104 research outputs found
Uniform interpolation via nested sequents
A modular proof-theoretic framework was recently developed to prove Craig interpolation for normal modal logics based on generalizations of sequent calculi (e.g., nested sequents, hypersequents, and labelled sequents). In this paper, we turn to uniform interpolation, which is stronger than Craig interpolation. We develop a constructive method for proving uniform interpolation via nested sequents and apply it to reprove the uniform interpolation property for normal modal logics K, D, and T. While our method is proof-theoretic, the definition of uniform interpolation for nested sequents also uses semantic notions, including bisimulation modulo an atomic proposition
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
Proving Craig and Lyndon Interpolation Using Labelled Sequent Calculi
We have recently presented a general method of proving the fundamental
logical properties of Craig and Lyndon Interpolation (IPs) by induction on
derivations in a wide class of internal sequent calculi, including sequents,
hypersequents, and nested sequents. Here we adapt the method to a more general
external formalism of labelled sequents and provide sufficient criteria on the
Kripke-frame characterization of a logic that guarantee the IPs. In particular,
we show that classes of frames definable by quantifier-free Horn formulas
correspond to logics with the IPs. These criteria capture the modal cube and
the infinite family of transitive Geach logics
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
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
Inducing syntactic cut-elimination for indexed nested sequents
The key to the proof-theoretic study of a logic is a proof calculus with a
subformula property. Many different proof formalisms have been introduced (e.g.
sequent, nested sequent, labelled sequent formalisms) in order to provide such
calculi for the many logics of interest. The nested sequent formalism was
recently generalised to indexed nested sequents in order to yield proof calculi
with the subformula property for extensions of the modal logic K by
(Lemmon-Scott) Geach axioms. The proofs of completeness and cut-elimination
therein were semantic and intricate. Here we show that derivations in the
labelled sequent formalism whose sequents are `almost treelike' correspond
exactly to indexed nested sequents. This correspondence is exploited to induce
syntactic proofs for indexed nested sequent calculi making use of the elegant
proofs that exist for the labelled sequent calculi. A larger goal of this work
is to demonstrate how specialising existing proof-theoretic transformations
alleviate the need for independent proofs in each formalism. Such coercion can
also be used to induce new cutfree calculi. We employ this to present the first
indexed nested sequent calculi for intermediate logics.Comment: This is an extended version of the conference paper [20
Extensions of K5: Proof Theory and Uniform Lyndon Interpolation
We introduce a Gentzen-style framework, called layered sequent calculi, for
modal logic K5 and its extensions KD5, K45, KD45, KB5, and S5 with the goal to
investigate the uniform Lyndon interpolation property (ULIP), which implies
both the uniform interpolation property and the Lyndon interpolation property.
We obtain complexity-optimal decision procedures for all logics and present a
constructive proof of the ULIP for K5, which to the best of our knowledge, is
the first such syntactic proof. To prove that the interpolant is correct, we
use model-theoretic methods, especially bisimulation modulo literals.Comment: 20-page conference paper + 5-page appendix with examples and proof
Nested Sequents for Quantified Modal Logics
This paper studies nested sequents for quantified modal logics. In
particular, it considers extensions of the propositional modal logics definable
by the axioms D, T, B, 4, and 5 with varying, increasing, decreasing, and
constant domains. Each calculus is proved to have good structural properties:
weakening and contraction are height-preserving admissible and cut is
(syntactically) admissible. Each calculus is shown to be equivalent to the
corresponding axiomatic system and, thus, to be sound and complete. Finally, it
is argued that the calculi are internal -- i.e., each sequent has a formula
interpretation -- whenever the existence predicate is expressible in the
language.Comment: accepted to TABLEAUX 202
Nested Sequents for Quantified Modal Logics
This paper studies nested sequents for quantified modal logics. In particular, it considers extensions of the propositional modal logics definable by the axioms D, T, B, 4, and 5 with varying, increasing, decreasing, and constant domains. Each calculus is proved to have good structural properties: weakening and contraction are height-preserving admissible and cut is (syntactically) admissible. Each calculus is shown to be equivalent to the corresponding axiomatic system and, thus, to be sound and complete. Finally, it is argued that the calculi are internal—i.e., each sequent has a formula interpretation—whenever the existence predicate is expressible in the language
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