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39080.011707
David Hilbert was promoting formalized mathematics, in which every real number with its infinite series of digits is a completed individual object. On the other side the Dutch mathematician, Luitzen Egbertus Jan Brouwer, was defending the view that each point on the line should be represented as a never-ending process that develops in time, a view known as intuitionistic mathematics (Box 1).
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278331.011888
In this paper, I motivate the addition of an actuality operator to relevant logics. Straightforward ways of doing this are in tension with standard motivations for relevant logics, but I show how to add the operator in a way that permits one to maintain the intuitions behind relevant logics. I close by exploring some of the philosophical consequences of the addition.
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308085.011931
I am indebted to my own teachers, Prof. Peter Koepke and Prof. Stefan Geschke, who taught me everything in these notes. Prof. Geschke’s scriptum for Einführung in die Logik und Modelltheorie (Bonn, Summer 2010) provided an invaluable basis for the compilation of these notes.
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345174.01195
We define mereologically invariant composition as the relation between a whole object and its parts when the object retains the same parts during a time interval. We argue that mereologically invariant composition is identity between a whole and its parts taken collectively. Our reason is that parts and wholes are equivalent measurements of a portion of reality at diferent scales in the precise sense employed by measurement theory. The purpose of these scales is the numerical representation of primitive relations between quantities of being. To show this, we prove representation and uniqueness theorems for composition. Thus, mereologically invariant composition is trans-scalar identity.
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630154.011965
Multiple ontology languages have been developed over the years, which brings afore two key components: how to select the appropriate language for the task at hand and language design itself. This engineering step entails examining the ontological ‘commitments’ embedded into the language, which, in turn, demands for an insight into what the effects of philosophical viewpoints may be on the design of a representation language. But what are the sort of commitments one should be able to choose from that have an underlying philosophical point of view, and which philosophical stances have a knock-on effect on the specification or selection of an ontology language? In this paper, we provide a first step towards answering these questions. We identify and analyse ontological commitments embedded in logics, or that could be, and show that they have been taken in well-known ontology languages. This contributes to reflecting on the language as enabler or inhibitor to formally characterising an ontology or an ontological investigation, as well as the design of new ontology languages following the proposed design process.
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846956.01198
We describe the translation invariant stationary states (TIS) of the one-dimensional facilitated asymmetric exclusion process in continuous time, in which a particle at site i ∈ Z jumps to site i + 1 (respectively i − 1) with rate p (resp. 1 − p), provided that site i − 1 (resp. i + 1) is occupied and site i + 1 (resp. i − 1) is empty. All TIS states with density ρ ≤ 1/2 are supported on trapped configurations in which no two adjacent sites are occupied; we prove that if in this case the initial state is Bernoulli then the final state is independent of p. This independence also holds for the system on a finite ring. For ρ > 1/2 there is only one TIS. It is the infinite volume limit of the probability distribution that gives uniform weight to all configurations in which no two holes are adjacent, and is isomorphic to the Gibbs measure for hard core particles with nearest neighbor exclusion.
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879688.011994
Bayesian personalism models learning from experience as the updating of an agent’s credence function on the information the agent acquires. The standard updating rules are hamstrung for zero probability events. The maneuvers that have been proposed to handle this problem are examined and found wanting: they offer only temporary relief but no satisfying and stable long term resolution. They do suggest a strategy for avoiding the problem altogether, but the price to be paid is a very crabbed account of learning from experience. I outline what Bayesians would need to do in order to come to grips with the problem rather than seeking to avoid it. Furthermore, I emphasize that an adequate treatment of the issues must work not only for classical probability but also for quantum probability as well, the latter of which is rarely discussed in the philosophical literature in the same breath with the updating problem. Since it is not obvious how the maneuvers applied to updating classical probability can be made to work for updating quantum probability a rethinking of the problem may be required. At the same time I indicate that in some special cases quantum probability theory has a self-contained solution to the problem of updating on zero probability events requiring no additional technical devices or rationality constraints.
