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Cutelimination and proofsearch for biintuitionistic logic using nested sequents
, 2008
"... We propose a new sequent calculus for biintuitionistic logic which sits somewhere between display calculi and traditional sequent calculi by using nested sequents. Our calculus enjoys a simple (purely syntactic) cutelimination proof as do display calculi. But it has an easily derivable variant cal ..."
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We propose a new sequent calculus for biintuitionistic logic which sits somewhere between display calculi and traditional sequent calculi by using nested sequents. Our calculus enjoys a simple (purely syntactic) cutelimination proof as do display calculi. But it has an easily derivable variant calculus which is amenable to automated proof search as are (some) traditional sequent calculi. We first present the initial calculus and its cutelimination proof. We then present the derived calculus, and then present a proofsearch strategy which allows it to be used for automated proof search. We prove that this search strategy is terminating and complete by showing how it can be used to mimic derivations obtained from an existing calculus GBiInt for biintuitionistic logic. As far as we know, our new calculus is the first sequent calculus for biintuitionistic logic which uses no semantic additions like labels, which has a purely syntactic cutelimination proof, and which can be used naturally for backwards proofsearch.
Proof search and countermodel construction for biintuitionistic propositional logic with labelled sequents
 In TABLEAUX
, 2009
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Theoremhood Preserving Maps As A Characterisation Of Cut Elimination For Provability Logics.
, 1999
"... We define cutfree display calculi for provability (modal) logics that are not properly displayable according to Kracht's analysis. We also show that a weak form of the cutelimination theorem (for these modal display calculi) is equivalent to the theoremhoodpreserving property of certain maps ..."
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Cited by 7 (5 self)
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We define cutfree display calculi for provability (modal) logics that are not properly displayable according to Kracht's analysis. We also show that a weak form of the cutelimination theorem (for these modal display calculi) is equivalent to the theoremhoodpreserving property of certain maps from the provability logics into properly displayable modal logics. 1 Australian Research Council International Research Fellow from Laboratoire LEIBNIZCNRS, Grenoble, France. 2 Supported by an Australian Research Council Queen Elizabeth II Fellowship. 1 Introduction Background. Display Logic (DL) is a prooftheoretical framework introduced by Belnap [Bel82] that generalises the structural language of Gentzen's sequents in a rather abstract way by using multiple complex structural connectives instead of Gentzen's comma. The term "display" comes from the nice property that any occurrence of a structure in a sequent can be displayed either as the entire antecedent or as the entire succedent...
A TERM ASSIGNMENT FOR DUAL INTUITIONISTIC LOGIC.
"... Abstract. We study the prooftheory of coHeyting algebras and present a calculus of continuations typed in the disjunctive–subtractive fragment of dual intuitionistic logic. We give a singleassumption multipleconclusions Natural Deduction system NJ � � for this logic: unlike the bestknown treatm ..."
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Abstract. We study the prooftheory of coHeyting algebras and present a calculus of continuations typed in the disjunctive–subtractive fragment of dual intuitionistic logic. We give a singleassumption multipleconclusions Natural Deduction system NJ � � for this logic: unlike the bestknown treatments of multipleconclusion systems (e.g., Parigot’s λ−µ calculus, or Urban and Bierman’s termcalculus) here the termassignment is distributed to all conclusions, and exhibits several features of calculi for concurrency, such as remote capture of variable and remote substitution. The present construction can be regarded as the construction of a free coCartesian
Kripke semantics and proof systems for combining intuitionistic logic and classical logic. Submitted
, 2011
"... We combine intuitionistic logic and classical logic into a new, firstorder logic called Polarized Intuitionistic Logic. This logic is based on a distinction between two dual polarities which we call red and green to distinguish them from other forms of polarization. The meaning of these polarities ..."
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We combine intuitionistic logic and classical logic into a new, firstorder logic called Polarized Intuitionistic Logic. This logic is based on a distinction between two dual polarities which we call red and green to distinguish them from other forms of polarization. The meaning of these polarities is defined modeltheoretically by a Kripkestyle semantics for the logic. Two proof systems are also formulated. The first system extends Gentzen’s intuitionistic sequent calculus LJ. In addition, this system also bears essential similarities to Girard’s LC proof system for classical logic. The second proof system is based on a semantic tableau and extends Dragalin’s multipleconclusion version of intuitionistic sequent calculus. We show that soundness and completeness hold for these notions of semantics and proofs, from which it follows that cut is admissible in the proof systems and that the propositional fragment of the logic is decidable. 1
Categorical Proof Theory of CoIntuitionistic Linear Logic
"... Summary. To provide a categorical semantics for cointuitionistic logic, one has to face the fact, noted by Tristan Crolard, that the definition of coexponents as adjuncts of coproducts does not work in the category Set, where coproducts are disjoint unions. Following the familiar construction of ..."
