Relating Sequent Calculi for Bi-intuitionistic Propositional Logic

Bi-intuitionistic 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 bi-intuitionistic propositional logic: (1) a basic standard-style sequent calculus that restricts the premises of implication-right and exclusion-left inferences to be single-conclusion resp. single-assumption and is incomplete without the cut rule, (2) the calculus with nested sequents by Gore et al., where a complete class of cuts is encapsulated into special "unnest" rules and (3) a cut-free 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 standard-style sequent calculus.Comment: In Proceedings CL&C 2010, arXiv:1101.520

Syntactic Interpolation for Tense Logics and Bi-Intuitionistic Logic via Nested Sequents

We provide a direct method for proving Craig interpolation for a range of modal and intuitionistic logics, including those containing a "converse" modality. We demonstrate this method for classical tense logic, its extensions with path axioms, and for bi-intuitionistic logic. These logics do not have straightforward formalisations in the traditional Gentzen-style sequent calculus, but have all been shown to have cut-free nested sequent calculi. The proof of the interpolation theorem uses these calculi and is purely syntactic, without resorting to embeddings, semantic arguments, or interpreted connectives external to the underlying logical language. A novel feature of our proof includes an orthogonality condition for defining duality between interpolants

Proof search and counter-model construction for bi-intuitionistic propositional logic with labelled sequents

Bi-intuitionistic logic is a conservative extension of intuitionistic logic with a connective dual to implication, called exclusion. We present a sound and complete cut-free labelled sequent calculus for bi-intuitionistic propositional logic, BiInt, following S. Negri's general method for devising sequent calculi for normal modal logics. Although it arises as a natural formalization of the Kripke semantics, it is does not directly support proof search. To describe a proof search procedure, we develop a more algorithmic version that also allows for counter-model extraction from a failed proof attempt.Estonian Science Foundation - grants no. 5567; 6940Fundação para a Ciência e a Tecnologia (FCT)RESCUE - no. PTDC/EIA/65862/2006TYPES - FP6 ISTCentro de matemática da Universidade do Minh

A proof-theoretic study of bi-intuitionistic propositional sequent calculus

Bi-intuitionistic logic is the conservative extension of intuitionistic logic with a connective dual to implication usually called ‘exclusion’. A standard-style sequent calculus for this logic is easily obtained by extending multiple-conclusion sequent calculus for intuitionistic logic with exclusion rules dual to the implication rules (in particular, the exclusion-left rule restricts the premise to be single-assumption). However, similarly to standard-style sequent calculi for non-classical logics like S5, this calculus is incomplete without the cut rule. Motivated by the problem of proof search for propositional bi-intuitionistic logic (BiInt), various cut-free calculi with extended sequents have been proposed, including (i) a calculus of nested sequents by Goré et al., which includes rules for creation and removal of nests (called ‘nest rules’, resp. ‘unnest rules’) and (ii) a calculus of labelled sequents by the authors, derived from the Kripke semantics of BiInt, which includes ‘monotonicity rules’ to propagate truth/falsehood between accessible worlds. In this paper, we develop a proof-theoretic study of these three sequent calculi for BiInt grounded on translations between them. We start by establishing the basic meta-theory of the labelled calculus (including cut-admissibility), and use then the translations to obtain results for the other two calculi. The translation of the nested calculus into the standard-style calculus explains how the unnest rules encapsulate cuts. The translations between the labelled and the nested calculi reveal the two formats to be very close, despite the former incorporating semantic elements, and the latter being syntax-driven. Indeed, we single out (i) a labelled calculus whose sequents have a ‘label in focus’ and which includes ‘refocusing rules’ and (ii) a nested calculus with monotonicity and refocusing rules, and prove these two calculi to be isomorphic (in a bijection both at the level of sequents and at the level of derivations).ERDF through the Estonian Centre of Excellence in Computer Science (EXCS), by the Estonian Science Foundation under grant no. 6940; COST action CA15123 EUTYPES.info:eu-repo/semantics/publishedVersio

A cut-free sequent calculus for the bi-intuitionistic logic 2Int

The purpose of this paper is to introduce a bi-intuitionistic sequent calculus and to give proofs of admissibility for its structural rules. The calculus I will present, called SC2Int, is a sequent calculus for the bi-intuitionistic logic 2Int, which Wansing presents in [2016a]. There he also gives a natural deduction system for this logic, N2Int, to which SC2Int is equivalent in terms of what is derivable. What is important is that these calculi represent a kind of bilateralist reasoning, since they do not only internalize processes of verifcation or provability but also the dual processes in terms of falsifcation or what is called dual provability. In [Wansing, 2017] a normal form theorem for N2Int is stated, here, I want to prove a cut-elimination theorem for SC2Int, i.e., if successful, this would extend the results existing so far

On an Intuitionistic Logic for Pragmatics

We reconsider the pragmatic interpretation of intuitionistic logic [21] regarded as a logic of assertions and their justications and its relations with classical logic. We recall an extension of this approach to a logic dealing with assertions and obligations, related by a notion of causal implication [14, 45]. We focus on the extension to co-intuitionistic logic, seen as a logic of hypotheses [8, 9, 13] and on polarized bi-intuitionistic logic as a logic of assertions and conjectures: looking at the S4 modal translation, we give a denition of a system AHL of bi-intuitionistic logic that correctly represents the duality between intuitionistic and co-intuitionistic logic, correcting a mistake in previous work [7, 10]. A computational interpretation of cointuitionism as a distributed calculus of coroutines is then used to give an operational interpretation of subtraction.Work on linear co-intuitionism is then recalled, a linear calculus of co-intuitionistic coroutines is dened and a probabilistic interpretation of linear co-intuitionism is given as in [9]. Also we remark that by extending the language of intuitionistic logic we can express the notion of expectation, an assertion that in all situations the truth of p is possible and that in a logic of expectations the law of double negation holds. Similarly, extending co-intuitionistic logic, we can express the notion of conjecture that p, dened as a hypothesis that in some situation the truth of p is epistemically necessary