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Articles 1 - 17 of 17
Full-Text Articles in Logic and Foundations
Soundness And Completeness Results For The Logic Of Evidence Aggregation And Its Probability Semantics, Eoin Moore
Soundness And Completeness Results For The Logic Of Evidence Aggregation And Its Probability Semantics, Eoin Moore
Dissertations, Theses, and Capstone Projects
The Logic of Evidence Aggregation (LEA), introduced in 2020, offers a solution to the problem of evidence aggregation, but LEA is not complete with respect to the intended probability semantics. This left open the tasks to find sound and complete semantics for LEA and a proper axiomatization for probability semantics. In this thesis we do both. We also develop the proof theory for some LEA-related logics and show surprising connections between LEA-related logics and Lax Logic.
Asymptotic Classes, Pseudofinite Cardinality And Dimension, Alexander Van Abel
Asymptotic Classes, Pseudofinite Cardinality And Dimension, Alexander Van Abel
Dissertations, Theses, and Capstone Projects
We explore the consequences of various model-theoretic tameness conditions upon the behavior of pseudofinite cardinality and dimension. We show that for pseudofinite theories which are either Morley Rank 1 or uncountably categorical, pseudofinite cardinality in ultraproducts satisfying such theories is highly well-behaved. On the other hand, it has been shown that pseudofinite dimension is not necessarily well-behaved in all ultraproducts of theories which are simple or supersimple; we extend such an observation by constructing simple and supersimple theories in which pseudofinite dimension is necessarily ill-behaved in all such ultraproducts. Additionally, we have novel results connecting various forms of asymptotic classes …
Introduction To Discrete Mathematics: An Oer For Ma-471, Mathieu Sassolas
Introduction To Discrete Mathematics: An Oer For Ma-471, Mathieu Sassolas
Open Educational Resources
The first objective of this book is to define and discuss the meaning of truth in mathematics. We explore logics, both propositional and first-order , and the construction of proofs, both formally and human-targeted. Using the proof tools, this book then explores some very fundamental definitions of mathematics through set theory. This theory is then put in practice in several applications. The particular (but quite widespread) case of equivalence and order relations is studied with detail. Then we introduces sequences and proofs by induction, followed by number theory. Finally, a small introduction to combinatorics is …
Covid-19 And Knowledge Based Computation, Rohit J. Parikh
Covid-19 And Knowledge Based Computation, Rohit J. Parikh
Publications and Research
The problem of dealing with Covid-19, until a vaccine is universally administered, is to decrease the rate of transmission while getting some social and economic activity going.
Infection passes from one person A to another person B when A is infected and B is susceptible. That is to say that B is not infected and not yet immune.
Social activity also takes place when one person interacts with another. Perhaps A is a taxpayer and B is a tax consultant. Then filing the tax return may take the form of the two of them meeting. Much can be done electronically …
Some Model Theory Of Free Groups, Christopher James Natoli
Some Model Theory Of Free Groups, Christopher James Natoli
Dissertations, Theses, and Capstone Projects
There are two main sets of results, both pertaining to the model theory of free groups. In the first set of results, we prove that non-abelian free groups of finite rank at least 3 or of countable rank are not A-homogeneous. We then build on the proof of this result to show that two classes of groups, namely finitely generated free groups and finitely generated elementary free groups, fail to form A-Fraisse classes and that the class of non-abelian limit groups fails to form a strong A-Fraisse class.
The second main result is that if a countable group is elementarily …
Alternative Cichoń Diagrams And Forcing Axioms Compatible With Ch, Corey B. Switzer
Alternative Cichoń Diagrams And Forcing Axioms Compatible With Ch, Corey B. Switzer
Dissertations, Theses, and Capstone Projects
This dissertation surveys several topics in the general areas of iterated forcing, infinite combinatorics and set theory of the reals. There are two parts. In the first half I consider alternative versions of the Cichoń diagram. First I show that for a wide variety of reduction concepts there is a Cichoń diagram for effective cardinal characteristics relativized to that reduction. As an application I investigate in detail the Cichoń diagram for degrees of constructibility relative to a fixed inner model of ZFC. Then I study generalizations of cardinal characteristics to the space of functions from Baire space to Baire space. …
Model Theory Of Groups And Monoids, Laura M. Lopez Cruz
Model Theory Of Groups And Monoids, Laura M. Lopez Cruz
Dissertations, Theses, and Capstone Projects
We first show that arithmetic is bi-interpretable (with parameters) with the free monoid and with partially commutative monoids with trivial center. This bi-interpretability implies that these monoids have the QFA property and that finitely generated submonoids of these monoids are definable. Moreover, we show that any recursively enumerable language in a finite alphabet X with two or more generators is definable in the free monoid. We also show that for metabelian Baumslag-Solitar groups and for a family of metabelian restricted wreath products, the Diophantine Problem is decidable. That is, we provide an algorithm that decides whether or not a given …
Math 220p Foundations Of Mathematics, Nicholas Vlamis
Math 220p Foundations Of Mathematics, Nicholas Vlamis
Open Educational Resources
No abstract provided.