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1227854.012009
Virtual bargaining tries to explain why some actions suggested in games by best response reasoning appear unreasonable and why real players often succeed easily in some coordination problems, although orthodox game theory is unable to resolve them. Yet the original account of virtual bargaining was lacking a proper mathematical formalism, a convincing motivation for players to follow the reasoning as well as a deliberate notion of the feasible agreements and the disagreement point. However, the ideas of two recent papers on virtual bargaining, one of them yet unpublished, can overcome some of the initial problems, while also raising new issues. I will argue here that the latest account rests on an implausible disagreement point, an in general unintuitive assumption of non-spiteful best responses and an inconsistent definition of the worst payoff function. Yet I propose a new disagreement point, which I argue to be convincing and which justifies a slightly weaker assumption than non-spitefulness for an arbitrary number of players such that virtual bargaining accounts for the phenomena, it tries to explain. Moreover, I will adequately generalize the worst payoff function and the virtual bargaining equilibrium (VBE) for n-player games, while also outlining the epistemic conditions for a VBE to be chosen.
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1227932.012023
Five conceptually distinct notions of symmetry in quantum theory are studied in the algebraic setting where a quantum system is characterized by a von Neumann algebra of observables and the set of normal states on the algebra.
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1239331.012048
We introduce a general theory of functions called Flow. We prove ZF, non-well founded ZF and ZFC can be immersed within Flow as a natural consequence from our framework. The existence of strongly inaccessible cardinals is entailed from our axioms. And our first important application is the introduction of a model of Zermelo-Fraenkel set theory where the Partition Principle (PP) holds but not the Axiom of Choice (AC). So, Flow allows us to answer to the oldest open problem in set theory: if PP entails AC.
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1500066.012087
This series of self-contained lectures on the philosophy of mathematics, offered for Oxford Michaelmas Term 2020, is intended for students preparing for philosophy exam paper 122, although all interested parties are welcome to join. …
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1771911.012131
A half-truth may be defined as a sentence that is true in one sense, but that fails to be true in another, hence as a sentence only true to some extent. This paper discusses some aspects in which the Liar may be considered a half-truth. Talk of half-truths, like talk of half-full containers, implies that truth is gradable, and moreover that some sentences can be true without being perfectly true. I review some evidence for the view that “true” and “false” are absolute gradable adjectives, and argue that both are moreover systematically ambiguous between a total and a partial interpretation supporting the strict-tolerant distinction. I use this evidence to revisit the strict-tolerant account of truth and the Liar. While the strict-tolerant account was originally conceived for vague predicates, its extension to the semantic paradoxes assumed that assertion, but not truth, comes in different degrees. I reconsider this claim, and argue that we get a more unified picture by treating “true” as a special type of vague predicate.
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1920316.012174
Max Born’s statistical interpretation made probabilities play a major role in quantum theory. Here we show that these quantum probabilities and the classical probabilities have very different origins. While the latter always result from an assumed probability measure, the first include transition probabilities with a purely algebraic origin. Moreover, the general definition of transition probability introduced here comprises not only the well-known quantum mechanical transition probabilities between pure states or wave functions, but further novel cases. A transition probability that differs from 0 and 1 manifests the typical quantum indeterminacy in a similar way as Heisenberg’s and others’ uncertainty relations and, furthermore, rules out deterministic states in the same way as the Bell-Kochen-Specker theorem. However, the transition probability defined here achieves a lot more beyond that: it demonstrates that the algebraic structure of the Hilbert space quantum logic dictates the precise values of certain probabilities and it provides an unexpected access to these quantum probabilities that does not rely on states or wave functions.
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2010290.012204
Peter Damian, an eleventh-century monastic leader and Church
reformer, has received a modest place in the historiography of early
medieval philosophy because of his little tract De divina
omnipotentia. In this work, Damian treats two questions related to
the limits of divine power: can God restore virginity to a woman who
has lost it, and, can God change the past? Damian has often been
depicted as a thinker who, in his defense of divine omnipotence, went
as far as denying the universal validity of the principle of
non-contradiction. For the most part, this depiction of Damian is
unfounded.
