120 research outputs found
The Logic of Joint Ability in Two-Player Tacit Games
Logics of joint strategic ability have recently received attention, with arguably the most influential being those in a family that includes Coalition Logic (CL) and Alternating-time Temporal Logic (ATL). Notably, both CL and ATL bypass the epistemic issues that underpin Schelling-type coordination problems, by apparently relying on the meta-level assumption of (perfectly reliable) communication between cooperating rational agents. Yet such epistemic issues arise naturally in settings relevant to ATL and CL: these logics are standardly interpreted on structures where agents move simultaneously, opening the possibility that an agent cannot foresee the concurrent choices of other agents. In this paper we introduce a variant of CL we call Two-Player Strategic Coordination Logic (SCL2). The key novelty of this framework is an operator for capturing coalitional ability when the cooperating agents cannot share strategic information. We identify significant differences in the expressive power and validities of SCL2 and CL2, and present a sound and complete axiomatization for SCL2. We briefly address conceptual challenges when shifting attention to games with more than two players and stronger notions of rationality
Knowability Relative to Information
We present a formal semantics for epistemic logic, capturing the notion of knowability relative to information (KRI). Like Dretske, we move from the platitude that what an agent can know depends on her (empirical) information. We treat operators of the form K_AB (âB is knowable on the basis of information Aâ) as variably strict quantifiers over worlds with a topic- or aboutness- preservation constraint. Variable strictness models the non-monotonicity of knowledge acquisition while allowing knowledge to be intrinsically stable. Aboutness-preservation models the topic-sensitivity of information, allowing us to invalidate controversial forms of epistemic closure while validating less controversial ones. Thus, unlike the standard modal framework for epistemic logic, KRI accommodates plausible approaches to the Kripke-Harman dogmatism paradox, which bear on non-monotonicity, or on topic-sensitivity. KRI also strikes a better balance between agent idealization and a non-trivial logic of knowledge ascriptions
Van Inwagen's modal skepticism
Abstract
In this research report, the author defends Peter van Inwagenâs modal
skepticism. Van Inwagen accepts that we have much basic, everyday modal
knowledge, but denies that we have the capacity to justify philosophically interesting
modal claims that are far removed from this basic knowledge. The
author also defends the argument by means of which van Inwagen supports
his modal skepticism. Van Inwagen argues that Stephen Yabloâs recent and
influential account of the relationship between conceivability and possibility
supports his skeptical claims. The authorâs defence involves a creative interpretation
and development of Yabloâs account, which results in a recursive
account of modal epistemology, what the author calls the âsafe explanationâ
model of modal epistemology. The defence of van Inwagenâs argument also
involves a rebuttal to objections offered to van Inwagen by Geirrson and
Sosa
An Acceptance Semantics for Stable Modal Knowledge
We observe some puzzling linguistic data concerning ordinary knowledge
ascriptions that embed an epistemic (im)possibility claim. We conclude that it
is untenable to jointly endorse both classical logic and a pair of intuitively
attractive theses: the thesis that knowledge ascriptions are always veridical
and a `negative transparency' thesis that reduces knowledge of a simple negated
`might' claim to an epistemic claim without modal content. We motivate a
strategy for answering the trade-off: preserve veridicality and (generalized)
negative transparency, while abandoning the general validity of contraposition.
We survey and criticize various approaches for incorporating veridicality into
domain semantics, a paradigmatic `information-sensitive' framework for
capturing negative transparency and, more generally, the non-classical behavior
of sentences with epistemic modals. We then present a novel
information-sensitive semantics that successfully executes our favored
strategy: stable acceptance semantics.Comment: In Proceedings TARK 2023, arXiv:2307.0400
Theories of Aboutness
Our topic is the theory of topics (that is, the theory of subject matter). My goal is to clarify and evaluate three competing traditions: what I call the way-based approach, the atom-based approach, and the subject-predicate approach. I develop (defeasible) criteria for adequacy using robust linguistic intuitions that feature prominently in the literature. Then I evaluate the extent to which various existing theories satisfy these constraints. I conclude that recent theories due to Parry, Perry, Lewis, and Yablo do not meet the constraints in total. I then introduce the issue-based theoryâa novel and natural entry in the atom-based tradition that meets our constraints. In a coda, I categorize a recent theory from Fine as atom-based, and contrast it to the issue-based theory, concluding that they are evenly matched, relative to our main criteria of adequacy. I offer tentative reasons to nevertheless favour the issue-based theory
Relevant Alternatives in Epistemology and Logic
The goal of the current paper is to provide an introduction to and survey of the diverse landscape of relevant alternatives theories of knowledge. Emphasis is placed throughout both on the abstractness of the relevant alternatives approach and its amenability to formalization through logical techniques. We present some of the important motivations for adopting the relevant alternatives approach; briefly explore the connections and contrasts between the relevant alternatives approach and related developments in logic, epistemology and philosophy of science; provide a schema for classifying and studying relevant alternatives theories at different levels of abstraction; and present a sample of relevant alternatives theories (contrasting what we call question-first and topic-first theories) that tie our discussion to on-going debates in the philosophical literature, as well as showcasing techniques for formalizing some of the important positions in these debates
Markov Operators on Banach Lattices
Student Number : 0108851W -
MSc Dissertation -
School of Mathematics -
Faculty of ScienceA brief search on www.ams.org with the keyword âMarkov operatorâ produces some
684 papers, the earliest of which dates back to 1959. This suggests that the term
âMarkov operatorâ emerged around the 1950âs, clearly in the wake of Andrey Markovâs
seminal work in the area of stochastic processes and Markov chains. Indeed, [17] and
[6], the two earliest papers produced by the ams.org search, study Markov processes
in a statistical setting and âMarkov operatorsâ are only referred to obliquely, with no
explicit definition being provided. By 1965, in [7], the situation has progressed to the
point where Markov operators are given a concrete definition and studied more directly.
