Menachem Magidor

What does a conditional knowledge base entail

Abstract 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, 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 (nonstandard) 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.

Martin's Maximum, saturated ideals and non-regular ultrafilters. Part II

We prove, assuming the existence of a huge cardinal, the consistency of fully non-regular ultrafilters on the successor of any regular cardinal. We also construct ultrafilters with ultraproducts of small cardinality. Part II is logically independent of Part I.

On the singular cardinals problem I

We show how to get a model of set theory in which ℵω is a strong limit cardinal which violates the generalized continuum hypothesis. Generalizations to other cardinals are also given.

Reflecting Stationary Sets

We prove that the statement “For every pair A, B , stationary subsets of ω 2 , composed of points of cofinality ω , there exists an ordinal α such that both A ∩ α and B ∩ α are stationary subsets of α is equiconsistent with the existence of weakly compact cardinal. (This completes results of Baumgartner and Harrington and Shelah.) We also prove, assuming the existence of infinitely many supercompact cardinals, the statement “Every stationary subset of ω ω+1 has a stationary initial segment.”

Distance Semantics for Belief Revision

A vast and interesting family of natural semantics for Belief Revision is defined. Suppose one is given a distance d between any two models. One may define the revision of a theory K by a formula a as the theory defined by the set of all those models of a that are closest, by d, to the set of models of K. This family is characterized by a set of rationality postulates that extends the AGM postulates. The new postulates describe properties of iterated revisions.

How large is the first strongly compact cardinal? or a study on identity crises

Abstract It is proved that if strongly compact cardinals are consistent, then it is consistent that the first such cardinal is the first measurable. On the other hand, if it is consistent to assume the existence of supercompact cardinal, then it is consistent to assume that it is the first strongly compact cardinal.

A very weak square principle

In this paper we explicate a very weak version of the principle □ discovered by Jensen who proved it holds in the constructible universe L . This principle is strong enough to include many of the known applications of □, but weak enough that it is consistent with the existence of very large cardinals. In this section we show that this principle is equivalent to a common combinatorial device, which we call a Jensen matrix. In the second section we show that our principle is consistent with a supercompact cardinal. In the third section of this paper we show that this principle is exactly equivalent to the statement that every torsion free Abelian group has a filtration into σ -balanced subgroups. In the fourth section of this paper we show that this principle fails if you assume the Changs Conjecture: In the fifth section of the paper we review the proofs that the various weak squares we consider are strictly decreasing in strength. Section 6 was added in an ad hoc manner after the rest of the paper was written, because the subject matter of Theorem 6.1 fit well with the rest of the paper. It deals with a principle dubbed “Not So Very Weak Square”, which appears close to Very Weak Square but turns out not to be equivalent.

Shelah's pcf theory and its applications

Abstract This is a survey paper giving a self-contained account of Shelahs theory of the pcf function pcf( a )={cf(Π a/D, D ): D is an ultrafilter on a }, where a is a set of regular cardinals such that | a | a ). We also give several applications of the theory to cardinal arithmetic, the existence of Jonsson algebras, and partition calculus.

On the role of supercompact and extendible cardinals in logic

It is proved that the existence of supercompact cardinal is equivalent to a certain Skolem-Löwenheim Theorem for second order logic, whereas the existence of extendible cardinal is equivalent to a certain compactness theorem for that logic. It is also proved that a certain axiom schema related to model theory implies the existence of many extendible cardinals.

Large cardinals and definable counterexamples to the continuum hypothesis

Abstract In this paper we consider whether L( R ) has “enough information” to contain a counterexample to the continuum hypothesis. We believe this question provides deep insight into the difficulties surrounding the continuum hypothesis. We show sufficient conditions for L( R ) not to contain such a counterexample. Along the way we establish many results about nonstationary towers, non-reflecting stationary sets, generalizations of proper and semiproper forcing and Changs conjecture.