Chromatic number of the product of graphs, graph homomorphisms, Antichains and cofinal subsets of posets without AC

We have observations concerning the set theoretic strength of the following combinatorial statements without the axiom of choice. 1. If in a partially ordered set, all chains are finite and all antichains are countable, then the set is countable. 2. If in a partially ordered set, all chains are finite and all antichains have size $\aleph_{\alpha}$, then the set has size $\aleph_{\alpha}$ for any regular $\aleph_{\alpha}$. 3. CS (Every partially ordered set without a maximal element has two disjoint cofinal subsets). 4. CWF (Every partially ordered set has a cofinal well-founded subset). 5. DT (Dilworth's decomposition theorem for infinite p.o.sets of finite width). 6. If the chromatic number of a graph $G_{1}$ is finite (say $k<\omega$), and the chromatic number of another graph $G_{2}$ is infinite, then the chromatic number of $G_{1}\times G_{2}$ is $k$. 7. For an infinite graph $G=(V_{G}, E_{G})$ and a finite graph $H=(V_{H}, E_{H})$, if every finite subgraph of $G$ has a homomorphism into $H$, then so has $G$. Further we study a few statements restricted to linearly-ordered structures without the axiom of choice.Comment: Revised versio

Standard Bayes logic is not finitely axiomatizable

In the paper [http://philsci-archive.pitt.edu/14136] a hierarchy of modal logics have been defined to capture the logical features of Bayesian belief revision. Elements in that hierarchy were distinguished by the cardinality of the set of elementary propositions. By linking the modal logics in the hierarchy to Medvedev's logic of (in)finite problems it has been shown that the modal logic of Bayesian belief revision determined by probabilities on a finite set of elementary propositions is not finitely axiomatizable. However, the infinite case remained open. In this paper we prove that the modal logic of Bayesian belief revision determined by standard Borel spaces (these cover probability spaces that occur in most of the applications) is also not finitely axiomatizable

Finite Jeffrey logic is not finitely axiomatizable

Bayes logics based on Bayes conditionalization as a probability updating mechanism have recently been introduced in [http://philsci-archive.pitt.edu/14136/]. It has been shown that the modal logic of Bayesian belief revision determined by probabilities on a finite set of elementary propositions or on a standard Borel space is not finitely axiomatizable [http://philsci-archive.pitt.edu/14136/]. Apart from Bayes conditionalization there are other methods, extensions of the standard one, of updating a probability measure. One such important method is Jeffrey's conditionalization. In this paper we consider the modal logic \JL_{<\omega} of probability updating based on Jeffrey's conditionalization where the underlying measurable space is finite. By relating this logic to the logic of absolute continuity and to Medvedev's logic of finite problems, we show that \JL_{<\omega} is not finitely axiomatizable. The result is significant because it indicates that axiomatic approaches to belief revision might be severely limited

Standard Bayes logic is not finitely axiomatizable

In the paper [http://philsci-archive.pitt.edu/14136] a hierarchy of modal logics have been defined to capture the logical features of Bayesian belief revision. Elements in that hierarchy were distinguished by the cardinality of the set of elementary propositions. By linking the modal logics in the hierarchy to Medvedev's logic of (in)finite problems it has been shown that the modal logic of Bayesian belief revision determined by probabilities on a finite set of elementary propositions is not finitely axiomatizable. However, the infinite case remained open. In this paper we prove that the modal logic of Bayesian belief revision determined by standard Borel spaces (these cover probability spaces that occur in most of the applications) is also not finitely axiomatizable

The ubiquitous defeaters: no admissibility troubles for Bayesian accounts of direct inference

In this paper we dispel the supposed ``admissibility troubles'' for Bayesian accounts of direct inference proposed by Wallmann and Hawthorne (2018), which concern the existence of surprising, unintuitive defeaters even for mundane cases of direct inference. We show that if one follows the majority of authors in the field in using classical probability spaces unimbued with any additional structure, one should expect similar phenomena to arise and should consider them unproblematic in themselves: defeaters abound! We then show that the framework of Higher Probability Spaces (Gaifman 1988) allows the natural modelling of the discussed cases which produces no troubles of this kind

Admissibility and Bayesian direct inference: no HOPe against ubiquitous defeaters

