284 research outputs found
Modular Focused Proof Systems for Intuitionistic Modal Logics
Focusing is a general technique for syntactically compartmentalizing
the non-deterministic choices in a proof system, which not only
improves proof search but also has the representational benefit of
distilling sequent proofs into synthetic normal forms. However, since
focusing is usually specified as a restriction of the sequent
calculus, the technique has not been transferred to logics that lack a
(shallow) sequent presentation, as is the case for some of the logics
of the modal cube. We have recently extended the focusing technique
to classical nested sequents, a generalization of ordinary sequents.
In this work we further extend focusing to intuitionistic nested
sequents, which can capture all the logics of the intuitionistic S5
cube in a modular fashion. We present an internal cut-elimination
procedure for the focused system which in turn is used to show its
completeness
On Nested Sequents for Constructive Modal Logics
We present deductive systems for various modal logics that can be obtained
from the constructive variant of the normal modal logic CK by adding
combinations of the axioms d, t, b, 4, and 5. This includes the constructive
variants of the standard modal logics K4, S4, and S5. We use for our
presentation the formalism of nested sequents and give a syntactic proof of cut
elimination.Comment: 33 page
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
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
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
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
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