•  249
    Sequent calculus in natural deduction style
    Journal 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
  •  123
    Proof Theory for Modal Logic
    Philosophy 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.
  •  107
    Decision methods for linearly ordered Heyting algebras
    with Roy Dyckhoff
    Archive for Mathematical Logic 45 (4): 411-422. 2005.
    The decision problem for positively quantified formulae in the theory of linearly ordered Heyting algebras is known, as a special case of work of Kreisel, to be solvable; a simple solution is here presented, inspired by related ideas in Gödel-Dummett logic
  •  103
    Proof Analysis in Modal Logic
    Journal of Philosophical Logic 34 (5-6): 507-544. 2005.
    A general method for generating contraction- and cut-free sequent calculi for a large family of normal modal logics is presented. The method covers all modal logics characterized by Kripke frames determined by universal or geometric properties and it can be extended to treat also Gödel-Löb provability logic. The calculi provide direct decision methods through terminating proof search. Syntactic proofs of modal undefinability results are obtained in the form of conservativity theorems.
  •  82
    Varieties of linear calculi
    Journal 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
  •  78
    Proof analysis in intermediate logics
    with Roy Dyckhoff
    Archive for Mathematical Logic 51 (1-2): 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
  •  63
    Cut Elimination in the Presence of Axioms
    with Sara Negri and Jan Von Plato
    Bulletin of Symbolic Logic 4 (4): 418-435. 1998.
    A way is found to add axioms to sequent calculi that maintains the eliminability of cut, through the representation of axioms as rules of inference of a suitable form. By this method, the structural analysis of proofs is extended from pure logic to free-variable theories, covering all classical theories, and a wide class of constructive theories. All results are proved for systems in which also the rules of weakening and contraction can be eliminated. Applications include a system of predicate l…Read more
  •  62
    Structural Proof Theory
    with Jan von Plato and Aarne Ranta
    Cambridge 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
  •  50
    Admissibility of structural rules for contraction-free systems of intuitionistic logic
    with Roy Dyckhoff
    Journal of Symbolic Logic 65 (4): 1499-1518. 2000.
    We give a direct proof of admissibility of cut and contraction for the contraction-free sequent calculus G4ip for intuitionistic propositional logic and for a corresponding multi-succedent calculus: this proof extends easily in the presence of quantifiers, in contrast to other, indirect, proofs. i.e., those which use induction on sequent weight or appeal to admissibility of rules in other calculi
  •  49
    The continuum as a formal space
    with Daniele Soravia
    Archive for Mathematical Logic 38 (7): 423-447. 1999.
    A constructive definition of the continuum based on formal topology is given and its basic properties studied. A natural notion of Cauchy sequence is introduced and Cauchy completeness is proved. Other results include elementary proofs of the Baire and Cantor theorems. From a classical standpoint, formal reals are seen to be equivalent to the usual reals. Lastly, the relation of real numbers as a formal space to other approaches to constructive real numbers is determined
  •  48
    Proofs and Countermodels in Non-Classical Logics
    Logica Universalis 8 (1): 25-60. 2014.
    Proofs and countermodels are the two sides of completeness proofs, but, in general, failure to find one does not automatically give the other. The limitation is encountered also for decidable non-classical logics in traditional completeness proofs based on Henkin’s method of maximal consistent sets of formulas. A method is presented that makes it possible to establish completeness in a direct way: For any given sequent either a proof in the given logical system or a countermodel in the correspon…Read more
  •  43
    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.
  •  37
    A normalizing system of natural deduction for intuitionistic linear logic
    Archive for Mathematical Logic 41 (8): 789-810. 2002.
    The main result of this paper is a normalizing system of natural deduction for the full language of intuitionistic linear logic. No explicit weakening or contraction rules for -formulas are needed. By the systematic use of general elimination rules a correspondence between normal derivations and cut-free derivations in sequent calculus is obtained. Normalization and the subformula property for normal derivations follow through translation to sequent calculus and cut-elimination
  •  37
    Sequent calculus proof theory of intuitionistic apartness and order relations
    Archive for Mathematical Logic 38 (8): 521-547. 1999.
