The Varieties of Ought-implies-Can and Deontic STIT Logic

STIT logic is a prominent framework for the analysis of multi-agent choice-making. In the available deontic extensions of STIT, the principle of Ought-implies-Can (OiC) fulfills a central role. However, in the philosophical literature a variety of alternative OiC interpretations have been proposed and discussed. This paper provides a modular framework for deontic STIT that accounts for a multitude of OiC readings. In particular, we discuss, compare, and formalize ten such readings. We provide sound and complete sequent-style calculi for all of the various STIT logics accommodating these OiC principles. We formally analyze the resulting logics and discuss how the different OiC principles are logically related. In particular, we propose an endorsement principle describing which OiC readings logically commit one to other OiC readings

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

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

On Deriving Nested Calculi for Intuitionistic Logics from Semantic Systems

This paper shows how to derive nested calculi from labelled calculi for propositional intuitionistic logic and first-order intuitionistic logic with constant domains, thus connecting the general results for labelled calculi with the more refined formalism of nested sequents. The extraction of nested calculi from labelled calculi obtains via considerations pertaining to the elimination of structural rules in labelled derivations. Each aspect of the extraction process is motivated and detailed, showing that each nested calculus inherits favorable proof-theoretic properties from its associated labelled calculus

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