1,044 research outputs found
A probabilistic analysis of selected notions of iterated conditioning under coherence
It is well known that basic conditionals satisfy some desirable basic logical and probabilistic properties, such as the compound probability theorem. However checking the validity of these becomes trickier when we switch to compound and iterated conditionals. Herein we consider de Finetti's notion of conditional both in terms of a three-valued object and as a conditional random quantity in the betting framework. We begin by recalling the notions of conjunction and disjunction among conditionals in selected trivalent logics. Then we analyze the notions of iterated conditioning in the frameworks of the specific three-valued logics introduced by Cooper-Calabrese, by de Finetti, and by Farrel. By computing some probability propagation rules we show that the compound probability theorem and other important properties are not always preserved by these formulations. Then, for each trivalent logic we introduce an iterated conditional as a suitable random quantity which satisfies the compound prevision theorem as well as some other desirable properties. We also check the validity of two generalized versions of Bayes' Rule for iterated conditionals. We study the p-validity of generalized versions of Modus Ponens and two-premise centering for iterated conditionals. Finally, we observe that all the basic properties are satisfied within the framework of iterated conditioning followed in recent papers by Gilio and Sanfilippo in the setting of conditional random quantities
On the cofinality of the least λ-strongly compact cardinal
In this paper, we characterize the possible cofinalities of the least λ-strongly compact cardinal.We show that, on the one hand, for any regular cardinal, δ, that carries a λ-complete uniform ultrafilter, it is consistent, relative to the existence of a supercompact cardinal above δ, that the least λ-strongly compact cardinal has cofinality δ. On the other hand, provably the cofinality of the least λ-strongly compact cardinal always carries a λ-complete uniform ultrafilter
Extender-based Magidor-Radin forcings without top extenders
Continuing \cite{GitJir22}, we develop a version of Extender-based
Magidor-Radin forcing where there are no extenders on the top ordinal. As an
application, we provide another approach to obtain a failure of SCH on a club
subset of an inaccessible cardinal, and a model where the cardinal arithmetic
behaviors are different on stationary classes, whose union is the club, is
provided. The cardinals and the cofinalities outside the clubs are not affected
by the forcings.Comment: 32 page
The Galvin property under the Ultrapower Axiom
We continue the study of the Galvin property. In particular, we deepen the
connection between certain diamond-like principles and non-Galvin ultrafilters.
We also show that any Dodd sound ultrafilter that is not a -point is
non-Galvin. We use these ideas to formulate an essentially optimal large
cardinal hypothesis that ensures the existence of a non-Galvin ultrafilter,
improving on results of Benhamou and Dobrinen. Finally, we use a strengthening
of the Ultrapower Axiom to prove that in all the known canonical inner models,
a -complete ultrafilter on has the Galvin property if and only
if it is an iterated sum of -points
Historical infinitesimalists and modern historiography of infinitesimals
In the history of infinitesimal calculus, we trace innovation from Leibniz to
Cauchy and reaction from Berkeley to Mansion and beyond. We explore 19th
century infinitesimal lores, including the approaches of Simeon-Denis Poisson,
Gaspard-Gustave de Coriolis, and Jean-Nicolas Noel. We examine contrasting
historiographic approaches to such lores, in the work of Laugwitz, Schubring,
Spalt, and others, and address a recent critique by Archibald et al. We argue
that the element of contingency in this history is more prominent than many
modern historians seem willing to acknowledge.Comment: 60 page
Generically extendible cardinals
In this paper, we study generically extendible cardinal, which is a generic
version of extendible cardinal. We prove that the generic extendibility of
or has small consistency strength, but of a cardinal
is not. We also consider some results concerning with generic
extendible cardinals, such as indestructibility, generic absoluteness of the
reals, and Boolean valued second order logic
Quantum Expanders and Quantifier Reduction for Tracial von Neumann Algebras
We provide a complete characterization of theories of tracial von Neumann
algebras that admit quantifier elimination. We also show that the theory of a
separable tracial von Neumann algebra is never model complete if
its direct integral decomposition contains factors
such that embeds into an ultrapower of
. The proof in the case of factors uses an
explicit construction based on random matrices and quantum expanders.Comment: 38 pages, comments are welcom
More Ramsey theory for highly connected monochromatic subgraphs
An infinite graph is said to be highly connected if the induced subgraph on
the complement of any set of vertices of smaller size is connected. We continue
the study of weaker versions of Ramsey Theorem on uncountable cardinals
asserting that if we color edges of the complete graph we can find a large
highly connected monochromatic subgraph. In particular, several questions of
Bergfalk, Hru\v{s}\'ak and Shelah are answered by showing that assuming the
consistency of suitable large cardinals the following are relatively consistent
with : for every regular
cardinal and . Building on a work of Lambie-Hanson, we also show that
is consistent with
. To prove these results, we use the existence of ideals with
strong combinatorial properties after collapsing suitable large cardinals.Comment: Number 1242 on Shelah's publication list. 18 page
A function-first approach to doubt
Doubt is a much-maligned state. We are racked by doubts, tormented by doubts, plagued by them, paralysed. Doubts can be troubling, consuming, agonising. But however ill-regarded is doubt, anxiety is more so. We recognise the significance of doubting in certain contexts, and allow ourselves to be guided by our doubts. For example, the criminal standard of proof operative in the U.K., U.S., as well as in most other anglophone countries, Germany, Italy, Sweden and Israel, requires for conviction to be permissible that the defendant’s guilt is proved beyond reasonable doubt; to feel a doubt about a defendant’s guilt, so long as it is reasonable, is reason to refrain from convicting. But our folk understanding of anxiety ascribes no value to that state. Anxiety is inherently unpleasant and irrational; it prevents us from being able to perform well when it is most important to us that we do; it is an emotion that, if we could, we’d eliminate from our emotional toolbox. Yet in this thesis, I offer a vindication of doubt – a defence of doubt in terms of what it does for us – on which it ultimately turns out to be a kind of anxiety. The basic idea is that the concept of doubt serves a function for us that we couldn’t do without: it signals when we should begin inquiry. I will argue that the concept doubt is able to serve this function because the state it picks out, the state of doubt, is a kind of anxiety: epistemic anxiety. I develop a picture of epistemic anxiety as an emotional response to epistemic risk: potential disvalue in the epistemic realm. Because doubt is a kind of anxiety, it has the right kind of representational and motivational profile to track epistemic risk in our environments, and motivate us to reduce or avoid that risk. This makes it hugely valuable for us, as knowledge-seeking creatures, given the incompatibility of knowledge with high levels of epistemic risk
An almost strong relation
We prove an almost strong polarized relation at successors of strong limit
singular cardinals, based on the assumption that . This relation
is preserved by the collapse of to , and then it becomes
optimal
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