76 research outputs found
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
Hypersequent calculi for non-normal modal and deontic logics: Countermodels and optimal complexity
We present some hypersequent calculi for all systems of the classical cube
and their extensions with axioms , , , and, for every , rule
. The calculi are internal as they only employ the language of the
logic, plus additional structural connectives. We show that the calculi are
complete with respect to the corresponding axiomatisation by a syntactic proof
of cut elimination. Then we define a terminating root-first proof search
strategy based on the hypersequent calculi and show that it is optimal for
coNP-complete logics. Moreover, we obtain that from every saturated leaf of a
failed proof it is possible to define a countermodel of the root hypersequent
in the bi-neighbourhood semantics, and for regular logics also in the
relational semantics. We finish the paper by giving a translation between
hypersequent rule applications and derivations in a labelled system for the
classical cube
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
Hypersequent Calculi for S5: The Methods of Cut Elimination
S5 is one of the most important modal logic with nice syntactic, semantic and algebraic properties. In spite of that, a successful (i.e. cut-free) formalization of S5 on the ground of standard sequent calculus (SC) was problematic and led to the invention of numerous nonstandard, generalized forms of SC. One of the most interesting framework which was very often used for this aim is that of hypersequent calculi (HC). The paper is a survey of HC for S5 proposed by Pottinger, Avron, Restall, Poggiolesi, Lahav and Kurokawa. We are particularly interested in examining different methods which were used for proving the eliminability/admissibility of cut in these systems and present our own variant of a system which admits relatively simple proof of cut elimination
Ecumenical modal logic
The discussion about how to put together Gentzen's systems for classical and
intuitionistic logic in a single unified system is back in fashion. Indeed,
recently Prawitz and others have been discussing the so called Ecumenical
Systems, where connectives from these logics can co-exist in peace. In Prawitz'
system, the classical logician and the intuitionistic logician would share the
universal quantifier, conjunction, negation, and the constant for the absurd,
but they would each have their own existential quantifier, disjunction, and
implication, with different meanings. Prawitz' main idea is that these
different meanings are given by a semantical framework that can be accepted by
both parties. In a recent work, Ecumenical sequent calculi and a nested system
were presented, and some very interesting proof theoretical properties of the
systems were established. In this work we extend Prawitz' Ecumenical idea to
alethic K-modalities
A Deep Inference System for the Modal Logic S5
We present a cut-admissible system for the modal logic S5 in a formalism that makes explicit and intensive use of deep inference. Deep inference is induced by the methods applied so far in conceptually pure systems for this logic. The system enjoys systematicity and modularity, two important properties that should be satisfied by modal systems. Furthermore, it enjoys a simple and direct design: the rules are few and the modal rules are in exact correspondence to the modal axiom
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