However, the way in which Markov operators originally entered mathematical
discourse, emerging from Statistics as various attempts to generalize Markov processes
and Markov chains, seems to have left its mark on the theory, with a notable
lack of cohesion amongst its propagators.
The study of Markov operators in the Lp setting has assumed a place of importance in
a variety of fields. Markov operators figure prominently in the study of densities, and
thus in the study of dynamical and deterministic systems, noise and other probabilistic
notions of uncertainty. They are thus of keen interest to physicists, biologists and
economists alike. They are also a worthy topic to a statistician, not least of all since
Markov chains are nothing more than discrete examples of Markov operators (indeed, Markov operators earned their name by virtue of this connection) and, more recently,
in consideration of the connection between copulas and Markov operators. In the
realm of pure mathematics, in particular functional analysis, Markov operators have
proven a critical tool in ergodic theory and a useful generalization of the notion of a
conditional expectation.
Considering the origin of Markov operators, and the diverse contexts in which they
are introduced, it is perhaps unsurprising that, to the uninitiated observer at least,
the theory of Markov operators appears to lack an overall unity. In the literature there
are many different definitions of Markov operators defined on L1(Ό) and/or L1(Ό)
spaces. See, for example, [13, 14, 26, 2], all of which manage to provide different
definitions. Even at a casual glance, although they do retain the same overall flavour,
it is apparent that there are substantial differences in these definitions. The situation
is not much better when it comes to the various discussions surrounding ergodic
Markov operators: we again see a variety of definitions for an ergodic operator (for
example, see [14, 26, 32]), and again the connections between these definitions are
not immediately apparent.
In truth, the situation is not as haphazard as it may at first appear. All the definitions
provided for Markov operator may be seen as describing one or other subclass of
a larger class of operators known as the positive contractions. Indeed, the theory
of Markov operators is concerned with either establishing results for the positive
contractions in general, or specifically for one of the aforementioned subclasses. The
confusion concerning the definition of an ergodic operator can also be rectified in
a fairly natural way, by simply viewing the various definitions as different possible
generalizations of the central notion of a ergodic point-set transformation (such a
transformation representing one of the most fundamental concepts in ergodic theory).
The first, and indeed chief, aim of this dissertation is to provide a coherent and
reasonably comprehensive literature study of the theory of Markov operators. This
theory appears to be uniquely in need of such an effort. To this end, we shall present a wealth of material, ranging from the classical theory of positive contractions; to a
variety of interesting results arising from the study of Markov operators in relation
to densities and point-set transformations; to more recent material concerning the
connection between copulas, a breed of bivariate function from statistics, and Markov
operators. Our goals here are two-fold: to weave various sources into a integrated
whole and, where necessary, render opaque material readable to the non-specialist.
Indeed, all that is required to access this dissertation is a rudimentary knowledge of
the fundamentals of measure theory, functional analysis and Riesz space theory. A
command of measure and integration theory will be assumed. For those unfamiliar
with the basic tenets of Riesz space theory and functional analysis, we have included
an introductory overview in the appendix.
The second of our overall aims is to give a suitable definition of a Markov operator on
Banach lattices and provide a survey of some results achieved in the Banach lattice
setting, in particular those due to [5, 44]. The advantage of this approach is that
the theory is order theoretic rather than measure theoretic. As we proceed through
the dissertation, definitions will be provided for a Markov operator, a conservative
operator and an ergodic operator on a Banach lattice. Our guide in this matter will
chiefly be [44], where a number of interesting results concerning the spectral theory of
conservative, ergodic, so-called âstochasticâ operators is studied in the Banach lattice
setting. We will also, and to a lesser extent, tentatively suggest a possible definition
for a Markov operator on a Riesz space. In fact, we shall suggest, as a topic for
further research, two possible approaches to the study of such objects in the Riesz
space setting.
We now offer a more detailed breakdown of each chapter.
In Chapter 2 we will settle on a definition for a Markov operator on an L1 space,
prove some elementary properties and introduce several other important concepts.
We will also put forward a definition for a Markov operator on a Banach lattice.
