st International Ludwig Wittgenstein Symposium, edited by Gabriele M. Mras, Paul Weingartner, and Bernhard Ritter, 313–25. Publications of the Austrian Ludwig Wittgenstein Society, New Series 26. Berlin: De Gruyter. https://doi.org/10.1515/9783110657883-018. The Curry-Howard isomorphism is a proof-theoretic result … Continue reading The significance of the Curry-Howard isomorphism

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Sets, Logic, Computation: An Open Introduction to Metalogic

Sets, Logic, Computation is an introductory textbook on metalogic. It covers naive set theory, first-order logic, sequent calculus and natural deduction, the completeness, compactness, and Löwenheim-Skolem theorems, Turing machines, and the undecidability of the halting problem and of first-order logic. The audience is undergraduate students with some background in formal logic, e.g., what is covered … Continue reading Sets, Logic, Computation: An Open Introduction to Metalogic

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forall x: Calgary. An Introduction to Formal Logic

forall x: Calgary is a full-featured textbook on formal logic. It covers key notions of logic such as consequence and validity of arguments, the syntax of truth-functional propositional logic TFL and truth-table semantics, the syntax of first-order (predicate) logic FOL with identity (first-order interpretations), translating (formalizing) English in TFL and FOL, and Fitch-style natural deduction … Continue reading forall x: Calgary. An Introduction to Formal Logic

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Rudolf Carnap: Early Writings

The Collected Works of Rudolf Carnap, Volume 1 Edited by A.W. Carus, Michael Friedman, Wolfgang Kienzler, Alan Richardson, and Sven Schlotter. With editorial assistance by Steve Awodey, Dirk Schlimm, and Richard Zach. Oxford: Oxford University Press, 2019. Publisher linkGoogle Books

Non-analytic tableaux for Chellas’s conditional logic CK and Lewis’s logic of counterfactuals VC

Zach, Richard. 2018. “Non-Analytic Tableaux for Chellas’s Conditional Logic CK and Lewis’s Logic of Counterfactuals VC.” Australasian Journal of Logic 15 (3): 609–28. https://doi.org/10.26686/ajl.v15i3.4780. Priest has provided a simple tableau calculus for Chellas's conditional logic Ck. We provide rules which, when added to Priest's system, result in tableau calculi for Chellas's CK and Lewis's VC. … Continue reading Non-analytic tableaux for Chellas’s conditional logic CK and Lewis’s logic of counterfactuals VC

Rumfitt on truth-grounds, negation, and vagueness

Zach, Richard. 2018. “Rumfitt on Truth-Grounds, Negation, and Vagueness.” Philosophical Studies 175 (8): 2079–89. https://doi.org/10.1007/s11098-018-1114-7. In The Boundary Stones of Thought (2015), Rumfitt defends classical logic against challenges from intuitionistic mathematics and vagueness, using a semantics of pre-topologies on possibilities, and a topological semantics on predicates, respectively. These semantics are suggestive but the characterizations of … Continue reading Rumfitt on truth-grounds, negation, and vagueness

Semantics and proof theory of the epsilon calculus

Zach, Richard. 2017. “Semantics and Proof Theory of the Epsilon Calculus.” In Logic and Its Applications. ICLA 2017, edited by Sujata Ghosh and Sanjiva Prasad, 27–47. LNCS 10119. Berlin, Heidelberg: Springer. DOI :10.1007/978-3-662-54069-5_4. The epsilon operator is a term-forming operator which replaces quantifiers in ordinary predicate logic. The application of this undervalued formalism has been … Continue reading Semantics and proof theory of the epsilon calculus

Natural deduction for the Sheffer stroke and Peirce’s arrow (and any other truth-functional connective)

Richard Zach, "Natural Deduction for the Sheffer Stroke and Peirce’s Arrow (and any Other Truth-Functional Connective)," Journal of Philosophical Logic 45(2) (2016), pp. 183–197. Methods available for the axiomatization of arbitrary finite-valued logics can be applied to obtain sound and complete intelim rules for all truth-functional connectives of classical logic including the Sheffer stroke (NAND) … Continue reading Natural deduction for the Sheffer stroke and Peirce’s arrow (and any other truth-functional connective)

Carnap’s early metatheory: Scope and limits

Georg Schiemer, Richard Zach, and Erich Reck. 2017. "Carnap's Early Metatheory: Scope and Limits," Synthese 194(1), 33–65 In his Untersuchungen zur allgemeinen Axiomatik (1928) and Abriss der Logistik (1929), Rudolf Carnap attempted to formulate the metatheory of axiomatic theories within a single, fully interpreted type-theoretic framework and to investigate a number of meta-logical notions in … Continue reading Carnap’s early metatheory: Scope and limits

Heinrich Behmann’s 1921 lecture on the decision problem and the algebra of logic

Paolo Mancosu and Richard Zach. "Heinrich Behmann's 1921 lecture on the decision problem and the algebra of logic," Bulletin of Symbolic Logic 21 (2015), 164–187 Heinrich Behmann (1891–1970) obtained his Habilitation under David Hilbert in Göttingen in 1921 with a thesis on the decision problem. In his thesis, he solved—independently of Löwenheim and Skolem’s earlier … Continue reading Heinrich Behmann’s 1921 lecture on the decision problem and the algebra of logic