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2025709.012224
This paper outlines an account of conditionals, the evidential account, which rests on the idea that a conditional is true just in case its antecedent supports its consequent. As we will show, the evidential account exhibits some distinctive logical features that deserve careful consideration. On the one hand, it departs from the material reading of ‘if then’ exactly in the way we would like it to depart from that reading. On the other, it significantly differs from the non-material accounts which hinge on the Ramsey Test, advocated by Adams, Stalnaker, Lewis, and others.
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2397725.012257
The answer to (1) is: no. One thing that is clear is that, if a model M extracted from a model of TZT is actually a model of TZTU, then each each level l + 1 of the extracted model is of size i of the size of level l, for some concrete n. For each n this is a first-order condition expressible in L(TZT). DEFINITION 1 • Let φl,m be the formula of L(TZT) that says that the cardinality of level l + 1 is in of the cardinality of level l. • For each l ∈ Z, let Σl be the type {¬φl,m : m ∈ IN}. • Let us give the name ‘TZT(Omit)’ to the smallest theory that locally omits all the Σl.
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2446021.012293
A. J. Ayer’s empiricist criterion of meaning was supposed to have sorted all statements into nonsense on the one hand, and tautologies or genuinely factual statements on the other. Unfortunately for Ayer, it follows from classical logic that his criterion is trivial – it classifies all statements as either tautologies or genuinely factual, but none as nonsense. However, in this paper I argue that Ayer’s criterion of meaning can be defended from classical proofs of its triviality by the adoption of a relevant logic – an idea which is motivated because, according to Ayer, the genuinely factual statements are those which observation is relevant to.
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2796649.012332
Recent work on epistemic modality appeals to nonclassical notions of logical consequence. On the classical conception (see e.g. Kaplan 1989a), logical consequence for natural language tracks preservation of truth (of a sentence, at a context). Many theorists have argued that this notion of consequence is inadequate for epistemic modal sentences and conditionals, like (1) and (2).
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2850341.012371
Michael Weiss and I have been carrying on a dialog on nonstandard models of arithmetic, and after a long break we’re continuing, here:
• Michael Weiss and John Baez, Non-standard models of arithmetic (Part 18). …
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2893576.012407
Axiom weakening is a technique that allows for a fine-grained repair of inconsistent ontologies. Its main advantage is that it repairs ontologies by making axioms less restrictive rather than by deleting them, employing the use of refinement operators. In this paper, we build on previously introduced axiom weakening for ALC, and make it much more irresistible by extending its definitions to deal with SROIQ, the expressive and decidable description logic underlying OWL 2 DL. We extend the definitions of refinement operator to deal with SROIQ constructs, in particular with role hierarchies, cardinality constraints and nominals, and illustrate its application. Finally, we discuss the problem of termination of an iterated weakening procedure.
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3147505.012447
Goodman and Lederman (2020) argue that the traditional Fregean strategy for preserving the validity of Leibniz’s Law of substitution fails when confronted with apparent counterexamples involving proper names embedded under propositional attitude verbs. We argue, on the contrary, that the Fregean strategy succeeds and that Goodman and Lederman’s argument misfires.
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3147618.012482
Substructural logics and their application to logical and semantic paradoxes have been extensively studied, but non-reflexive systems have been somewhat neglected. Here, we aim to (at least partly) fill this lacuna, by presenting a non-reflexive logic and theory of naïve consequence (and truth). We also investigate the semantics and the proof-theory of the system. Finally, we develop a compositional theory of truth (and consequence) in our non-reflexive framework.
