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Relating Sequent Calculi for Biintuitionistic Propositional Logic
"... Abstract. Biintuitionistic logic is the conservative extension of intuitionistic logic with a connective dual to implication. It is sometimes presented as a symmetric constructive subsystem of classical logic. In this paper, we compare three sequent calculi for biintuitionistic propositional logic ..."
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Abstract. Biintuitionistic logic is the conservative extension of intuitionistic logic with a connective dual to implication. It is sometimes presented as a symmetric constructive subsystem of classical logic. In this paper, we compare three sequent calculi for biintuitionistic propositional logic: (1) a basic standardstyle sequent calculus that restricts the premises of implicationright and exclusionleft inferences to be singleconclusion resp. singleassumption and is incomplete without the cut rule, (2) the calculus with nested sequents by Goré et al., where a complete class of cuts is encapsulated into special “unnest ” rules and (3) a cutfree labelled sequent calculus derived from the Kripke semantics of the logic. We show that these calculi can be translated into each other and discuss the ineliminable cuts of the standardstyle sequent calculus. 1
A Connectionbased Characterization of Biintuitionistic Validity
"... Abstract. We give a connectionbased characterization of validity in propositional biintuitionistic logic in terms of speci c directed graphs called Rgraphs. Such a characterization is wellsuited for deriving labelled proofsystems with countermodel construction facilities. We rst de ne the noti ..."
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Abstract. We give a connectionbased characterization of validity in propositional biintuitionistic logic in terms of speci c directed graphs called Rgraphs. Such a characterization is wellsuited for deriving labelled proofsystems with countermodel construction facilities. We rst de ne the notion of biintuitionistic Rgraph from which we then obtain a connectionbased characterization of propositional biintuitionistic validity and derive a sound and complete freevariable labelled sequent calculus that admits cutelimination and also variable splitting. 1
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
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"... Quantifiers are not interdefinable in the secondorder propositional constant domain logic ..."
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Quantifiers are not interdefinable in the secondorder propositional constant domain logic
DOI: 10.1007/9783642216916_6 Classical Callbyneed and duality
, 2011
"... Abstract. We study callbyneed from the point of view of the duality between callbyname and callbyvalue. We develop sequentcalculus style versions of callbyneed both in the minimal and classical case. As a result, we obtain a natural extension of callbyneed with control operators. This lea ..."
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Abstract. We study callbyneed from the point of view of the duality between callbyname and callbyvalue. We develop sequentcalculus style versions of callbyneed both in the minimal and classical case. As a result, we obtain a natural extension of callbyneed with control operators. This leads us to introduce a callbyneed λµcalculus. Finally, by using the dualities principles of λµ˜µcalculus, we show the existence of a new callbyneed calculus, which is distinct from callbyname, callbyvalue and usual callbyneed theories.
A pragmatic framework for intuitionistic modalities: Classical logic and Lax logic.
"... Summary. We reconsider Dalla Pozza and Garola pragmatic interpretation of intuitionistic logic [13] where sentences and proofs formalize assertions and their justifications and revise it so that the costruction is done within an intuitionistic metatheory. We reconsider also the extension of Dalla Po ..."
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Summary. We reconsider Dalla Pozza and Garola pragmatic interpretation of intuitionistic logic [13] where sentences and proofs formalize assertions and their justifications and revise it so that the costruction is done within an intuitionistic metatheory. We reconsider also the extension of Dalla Pozza and Garola’a approach to cointuitionistic logic, seen as a logic of hypotheses [5, 9, 4] and the duality between assertions and hypotheses represented by two negations, the assertive and the hypothetical ones. By adding illocutionary forces of conjecture, defined as a hypothesis that an assertion is justified and of expectation, an assertion that a hypothesis is justified we obtain pragmatic counterparts of the modalities of classical S4, but also a framework for different interpretations of intuitionistic modalities necessity and possibility. We consider two applications: one is typing Parigot’s λµ calculus in a biintuitionistic logic of expectations. The second is an interpretation of Fairtlough and Mendler’s Propositional Lax Logic as an extension of intuitionistic logic with a cointuitionistic operator of empirical possibility. 1 Preface: intuitionistic pragmatics and its extensions.
A pragmatic logic of hypotheses
"... Summary. In this paper we consider Dalla Pozza and Garola pragmatic interpretation of intuitionistic logic [13] focussing on the role of illocutionary forces and justification conditions and we extend their approach by interpreting intuitionistic and cointuitionistic logic through an “intended inte ..."
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Summary. In this paper we consider Dalla Pozza and Garola pragmatic interpretation of intuitionistic logic [13] focussing on the role of illocutionary forces and justification conditions and we extend their approach by interpreting intuitionistic and cointuitionistic logic through an “intended interpretation ” in the style of a gametheoretic semantics and a notion of duality between assertions and hypotheses. The compatibility of these constructions from the viewpoint of an intuitionistic philosophy is shown by checking that the construction can be performed in an intuitionistic metatheory. 1
Languages, Theory
"... Every functional programmer knows about sum and product types, a+b and a×b respectively. Negative and fractional types, a−b and a/b respectively, are much less known and their computational interpretation is unfamiliar and often complicated. We show that in a programming model in which information i ..."
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Every functional programmer knows about sum and product types, a+b and a×b respectively. Negative and fractional types, a−b and a/b respectively, are much less known and their computational interpretation is unfamiliar and often complicated. We show that in a programming model in which information is preserved (such as the model introduced in our recent paper on Information Effects), these types have particularly natural computational interpretations. Intuitively, values of negative types are values that flow “backwards ” to satisfy demands and values of fractional types are values that impose constraints on their context. The combination of these negative and fractional types enables greater flexibility in programming by breaking global invariants into local ones that can be autonomously satisfied by a subcomputation. Theoretically, these types give rise to two function spaces and to two notions of continuations, suggesting that the previously observed duality of computation conflated two orthogonal notions: an additive duality that corresponds to backtracking and a multiplicative duality that corresponds to constraint propagation.