77 research outputs found
What does a conditional knowledge base entail?
This paper presents a logical approach to nonmonotonic reasoning based on the
notion of a nonmonotonic consequence relation. A conditional knowledge base,
consisting of a set of conditional assertions of the type "if ... then ...",
represents the explicit defeasible knowledge an agent has about the way the
world generally behaves. We look for a plausible definition of the set of all
conditional assertions entailed by a conditional knowledge base. In a previous
paper, S. Kraus and the authors defined and studied "preferential" consequence
relations. They noticed that not all preferential relations could be considered
as reasonable inference procedures. This paper studies a more restricted class
of consequence relations, "rational" relations. It is argued that any
reasonable nonmonotonic inference procedure should define a rational relation.
It is shown that the rational relations are exactly those that may be
represented by a "ranked" preferential model, or by a (non-standard)
probabilistic model. The rational closure of a conditional knowledge base is
defined and shown to provide an attractive answer to the question of the title.
Global properties of this closure operation are proved: it is a cumulative
operation. It is also computationally tractable. This paper assumes the
underlying language is propositional.Comment: Preliminary version presented at KR'89. Minor corrections of the
Journal Versio
Independence of the existence of Pitowsky spin models
In 1982 I. Pitowsky used Continuum Hypothesis to construct hidden variable
models for spin-1/2 and spin-1 particles in quantum mechanics. We show that the
existence of Pitowsky models is independent from ZFC
Bext^2(G,T) can be nontrivial, even assuming GCH
Using the consistency of some large cardinals we produce a model of Set
Theory in which the generalized continuum hypothesis holds and for some
torsion-free abelian group G of cardinality aleph_{omega +1} and for some
torsion group T, Bext^2(G,T) not =0
Destructibility of the tree property at
We construct a model in which the tree property holds in and it is destructible under . On the other
hand we discuss some cases in which the tree property is indestructible under
small or closed forcings
On properties of compacta that do not reflect in small continuous images
Assuming that there is a stationary set in of ordinals of
countable cofinality that does not reflect, we prove that there exists a
compact space which is not Corson compact and whose all continuous images of
weight at most are Eberlein compacta. This yields an example of a
Banach space of density which is not weakly compactly generated
but all its subspaces of density are weakly compactly generated.
We also prove that under Martin's axiom countable functional tightness does not
reflect in small continuous images of compacta.Comment: 10 pages, version of Aug 8, 201
On Boolean algebras with strictly positive measures
We investigate reflection-type problems on the class SPM, of Boolean algebras
carrying strictly positive finitely additive measures. We show, in particular,
that in the constructible universe there is a Boolean algebra
which is not in SPM but every subalgebra of of cardinality
admits a strictly positive measure. This result is essentially
due to Farah and Velickovic.Comment: 8 pages; version of October 5, 201
On supercompactness and the continuum function
Given a cardinal that is -supercompact for some regular
cardinal and assuming \GCH, we show that one can force
the continuum function to agree with any function
F:[\kappa,\lambda]\cap\REG\to\CARD satisfying \forall\alpha,\beta\in\dom(F)
\alpha<\cf(F(\alpha)) and ,
while preserving the -supercompactness of from a hypothesis
that is of the weakest possible consistency strength, namely, from the
hypothesis that there is an elementary embedding with critical point
such that and . Our
argument extends Woodin's technique of surgically modifying a generic filter to
a new case: Woodin's key lemma applies when modifications are done on the range
of , whereas our argument uses a new key lemma to handle modifications done
off of the range of on the ghost coordinates. This work answers a question
of Friedman and Honzik [FH2012]. We also discuss several related open
questions.Comment: 12 page
Corson reflections
A reflection principle for Corson compacta holds in the forcing extension
obtained by Levy-collapsing a supercompact cardinal to~. In this
model, a compact Hausdorff space is Corson if and only if all of its continuous
images of weight~ are Corson compact.
We use the Gelfand--Naimark duality, and our results are stated in terms of
unital abelian \cstar-algebras.Comment: Some corrections, mostly mino
The tree property at successors of singular cardinals
Assuming some large cardinals, a model of ZFC is obtained in which
aleph_{omega+1} carries no Aronszajn trees. It is also shown that if lambda is
a singular limit of strongly compact cardinals, then lambda^+ carries no
Aronszajn trees
Nonmonotonic Reasoning, Preferential Models and Cumulative Logics
Many systems that exhibit nonmonotonic behavior have been described and
studied already in the literature. The general notion of nonmonotonic
reasoning, though, has almost always been described only negatively, by the
property it does not enjoy, i.e. monotonicity. We study here general patterns
of nonmonotonic reasoning and try to isolate properties that could help us map
the field of nonmonotonic reasoning by reference to positive properties. We
concentrate on a number of families of nonmonotonic consequence relations,
defined in the style of Gentzen. Both proof-theoretic and semantic points of
view are developed in parallel. The former point of view was pioneered by D.
Gabbay, while the latter has been advocated by Y. Shoham in. Five such families
are defined and characterized by representation theorems, relating the two
points of view. One of the families of interest, that of preferential
relations, turns out to have been studied by E. Adams. The "preferential"
models proposed here are a much stronger tool than Adams' probabilistic
semantics. The basic language used in this paper is that of propositional
logic. The extension of our results to first order predicate calculi and the
study of the computational complexity of the decision problems described in
this paper will be treated in another paper.Comment: Presented at JELIA, June 1988. Some misprints in the Journal paper
have been correcte
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