    Contraction-free sequent calculi for intuitionistic theories of apartness and order are given and cut-elimination for the calculi proved. Among the consequences of the result is the disjunction property for these theories. Through methods of proof analysis and permutation of rules, we establish conservativity of the theory of apartness over the theory of equality defined as the negation of apartness, for sequents in which all atomic formulas appear negated. The proof extends to conservativity re…Read more
  •  36
    Machine generated contents note: Prologue: Hilbert's Last Problem; 1. Introduction; Part I. Proof Systems Based on Natural Deduction: 2. Rules of proof: natural deduction; 3. Axiomatic systems; 4. Order and lattice theory; 5. Theories with existence axioms; Part II. Proof Systems Based on Sequent Calculus: 6. Rules of proof: sequent calculus; 7. Linear order; Part III. Proof Systems for Geometric Theories: 8. Geometric theories; 9. Classical and intuitionistic axiomatics; 10. Proof analysis in e…Read more
  •  33
    Proof-theoretical analysis of order relations
    with Jan von Plato and Thierry Coquand
    Archive 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
  •  31
    Kripke completeness revisited
    In Giuseppe Primiero (ed.), Acts of Knowledge: History, Philosophy and Logic, College Publications. pp. 233--266. 2009.
  •  29
    Geometrisation of first-order logic
    with Roy Dyckhoff
    Bulletin of Symbolic Logic 21 (2): 123-163. 2015.
    That every first-order theory has a coherent conservative extension is regarded by some as obvious, even trivial, and by others as not at all obvious, but instead remarkable and valuable; the result is in any case neither sufficiently well-known nor easily found in the literature. Various approaches to the result are presented and discussed in detail, including one inspired by a problem in the proof theory of intermediate logics that led us to the proof of the present paper. It can be seen as a …Read more
  •  27
    Glivenko sequent classes in the light of structural proof theory
    Archive for Mathematical Logic 55 (3-4): 461-473. 2016.
    In 1968, Orevkov presented proofs of conservativity of classical over intuitionistic and minimal predicate logic with equality for seven classes of sequents, what are known as Glivenko classes. The proofs of these results, important in the literature on the constructive content of classical theories, have remained somehow cryptic. In this paper, direct proofs for more general extensions are given for each class by exploiting the structural properties of G3 sequent calculi; for five of the seven …Read more
  •  20
    Conditional beliefs: From neighbourhood semantics to sequent calculus
    with Marianna Girlando, Nicola Olivetti, and Vincent Risch
    Review 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
  •  16
    Proof theory for quantified monotone modal logics
    with Eugenio Orlandelli
    Logic 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
  •  15
    Tychonoff's Theorem in the Framework of Formal Topologies
    with Silvio Valentini
    Journal of Symbolic Logic 62 (4): 1315-1332. 1997.
  •  14
    University of Azores, Ponta Delgada, Azores, Portugal June 30–July 4, 2010
    with Eric Allender, José L. Balcázar, Shafi Goldwasser, Denis Hirschfeldt, Toniann Pitassi, and Ronald de Wolf
    Bulletin of Symbolic Logic 17 (3). 2011.
  •  13
    Admissibility of Structural Rules for Contraction-Free Systems of Intuitionistic Logic
    with Roy Dyckhoff
    Journal of Symbolic Logic 65 (4): 1499-1518. 2000.
    We give a direct proof of admissibility of cut and contraction for the contraction-free sequent calculus G4ip for intuitionistic propositional logic and for a corresponding multi-succedent calculus: this proof extends easily in the presence of quantifiers, in contrast to other, indirect, proofs. i.e., those which use induction on sequent weight or appeal to admissibility of rules in other calculi.
  •  12
    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