124,456 research outputs found
The modal logic of forcing
What are the most general principles in set theory relating forceability and
truth? As with Solovay's celebrated analysis of provability, both this question
and its answer are naturally formulated with modal logic. We aim to do for
forceability what Solovay did for provability. A set theoretical assertion psi
is forceable or possible, if psi holds in some forcing extension, and
necessary, if psi holds in all forcing extensions. In this forcing
interpretation of modal logic, we establish that if ZFC is consistent, then the
ZFC-provable principles of forcing are exactly those in the modal theory known
as S4.2.Comment: 31 page
Logic of Negation-Complete Interactive Proofs (Formal Theory of Epistemic Deciders)
We produce a decidable classical normal modal logic of internalised
negation-complete and thus disjunctive non-monotonic interactive proofs (LDiiP)
from an existing logical counterpart of non-monotonic or instant interactive
proofs (LiiP). LDiiP internalises agent-centric proof theories that are
negation-complete (maximal) and consistent (and hence strictly weaker than, for
example, Peano Arithmetic) and enjoy the disjunction property (like
Intuitionistic Logic). In other words, internalised proof theories are
ultrafilters and all internalised proof goals are definite in the sense of
being either provable or disprovable to an agent by means of disjunctive
internalised proofs (thus also called epistemic deciders). Still, LDiiP itself
is classical (monotonic, non-constructive), negation-incomplete, and does not
have the disjunction property. The price to pay for the negation completeness
of our interactive proofs is their non-monotonicity and non-communality (for
singleton agent communities only). As a normal modal logic, LDiiP enjoys a
standard Kripke-semantics, which we justify by invoking the Axiom of Choice on
LiiP's and then construct in terms of a concrete oracle-computable function.
LDiiP's agent-centric internalised notion of proof can also be viewed as a
negation-complete disjunctive explicit refinement of standard KD45-belief, and
yields a disjunctive but negation-incomplete explicit refinement of
S4-provability.Comment: Expanded Introduction. Added Footnote 4. Corrected Corollary 3 and 4.
Continuation of arXiv:1208.184
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
A modal logic for reasoning on consistency and completeness of regulations
In this paper, we deal with regulations that may exist in multi-agent systems in order to regulate agent behaviour and we discuss two properties of regulations, that is consistency and completeness. After defining what consistency and completeness mean, we propose a way to consistently complete incomplete regulations. In this contribution, we extend previous works and we consider that regulations are expressed in a first order modal deontic logic
Multi-Agent Only-Knowing Revisited
Levesque introduced the notion of only-knowing to precisely capture the
beliefs of a knowledge base. He also showed how only-knowing can be used to
formalize non-monotonic behavior within a monotonic logic. Despite its appeal,
all attempts to extend only-knowing to the many agent case have undesirable
properties. A belief model by Halpern and Lakemeyer, for instance, appeals to
proof-theoretic constructs in the semantics and needs to axiomatize validity as
part of the logic. It is also not clear how to generalize their ideas to a
first-order case. In this paper, we propose a new account of multi-agent
only-knowing which, for the first time, has a natural possible-world semantics
for a quantified language with equality. We then provide, for the propositional
fragment, a sound and complete axiomatization that faithfully lifts Levesque's
proof theory to the many agent case. We also discuss comparisons to the earlier
approach by Halpern and Lakemeyer.Comment: Appears in Principles of Knowledge Representation and Reasoning 201
On Modal Logics of Partial Recursive Functions
The classical propositional logic is known to be sound and complete with
respect to the set semantics that interprets connectives as set operations. The
paper extends propositional language by a new binary modality that corresponds
to partial recursive function type constructor under the above interpretation.
The cases of deterministic and non-deterministic functions are considered and
for both of them semantically complete modal logics are described and
decidability of these logics is established
Logics for modelling collective attitudes
We introduce a number of logics to reason about collective propositional
attitudes that are defined by means of the majority rule. It is well known that majoritarian
aggregation is subject to irrationality, as the results in social choice theory and judgment
aggregation show. The proposed logics for modelling collective attitudes are based on
a substructural propositional logic that allows for circumventing inconsistent outcomes.
Individual and collective propositional attitudes, such as beliefs, desires, obligations, are
then modelled by means of minimal modalities to ensure a number of basic principles. In
this way, a viable consistent modelling of collective attitudes is obtained
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