In Chapter 3 we will examine the notion of a conservative positive contraction. Conservative operators will be shown to demonstrate a number of interesting properties,
not least of all the fact that a conservative positive contraction is automatically a
Markov operator. The notion of conservative operator will follow from the Hopf decomposition,
a fundmental result in the classical theory of positive contractions and
one we will prove via [13]. We will conclude the chapter with a Banach lattice/Riesz
space definition for a conservative operator, and a generalization of an important
property of such operators in the L1 case.
In Chapter 4 we will discuss another well-known result from the classical theory of
positive contractions: the Chacon-Ornstein Theorem. Not only is this a powerful
convergence result, but it also provides a connection between Markov operators and
conditional expectations (the latter, in fact, being a subclass of theMarkov operators).
To be precise, we will prove the result for conservative operators, following [32].
In Chapter 5 we will tie the study of Markov operators into classical ergodic theory,
with the introduction of the Frobenius-Perron operator, a specific type of Markov
operator which is generated from a given nonsingular point-set transformation. The
Frobenius-Perron operator will provide a bridge to the general notion of an ergodic
operator, as the definition of an ergodic Frobenius-Perron operator follows naturally
from that of an ergodic transformation.
In Chapter 6 will discuss two approaches to defining an ergodic operator, and establish
some connections between the various definitions of ergodicity. The second definition,
a generalization of the ergodic Frobenius-Perron operator, will prove particularly
useful, and we will be able to tie it, following [26], to several interesting results
concerning the asymptotic properties of Markov operators, including the asymptotic
periodicity result of [26, 27]. We will then suggest a definition of ergodicity in the
Banach lattice setting and conclude the chapter with a version, due to [5], of the
aforementioned asymptotic periodicity result, in this case for positive contractions on
a Banach lattice.
In Chapter 7 we will move into more modern territory with the introduction of the copulas of [39, 40, 41, 42, 16]. After surveying the basic theory of copulas, including
introducing a multiplication on the set of copulas, we will establish a one-to-one
correspondence between the set of copulas and a subclass of Markov operators.
In Chapter 8 we will carry our study of copulas further by identifying them as a
Markov algebra under their aforementioned multiplication. We will establish several
interesting properties of this Markov algebra, in parallel to a second Markov algebra,
the set of doubly stochastic matrices. This chapter is chiefly for the sake of interest
and, as such, diverges slightly from our main investigation of Markov operators.
In Chapter 9, we will present the results of [44], in slightly more detail than the original
source. As has been mentioned previously, these concern the spectral properties of
ergodic, conservative, stochastic operators on a Banach lattice, a subclass of the
Markov operators on a Banach lattice.
Finally, as a conclusion to the dissertation, we present in Chapter 10 two possible
routes to the study of Markov operators in a Riesz space setting. The first definition
will be directly analogous to the Banach lattice case; the second will act as an analogue
to the submarkovian operators to be introduced in Chapter 2. We will not attempt
to develop any results from these definitions: we consider them a possible starting
point for further research on this topic.
In the interests of both completeness, and in order to aid those in need of more
background theory, the reader may find at the back of this dissertation an appendix
which catalogues all relevant results from Riesz space theory and operator theory
Rapid Geometry Creation for Computer-Aided Engineering Parametric Analyses: A Case Study Using ComGeom2 for Launch Abort System Design
ComGeom2, a tool developed to generate Common Geometry representation for multidisciplinary analysis, has been used to create a large set of geometries for use in a design study requiring analysis by two computational codes. This paper describes the process used to generate the large number of configurations and suggests ways to further automate the process and make it more efficient for future studies. The design geometry for this study is the launch abort system of the NASA Crew Launch Vehicle
Relevant Alternatives and Missed Clues: Redux
I construe Relevant Alternatives Theory (RAT) as an abstract combination of anti-skepticism and epistemic modesty, then re-evaluate the challenge posed to it by the missed clue counter-examples of Schaffer [2001]. The import of this challenge has been underestimated, as Schafferâs specific argument invites distracting objections. I offer a novel formalization of RAT, accommodating a suitably wide class of concrete theories of knowledge. Then, I introduce abstract missed clue cases and prove that every RA theory, as formalized, admits such a case. This yields an argument - in Schafferâs spirit - that resists easy dismissal
Stable Acceptance for Mighty Knowledge
Drawing on the puzzling behavior of ordinary knowledge ascriptions that embed an epistemic (im)possibility claim, we tentatively conclude that it is untenable to jointly endorse (i) an unfettered classical logic for epistemic language, (ii) the general veridicality of knowledge ascription, and (iii) an intuitive ânegative transparencyâ thesis that reduces knowledge of a simple negated âmightâ claim to an epistemic claim without modal content. We motivate a strategic trade-off: preserve veridicality and (generalized) negative transparency, while abandoning the general validity of contraposition. We criticize various approaches to incorporating veridicality into domain semantics, a paradigmatic âinformation-sensitiveâ framework for capturing negative transparency and, more generally, the non-classical behavior of sentences with epistemic modals. We then present a novel information-sensitive semantics that successfully executes our favored strategy: stable acceptance semantics, extending a vanilla bilateral state-based semantics for epistemic modals with a knowledge operator loosely inspired by the defeasibility theory of knowledge
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