1,044 research outputs found

    A probabilistic analysis of selected notions of iterated conditioning under coherence

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    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

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    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

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    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

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    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 pp-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 κ\kappa-complete ultrafilter on κ\kappa has the Galvin property if and only if it is an iterated sum of pp-points

    Historical infinitesimalists and modern historiography of infinitesimals

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    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

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    In this paper, we study generically extendible cardinal, which is a generic version of extendible cardinal. We prove that the generic extendibility of ω1\omega_1 or ω2\omega_2 has small consistency strength, but of a cardinal >ω2>\omega_2 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

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    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 N\mathcal{N} is never model complete if its direct integral decomposition contains II1\mathrm{II}_1 factors M\mathcal{M} such that M2(M)M_2(\mathcal{M}) embeds into an ultrapower of M\mathcal{M}. The proof in the case of II1\mathrm{II}_1 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

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    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 ZFC\mathsf{ZFC}: κhc(κ)ω2\kappa\to_{hc} (\kappa)^2_\omega for every regular cardinal κ2\kappa\geq \aleph_2 and ¬CH+2hc(1)ω2\neg\mathsf{CH}+ \aleph_2 \to_{hc} (\aleph_1)^2_\omega. Building on a work of Lambie-Hanson, we also show that 2hc[2]ω,22\aleph_2 \to_{hc} [\aleph_2]^2_{\omega,2} is consistent with ¬CH\neg\mathsf{CH}. 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

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    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

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    We prove an almost strong polarized relation at successors of strong limit singular cardinals, based on the assumption that 2μ>μ+2^\mu>\mu^+. This relation is preserved by the collapse of 2μ2^\mu to μ+\mu^+, and then it becomes optimal
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