8,190 research outputs found
Logics without the contraction rule and residuated lattices
In this paper, we will develop an algebraic study of substructural propositional logics over FLew, i.e. the logic which is obtained from intuitionistic logics by eliminating the contraction rule. Our main technical tool is to use residuated lattices as the algebraic semantics for them. This enables us to study different kinds of nonclassical logics, including intermediate logics, BCK-logics, Lukasiewicz’s many-valued logics and fuzzy logics, within a uniform framework
Generic Modal Cut Elimination Applied to Conditional Logics
We develop a general criterion for cut elimination in sequent calculi for
propositional modal logics, which rests on absorption of cut, contraction,
weakening and inversion by the purely modal part of the rule system. Our
criterion applies also to a wide variety of logics outside the realm of normal
modal logic. We give extensive example instantiations of our framework to
various conditional logics. For these, we obtain fully internalised calculi
which are substantially simpler than those known in the literature, along with
leaner proofs of cut elimination and complexity. In one case, conditional logic
with modus ponens and conditional excluded middle, cut elimination and
complexity were explicitly stated as open in the literature
On the Correspondence between Display Postulates and Deep Inference in Nested Sequent Calculi for Tense Logics
We consider two styles of proof calculi for a family of tense logics,
presented in a formalism based on nested sequents. A nested sequent can be seen
as a tree of traditional single-sided sequents. Our first style of calculi is
what we call "shallow calculi", where inference rules are only applied at the
root node in a nested sequent. Our shallow calculi are extensions of Kashima's
calculus for tense logic and share an essential characteristic with display
calculi, namely, the presence of structural rules called "display postulates".
Shallow calculi enjoy a simple cut elimination procedure, but are unsuitable
for proof search due to the presence of display postulates and other structural
rules. The second style of calculi uses deep-inference, whereby inference rules
can be applied at any node in a nested sequent. We show that, for a range of
extensions of tense logic, the two styles of calculi are equivalent, and there
is a natural proof theoretic correspondence between display postulates and deep
inference. The deep inference calculi enjoy the subformula property and have no
display postulates or other structural rules, making them a better framework
for proof search
PSPACE Bounds for Rank-1 Modal Logics
For lack of general algorithmic methods that apply to wide classes of logics,
establishing a complexity bound for a given modal logic is often a laborious
task. The present work is a step towards a general theory of the complexity of
modal logics. Our main result is that all rank-1 logics enjoy a shallow model
property and thus are, under mild assumptions on the format of their
axiomatisation, in PSPACE. This leads to a unified derivation of tight
PSPACE-bounds for a number of logics including K, KD, coalition logic, graded
modal logic, majority logic, and probabilistic modal logic. Our generic
algorithm moreover finds tableau proofs that witness pleasant proof-theoretic
properties including a weak subformula property. This generality is made
possible by a coalgebraic semantics, which conveniently abstracts from the
details of a given model class and thus allows covering a broad range of logics
in a uniform way
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
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