In this paper we discuss the ``admissibility troubles'' for Bayesian accounts of direct inference proposed in (Wallmann, 2018), which concern the existence of surprising, unintuitive defeaters even for mundane cases of direct inference. We first show that one could reasonably suspect that the source of these troubles was informal talk about higher-order probabilities: for cardinality-related reasons, classical probability spaces abound in defeaters for direct inference. We proceed to discuss the issues in the context of the rigorous framework of Higher Probability Spaces (Gaifman, 1988). However, we show that the issues persist; we prove a few facts which pertain both to classical probability spaces and to HOPs, in our opinion capturing the essence of the problem. In effect we strengthen the message from the admissibility troubles: they arise not only for approaches using classical probability spaces---which are thus necessarily informal about metaprobabilistic phenomena like agents having credences in propositions about chances---but also for at least one respectable framework specifically tailored for rigorous discussion of higher-order probabilities

Admissibility and Bayesian direct inference: no HOPe against ubiquitous defeaters

In this paper we discuss the ``admissibility troubles'' for Bayesian accounts of direct inference proposed in (Wallmann, 2018), which concern the existence of surprising, unintuitive defeaters even for mundane cases of direct inference. We first show that one could reasonably suspect that the source of these troubles was informal talk about higher-order probabilities: for cardinality-related reasons, classical probability spaces abound in defeaters for direct inference. We proceed to discuss the issues in the context of the rigorous framework of Higher Probability Spaces (Gaifman, 1988). However, we show that the issues persist; we prove a few facts which pertain both to classical probability spaces and to HOPs, in our opinion capturing the essence of the problem. In effect we strengthen the message from the admissibility troubles: they arise not only for approaches using classical probability spaces---which are thus necessarily informal about metaprobabilistic phenomena like agents having credences in propositions about chances---but also for at least one respectable framework specifically tailored for rigorous discussion of higher-order probabilities

Categorial subsystem independence as morphism co-possibility

This paper formulates a notion of independence of subobjects of an object in a general (i.e. not necessarily concrete) category. Subobject independence is the categorial generalization of what is known as subsystem independence in the context of algebraic relativistic quantum field theory. The content of subobject independence formulated in this paper is morphism co-possibility: two subobjects of an object will be defined to be independent if any two morphisms on the two subobjects of an object are jointly implementable by a single morphism on the larger object. The paper investigates features of subobject independence in general, and subobject independence in the category of C∗ - algebras with respect to operations (completely positive unit preserving linear maps on C∗ - algebras)as morphisms is suggested as a natural subsystem independence axiom to express relativistic locality of the covariant functor in the categorial approach to quantum field theory

Defusing Bertrand's Paradox

The classical interpretation of probability together with the Principle of Indifference are formulated in terms of probability measure spaces in which the probability is given by the Haar measure. A notion called Labeling Invariance is defined in the category of Haar probability spaces, it is shown that Labeling Invariance is violated and Bertrand's Paradox is interpreted as the very proof of violation of Labeling Invariance. It is shown that Bangu's attempt (Bangu 2010) to block the emergence of Bertrand's Paradox by requiring the re-labeling of random events to preserve randomness cannot succeed non-trivially. A non-trivial strategy to preserve Labeling Invariance is identified and it is argued that, under the interpretation of Bertrand's Paradox suggested in the paper, the paradox does not undermine either the Principle of Indifference or the classical interpretation and is in complete harmony with how mathematical probability theory is used in the sciences to model phenomena; it is shown in particular that violation of Labeling Invariance does not entail that labeling of random events affects the probabilities of random events. It also is argued however that the content of the Principle of Indifference cannot be specified in such a way that it can establish the classical interpretation of probability as descriptively accurate or predictively successful

The Bayes Blind Spot of a finite Bayesian Agent is a large set

The Bayes Blind Spot of a Bayesian Agent is the set of probability measures on a Boolean algebra that are absolutely continuous with respect to the background probability measure (prior) of a Bayesian Agent on the algebra and which the Bayesian Agent cannot learn by conditionalizing no matter what (possibly uncertain) evidence he has about the elements in the Boolean algebra. It is shown that if the Boolean algebra is finite, then the Bayes Blind Spot is a very large set: it has the same cardinality as the set of all probability measures (continuum); it has the same measure as the measure of the set of all probability measures (in the natural measure on the set of measures); and is a ``fat'' (second Baire category) set in topological sense in the set of all probability measures taken with its natural topology