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A proof theoretical perspective on public announcement logicLogic and Philosophy of Science. forthcoming.
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Natural deduction and normal form for intuitionistic linear logicArchive for Mathematical Logic. forthcoming.
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53Proof-theoretical analysis of order relationsArchive for Mathematical Logic 43 (3): 297-309. 2004.A proof-theoretical analysis of elementary theories of order relations is effected through the formulation of order axioms as mathematical rules added to contraction-free sequent calculus. Among the results obtained are proof-theoretical formulations of conservativity theorems corresponding to Szpilrajn’s theorem on the extension of a partial order into a linear one. Decidability of the theories of partial and linear order for quantifier-free sequents is shown by giving terminating methods of pr…Read more
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25Alternative Axiomatization for Logics of Agency in a G3 CalculusFoundations of Science 28 (1): 205-224. 2021.In a recent paper, Negri and Pavlović (Studia Logica 1–35, 2020) have formulated a decidable sequent calculus for the logic of agency, specifically for a deliberative see-to-it-that modality, or dstit. In that paper the adequacy of the system is demonstrated by showing the derivability of the axiomatization of dstit from Belnap et al. (Facing the future: agents and choices in our indeterminist world. Oxford University Press, Oxford, 2001). And while the influence of the latter book on the study …Read more
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121Reasoning About Collectively Accepted Group BeliefsJournal of Philosophical Logic 40 (4): 531-555. 2011.A proof-theoretical treatment of collectively accepted group beliefs is presented through a multi-agent sequent system for an axiomatization of the logic of acceptance. The system is based on a labelled sequent calculus for propositional multi-agent epistemic logic with labels that correspond to possible worlds and a notation for internalized accessibility relations between worlds. The system is contraction- and cut-free. Extensions of the basic system are considered, in particular with rules th…Read more
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285Does the deduction theorem fail for modal logic?Synthese 187 (3): 849-867. 2012.Various sources in the literature claim that the deduction theorem does not hold for normal modal or epistemic logic, whereas others present versions of the deduction theorem for several normal modal systems. It is shown here that the apparent problem arises from an objectionable notion of derivability from assumptions in an axiomatic system. When a traditional Hilbert-type system of axiomatic logic is generalized into a system for derivations from assumptions, the necessitation rule has to be m…Read more
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191The Church–Fitch knowability paradox in the light of structural proof theorySynthese 190 (14): 2677-2716. 2012.Anti-realist epistemic conceptions of truth imply what is called the knowability principle: All truths are possibly known. The principle can be formalized in a bimodal propositional logic, with an alethic modality ${\diamondsuit}$ and an epistemic modality ${\mathcal{K}}$, by the axiom scheme ${A \supset \diamondsuit \mathcal{K} A}$. The use of classical logic and minimal assumptions about the two modalities lead to the paradoxical conclusion that all truths are known, ${A \supset \mathcal{K} A}…Read more
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8Admissibility of structural rules for extensions of contraction-free sequent calculiLogic Journal of the IGPL 9 (4): 541-548. 2001.The contraction-free sequent calculus G4 for intuitionistic logic is extended by rules following a general rule-scheme for nonlogical axioms. Admissibility of structural rules for these extensions is proved in a direct way by induction on derivations. This method permits the representation of various applied logics as complete, contraction- and cut-free sequent calculus systems with some restrictions on the nature of the derivations. As specific examples, intuitionistic theories of apartness and…Read more
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31The Gödel-McKinsey-Tarski embedding for infinitary intuitionistic logic and its extensionsAnnals of Pure and Applied Logic 174 (8): 103285. 2023.
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39Mathesis Universalis, Computability and Proof (edited book)Springer Verlag. 2019.In a fragment entitled Elementa Nova Matheseos Universalis Leibniz writes “the mathesis [...] shall deliver the method through which things that are conceivable can be exactly determined”; in another fragment he takes the mathesis to be “the science of all things that are conceivable.” Leibniz considers all mathematical disciplines as branches of the mathesis and conceives the mathesis as a general science of forms applicable not only to magnitudes but to every object that exists in our imaginat…Read more
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22Glivenko sequent classes and constructive cut elimination in geometric logicsArchive for Mathematical Logic 62 (5): 657-688. 2023.A constructivisation of the cut-elimination proof for sequent calculi for classical, intuitionistic and minimal infinitary logics with geometric rules—given in earlier work by the second author—is presented. This is achieved through a procedure where the non-constructive transfinite induction on the commutative sum of ordinals is replaced by two instances of Brouwer’s Bar Induction. The proof of admissibility of the structural rules is made ordinal-free by introducing a new well-founded relation…Read more
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9Geometric Rules in Infinitary LogicIn Ofer Arieli & Anna Zamansky (eds.), Arnon Avron on Semantics and Proof Theory of Non-Classical Logics, Springer Verlag. pp. 265-293. 2021.Large portions of mathematics such as algebra and geometry can be formalized using first-order axiomatizations. In many cases it is even possible to use a very well-behaved class of first-order axioms, namely, what are called coherent or geometric implications. Such class of axioms can be translated to inference rules that can be added to a sequent calculus while preserving its structural properties. In this work, this fundamental result is extended to their infinitary generalizations as extensi…Read more
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16Proof-Theoretic Analysis of the Logics of Agency: The Deliberative STITStudia Logica 109 (3): 473-507. 2020.A sequent calculus methodology for systems of agency based on branching-time frames with agents and choices is proposed, starting with a complete and cut-free system for multi-agent deliberative STIT; the methodology allows a transparent justification of the rules, good structural properties, analyticity, direct completeness and decidability proofs.