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Summary. To provide a categorical semantics for cointuitionistic logic, one has to face the fact, noted by Tristan Crolard, that the definition of coexponents as adjuncts of coproducts does not work in the category Set, where coproducts are disjoint unions. Following the familiar construction of models of intuitionistic linear logic with exponent!, we build models of cointuitionistic logic in symmetric monoidal closed categories with additional structure, using a variant of Crolard’s term assignment to cointuitionistic logic in the construction of a free category. 1
hypotheses, conjectures and expectations. Rough set semantics and prooftheory
 Advances in Natural Deduction
, 2013
"... Summary. In this paper biintuitionism is interpreted as an intensional logic which is about the justification conditions of assertions and hypotheses, extending C. Dalla Pozza and C. Garola’s pragmatic interpretation [18] of intuitionism, seen as a logic of assertions according to a suggestion by M ..."
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Summary. In this paper biintuitionism is interpreted as an intensional logic which is about the justification conditions of assertions and hypotheses, extending C. Dalla Pozza and C. Garola’s pragmatic interpretation [18] of intuitionism, seen as a logic of assertions according to a suggestion by M. Dummett. Revising our previous work on this matter [5], we consider two additional illocutionary forces, (i) conjecturing, seen as making the hypotheses that a proposition is epistemically necessary, and (ii) expecting, regarded as asserting that a propostion is epistemically possible; we show that a logic of expectations justifies the double negation law. We formalize our logic in a calculus of sequents and study bimodal Kripke semantics of biintuitionism based on translations in S4. We look at rough set semantics following P. Pagliani’s analysis of “intrinsic coHeyting boundaries ” [40] (after Lawvere). A Natural Deduction system for cointuitionistic logic is given where proofs are represented as upside down Prawitz trees. We give a computational interpretation of cointuitionism, based on T. Crolard’s notion of coroutine [16] as the programming construction corresponding to subtraction introduction. Our typed calculus of coroutines is dual to
NATURAL DEDUCTION AND TERM ASSIGNMENT FOR COHEYTING ALGEBRAS IN POLARIZED BIINTUITIONISTIC LOGIC.
"... Abstract. We reconsider Rauszer’s biintuitionistic logic in the framework of the logic for pragmatics: every formula is regarded as expressing an act of assertion or conjecture, where conjunction and implication are assertive and subtraction and disjunction are conjectural. The resulting system of ..."
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Abstract. We reconsider Rauszer’s biintuitionistic logic in the framework of the logic for pragmatics: every formula is regarded as expressing an act of assertion or conjecture, where conjunction and implication are assertive and subtraction and disjunction are conjectural. The resulting system of polarized biintuitionistic logic (PBL) consists of two fragments, positive intuitionistic logic LJ⊃ ∩ and its dual LJ� � , extended with two negations partially internalizing the duality between LJ⊃ ∩ and LJ� �. Modal interpretations and Kripke’s semantics over bimodal preordered frames are considered and a Natural Deduction system PBN is sketched for the whole system. A stricter interpretation of the duality and a simpler natural deduction system is obtained when polarized biintuitionistic logic is interpreted over S4 rather than bimodal S4 (a logic called intuitionistic logic for pragmatics of assertions and conjectures ILPAC). The term assignment for the conjectural fragment LJ� � exhibits several features of calculi for concurrency, such as remote capture of variable and remote substitution. The duality is extended from formulas to proofs and it is shown that every computation in our calculus is isomorphic to a computation in the simply typed λcalculus. §1. Preface. We present a natural deduction system for propositional polarized biintuitionistic logic PBL, (a variant of) intuitionistic logic extended with a connective of subtraction A � B, read as “A but not B”, which is dual to implication. 1 The logic PBL is polarized in the sense that its expressions are regarded as expressing acts of assertion or of conjecture; implications and conjunctions are assertive, subtractions and disjunctions are conjectural. Assertions and conjectures are regarded as dual; moreover there are two negations, transforming assertions into conjectures and viceversa, in some sense internalizing the duality. Our notion of polarity isn’t just a technical device: it is rooted in an analysis of the structure of speechacts, following the viewpoint of the
ComplementTopoi and Dual Intuitionistic Logic
, 2010
"... Abstract: Mortensen in [11] studies dual intuitionistic logic by dualizing topos internal logic, but he did not study a sequent calculus. In this paper I present a sequent calculus for complementtopos logic, which throws some light on the problem of giving a dualization for LJ. 1 introductory remar ..."
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Abstract: Mortensen in [11] studies dual intuitionistic logic by dualizing topos internal logic, but he did not study a sequent calculus. In this paper I present a sequent calculus for complementtopos logic, which throws some light on the problem of giving a dualization for LJ. 1 introductory remarks “Dual intuitionistic logic ” has been investigated to varying degrees of success through different perspectives. McKinsey and Tarski in [8] studied it from an algebraic point of view through the algebraic properties of “closure ” or Brouwerian algebras, the algebraic duals to Heyting algebras. 1 Curry in [2] presented what he called “Skolem lattices”, which comprise “absolute implicational lattices” and “absolute subtractive lattices”. Rauszer (cf. [14], [15]) used algebraic, Hilbertstyle and relational methods, but not Gentzen calculi, to investigate “intuitionistic logic with dual operators”, “pseudodifference ” being the dual to intuitionistic implication. From then the problem of dualizing intuitionistic logic has been carried