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3344853.012499
J. D. Hamkins and R. Solberg, “Categorical large cardinals and the tension between categoricity and set-theoretic reflection,” Mathematics arXiv, 2020. (Under review) Citation arχiv @ARTICLE{HamkinsSolberg:Categorical-large-cardinals,
author = {Joel David Hamkins and Robin Solberg},
title = {Categorical large cardinals and the tension between categoricity and set-theoretic reflection},
journal = {Mathematics arXiv},
year = {2020},
volume = {},
number = {},
pages = {},
month = {},
note = {Under review},
abstract = {},
keywords = {under-review},
url = {http://jdh.hamkins.org/categorical-large-cardinals/},
source = {},
doi = {},
eprint = {2009.07164},
archivePrefix ={arXiv},
primaryClass = {math.LO}
}
Abstract. …
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3364274.012538
Open Systems: A Double Categorical Perspective (Part 2)
Back to Kenny Courser’s thesis:
• Kenny Courser, Open Systems: A Double Categorical Perspective, Ph.D. thesis, U. C. Riverside, 2020. Kenny explains the flaws in a well-known framework for studying open systems: decorated cospans. …
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3444109.01257
This paper is concerned with learners who aim to learn patterns in infinite binary sequences: shown longer and longer initial segments of a binary sequence, they either attempt to predict whether the next bit will be a 0 or will be a 1 or they issue forecast probabilities for these events. Several variants of this problem are considered. In each case, a no-free-lunch result of the following form is established: the problem of learning is a formidably difficult one, in that no matter what method is pursued, failure is incomparably more common that success; and difficult choices must be faced in choosing a method of learning, since no approach dominates all others in its range of success. In the simplest case, the comparison of the set of situations in which a method fails and the set of situations in which it succeeds is a matter of cardinality (countable vs. uncountable); in other cases, it is a topological matter (meagre vs. co-meagre) or a hybrid computational-topological matter (effectively meagre vs. effectively co-meagre).
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3710494.012587
We provide a logical matrix semantics and a Gentzen-style sequent calculus for the first-degree entailments valid in W. T. Parry’s logic of Analytic Implication. We achieve the former by introducing a logical matrix closely related to that inducing paracomplete weak Kleene logic, and the latter by presenting a calculus where the initial sequents and the left and right rules for negation are subject to linguistic constraints.
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3710565.012601
The aim of this article is to discuss the extent to which certain sub-structural logics are related through the phenomenon of duality. Roughly speaking, metainferences are inferences between collections of inferences, and thus substructural logics can be regarded as those logics which have fewer valid metainferences that Classical Logic. In order to investigate duality in substructural logics, we will focus on the case study of the logics ST and TS, the former lacking Cut, the latter Reflexivity. The sense in which these logics, and these metainferences, are dual has yet to be explained in the context of a thorough and detailed exposition of duality for frameworks of this sort. Thus, our intent here is to try to elucidate whether or not this way of talking holds some ground—specially generalizing one notion of duality available in the specialized literature, the so-called notion of negation duality. In doing so, we hope to hint at broader points that might need to be addressed when studying duality in relation to substructural logics.
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3718436.012615
The dynamic view on the semantics of natural language, though stemming already from the seventies, has developed into a widely studied subject in the second half of the eighties. At present, the unification of various dynamic theories constitutes an important issue. In this paper, two theories are compared, viz. update semantics, and dynamic predicate logic. In section 1 a general characterization of the idea of a dynamic semantics for natural language is given which subsumes these two theories. Sections 2 and 3 are devoted to short expositions of each of them. In the final section 4 a comparison is made.
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3744161.01263
I’d like to introduce and discuss the otherworldly cardinals, a large cardinal notion that frequently arises in set-theoretic analysis, but which until now doesn’t seem yet to have been given its own special name. …
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3904917.012644
The last decade has seen the rise of neuromorphic architectures based on artificial spiking neural networks, such as the SpiNNaker, TrueNorth, and Loihi systems. The massive parallelism and co-locating of computation and memory in these architectures potentially allows for an energy usage that is orders of magnitude lower compared to traditional Von Neumann architectures. However, to date a comparison with more traditional computational architectures (particularly with respect to energy usage) is hampered by the lack of a formal machine model and a computational complexity theory for neuromorphic computation. In this paper we take the first steps towards such a theory. We introduce spiking neural networks as a machine model where—in contrast to the familiar Turing machine—information and the manipulation thereof are co-located in the machine. We introduce canonical problems, define hierarchies of complexity classes and provide some first completeness results.