1,838 research outputs found
Grafting Hypersequents onto Nested Sequents
We introduce a new Gentzen-style framework of grafted hypersequents that
combines the formalism of nested sequents with that of hypersequents. To
illustrate the potential of the framework, we present novel calculi for the
modal logics and , as well as for extensions of the
modal logics and with the axiom for shift
reflexivity. The latter of these extensions is also known as
in the context of deontic logic. All our calculi enjoy syntactic cut
elimination and can be used in backwards proof search procedures of optimal
complexity. The tableaufication of the calculi for and
yields simplified prefixed tableau calculi for these logic
reminiscent of the simplified tableau system for , which might be
of independent interest
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
Proof Theory for Intuitionistic Strong L\"ob Logic
This paper introduces two sequent calculi for intuitionistic strong L\"ob
logic : a terminating sequent calculus based
on the terminating sequent calculus for intuitionistic
propositional logic and an extension of the
standard cut-free sequent calculus without structural rules for
. One of the main results is a syntactic proof of the
cut-elimination theorem for . In addition, equivalences
between the sequent calculi and Hilbert systems for are
established. It is known from the literature that is complete
with respect to the class of intuitionistic modal Kripke models in which the
modal relation is transitive, conversely well-founded and a subset of the
intuitionistic relation. Here a constructive proof of this fact is obtained by
using a countermodel construction based on a variant of . The
paper thus contains two proofs of cut-elimination, a semantic and a syntactic
proof.Comment: 29 pages, 4 figures, submitted to the Special Volume of the Workshop
Proofs! held in Paris in 201
Kripke Models for Classical Logic
We introduce a notion of Kripke model for classical logic for which we
constructively prove soundness and cut-free completeness. We discuss the
novelty of the notion and its potential applications
The Basics of Display Calculi
The aim of this paper is to introduce and explain display calculi for a variety of logics. We provide a survey of key results concerning such calculi, though we focus mainly on the global cut elimination theorem. Propositional, first-order, and modal display calculi are considered and their properties detailed
Syntactic completeness of proper display calculi
A recent strand of research in structural proof theory aims at exploring the
notion of analytic calculi (i.e. those calculi that support general and modular
proof-strategies for cut elimination), and at identifying classes of logics
that can be captured in terms of these calculi. In this context, Wansing
introduced the notion of proper display calculi as one possible design
framework for proof calculi in which the analiticity desiderata are realized in
a particularly transparent way. Recently, the theory of properly displayable
logics (i.e. those logics that can be equivalently presented with some proper
display calculus) has been developed in connection with generalized Sahlqvist
theory (aka unified correspondence). Specifically, properly displayable logics
have been syntactically characterized as those axiomatized by analytic
inductive axioms, which can be equivalently and algorithmically transformed
into analytic structural rules so that the resulting proper display calculi
enjoy a set of basic properties: soundness, completeness, conservativity, cut
elimination and subformula property. In this context, the proof that the given
calculus is complete w.r.t. the original logic is usually carried out
syntactically, i.e. by showing that a (cut free) derivation exists of each
given axiom of the logic in the basic system to which the analytic structural
rules algorithmically generated from the given axiom have been added. However,
so far this proof strategy for syntactic completeness has been implemented on a
case-by-case base, and not in general. In this paper, we address this gap by
proving syntactic completeness for properly displayable logics in any normal
(distributive) lattice expansion signature. Specifically, we show that for
every analytic inductive axiom a cut free derivation can be effectively
generated which has a specific shape, referred to as pre-normal form.Comment: arXiv admin note: text overlap with arXiv:1604.08822 by other author
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