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51Proof theory for quantified monotone modal logicsLogic Journal of the IGPL 27 (4): 478-506. 2019.This paper provides a proof-theoretic study of quantified non-normal modal logics. It introduces labelled sequent calculi based on neighbourhood semantics for the first-order extension, with both varying and constant domains, of monotone NNML, and studies the role of the Barcan formulas in these calculi. It will be shown that the calculi introduced have good structural properties: invertibility of the rules, height-preserving admissibility of weakening and contraction and syntactic cut eliminati…Read more
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45Conditional beliefs: From neighbourhood semantics to sequent calculusReview of Symbolic Logic 11 (4): 736-779. 2018.The logic of Conditional Beliefs has been introduced by Board, Baltag, and Smets to reason about knowledge and revisable beliefs in a multi-agent setting. In this article both the semantics and the proof theory for this logic are studied. First, a natural semantics forCDLis defined in terms of neighbourhood models, a multi-agent generalisation of Lewis’ spheres models, and it is shown that the axiomatization ofCDLis sound and complete with respect to this semantics. Second, it is shown that the …Read more
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18The Logic of Conditional Beliefs: Neighbourhood Semantics and Sequent CalculusIn Lev Beklemishev, Stéphane Demri & András Máté (eds.), Advances in Modal Logic, Volume 11, Csli Publications. pp. 322-341. 2016.
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19Recent Advances in Proof Systems for Modal LogicIn Rajeev Goré, Barteld Kooi & Agi Kurucz (eds.), Advances in Modal Logic, Volume 10, Csli Publications. pp. 421-422. 2014.
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7Sequent Calculus in Natural Deduction StyleJournal of Symbolic Logic 66 (4): 1803-1816. 2001.A sequent calculus is given in which the management of weakening and contraction is organized as in natural deduction. The latter has no explicit weakening or contraction, but vacuous and multiple discharges in rules that discharge assumptions. A comparison to natural deduction is given through translation of derivations between the two systems. It is proved that if a cut formula is never principal in a derivation leading to the right premiss of cut, it is a subformula of the conclusion. Therefo…Read more
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26The intensional side of algebraic-topological representation theoremsSynthese 198 (Suppl 5): 1121-1143. 2017.Stone representation theorems are a central ingredient in the metatheory of philosophical logics and are used to establish modal embedding results in a general but indirect and non-constructive way. Their use in logical embeddings will be reviewed and it will be shown how they can be circumvented in favour of direct and constructive arguments through the methods of analytic proof theory, and how the intensional part of the representation results can be recovered from the syntactic proof of those…Read more
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69Contraction-free sequent calculi for geometric theories with an application to Barr's theoremArchive for Mathematical Logic 42 (4): 389-401. 2003.Geometric theories are presented as contraction- and cut-free systems of sequent calculi with mathematical rules following a prescribed rule-scheme that extends the scheme given in Negri and von Plato. Examples include cut-free calculi for Robinson arithmetic and real closed fields. As an immediate consequence of cut elimination, it is shown that if a geometric implication is classically derivable from a geometric theory then it is intuitionistically derivable.
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21Equality in the Presence of Apartness: An Application of Structural Proof Analysis to Intuitionistic AxiomaticsPhilosophia Scientiae 61-79. 2006.The theories of apartness, equality, and n-stable equality are presented through contraction- and cut-free sequent calculi. By methods of proof analysis, a purely proof-theoretic characterization of the equality fragment of apartness is obtained.
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98Structural Proof TheoryCambridge University Press. 2001.Structural proof theory is a branch of logic that studies the general structure and properties of logical and mathematical proofs. This book is both a concise introduction to the central results and methods of structural proof theory, and a work of research that will be of interest to specialists. The book is designed to be used by students of philosophy, mathematics and computer science. The book contains a wealth of results on proof-theoretical systems, including extensions of such systems fro…Read more
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53Kripke completeness revisitedIn Giuseppe Primiero (ed.), Acts of Knowledge: History, Philosophy and Logic, College Publications. pp. 233--266. 2009.
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109Proof analysis in intermediate logicsArchive for Mathematical Logic 51 (1): 71-92. 2012.Using labelled formulae, a cut-free sequent calculus for intuitionistic propositional logic is presented, together with an easy cut-admissibility proof; both extend to cover, in a uniform fashion, all intermediate logics characterised by frames satisfying conditions expressible by one or more geometric implications. Each of these logics is embedded by the Gödel–McKinsey–Tarski translation into an extension of S4. Faithfulness of the embedding is proved in a simple and general way by constructive…Read more
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97Varieties of linear calculiJournal of Philosophical Logic 31 (6): 569-590. 2002.A uniform calculus for linear logic is presented. The calculus has the form of a natural deduction system in sequent calculus style with general introduction and elimination rules. General elimination rules are motivated through an inversion principle, the dual form of which gives the general introduction rules. By restricting all the rules to their single-succedent versions, a uniform calculus for intuitionistic linear logic is obtained. The calculus encompasses both natural deduction and seque…Read more
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182Proof Theory for Modal LogicPhilosophy Compass 6 (8): 523-538. 2011.The axiomatic presentation of modal systems and the standard formulations of natural deduction and sequent calculus for modal logic are reviewed, together with the difficulties that emerge with these approaches. Generalizations of standard proof systems are then presented. These include, among others, display calculi, hypersequents, and labelled systems, with the latter surveyed from a closer perspective.