Modest Automorphisms Of Presburger Arithmetic, Simon Heller
Modest Automorphisms Of Presburger Arithmetic, Simon Heller
Dissertations, Theses, and Capstone Projects
It is interesting to consider whether a structure can be expanded by an automorphism so that one obtains a nice description of the expanded structure's first-order properties. In this dissertation, we study some such expansions of models of Presburger arithmetic. Building on some of the work of Harnik (1986) and Llewellyn-Jones (2001), in Chapter 2 we use a back-and-forth construction to obtain two automorphisms of sufficiently saturated models of Presburger arithmetic. These constructions are done first in the quotient of the Presburger structure by the integers (which is a divisible ordered abelian group with some added structure), and then lifted …
The Philosophical Foundations Of Plen: A Protocol-Theoretic Logic Of Epistemic Norms, Ralph E. Jenkins
The Philosophical Foundations Of Plen: A Protocol-Theoretic Logic Of Epistemic Norms, Ralph E. Jenkins
Dissertations, Theses, and Capstone Projects
In this dissertation, I defend the protocol-theoretic account of epistemic norms. The protocol-theoretic account amounts to three theses: (i) There are norms of epistemic rationality that are procedural; epistemic rationality is at least partially defined by rules that restrict the possible ways in which epistemic actions and processes can be sequenced, combined, or chosen among under varying conditions. (ii) Epistemic rationality is ineliminably defined by procedural norms; procedural restrictions provide an irreducible unifying structure for even apparently non-procedural prescriptions and normative expressions, and they are practically indispensable in our cognitive lives. (iii) These procedural epistemic norms are best analyzed in …
The Structure Of Models Of Second-Order Set Theories, Kameryn J. Williams
The Structure Of Models Of Second-Order Set Theories, Kameryn J. Williams
Dissertations, Theses, and Capstone Projects
This dissertation is a contribution to the project of second-order set theory, which has seen a revival in recent years. The approach is to understand second-order set theory by studying the structure of models of second-order set theories. The main results are the following, organized by chapter. First, I investigate the poset of T-realizations of a fixed countable model of ZFC, where T is a reasonable second-order set theory such as GBC or KM, showing that it has a rich structure. In particular, every countable partial order embeds into this structure. Moreover, we can arrange so that these embedding preserve …
Coincidence Of Bargaining Solutions And Rationalizability In Epistemic Games, Todd Stambaugh
Coincidence Of Bargaining Solutions And Rationalizability In Epistemic Games, Todd Stambaugh
Dissertations, Theses, and Capstone Projects
Chapter 1: In 1950, John Nash proposed the Bargaining Problem, for which a solution is a function that assigns to each space of possible utility assignments a single point in the space, in some sense representing the ’fair’ deal for the agents involved. Nash provided a solution of his own, and several others have been presented since then, including a notable solution by Ehud Kalai and Meir Smorodinsky. In chapter 1, a complete account is given for the conditions under which the two solutions will coincide for two player bargaining scenarios.
Chapter 2: In the same year, Nash …
Interstructure Lattices And Types Of Peano Arithmetic, Athar Abdul-Quader
Interstructure Lattices And Types Of Peano Arithmetic, Athar Abdul-Quader
Dissertations, Theses, and Capstone Projects
The collection of elementary substructures of a model of PA forms a lattice, and is referred to as the substructure lattice of the model. In this thesis, we study substructure and interstructure lattices of models of PA. We apply techniques used in studying these lattices to other problems in the model theory of PA.
In Chapter 2, we study a problem that had its origin in Simpson, who used arithmetic forcing to show that every countable model of PA has an expansion to PA∗ that is pointwise definable. Enayat later showed that there are 2ℵ0 models with …
Joint Laver Diamonds And Grounded Forcing Axioms, Miha Habič
Joint Laver Diamonds And Grounded Forcing Axioms, Miha Habič
Dissertations, Theses, and Capstone Projects
In chapter 1 a notion of independence for diamonds and Laver diamonds is investigated. A sequence of Laver diamonds for κ is joint if for any sequence of targets there is a single elementary embedding j with critical point κ such that each Laver diamond guesses its respective target via j. In the case of measurable cardinals (with similar results holding for (partially) supercompact cardinals) I show that a single Laver diamond for κ yields a joint sequence of length κ, and I give strict separation results for all larger lengths of joint sequences. Even though the principles get …
The Proscriptive Principle And Logics Of Analytic Implication, Thomas M. Ferguson
The Proscriptive Principle And Logics Of Analytic Implication, Thomas M. Ferguson
Dissertations, Theses, and Capstone Projects
The analogy between inference and mereological containment goes at least back to Aristotle, whose discussion in the Prior Analytics motivates the validity of the syllogism by way of talk of parts and wholes. On this picture, the application of syllogistic is merely the analysis of concepts, a term that presupposes—through the root ἀνά + λύω —a mereological background.
In the 1930s, such considerations led William T. Parry to attempt to codify this notion of logical containment in his system of analytic implication AI. Parry’s original system AI was later expanded to the system PAI. The hallmark of Parry’s systems—and of …
Set-Theoretic Mereology, Joel David Hamkins, Makoto Kikuchi
Set-Theoretic Mereology, Joel David Hamkins, Makoto Kikuchi
Publications and Research
We consider a set-theoretic version of mereology based on the inclusion relation ⊆ and analyze how well it might serve as a foundation of mathematics. After establishing the non-definability of ∈ from ⊆, we identify the natural axioms for ⊆-based mereology, which constitute a finitely axiomatizable, complete, decidable theory. Ultimately, for these reasons, we conclude that this form of set-theoretic mereology cannot by itself serve as a foundation of mathematics. Meanwhile, augmented forms of set-theoretic mereology, such as that obtained by adding the singleton operator, are foundationally robust.
Epistemic Considerations On Extensive-Form Games, Cagil Tasdemir
Epistemic Considerations On Extensive-Form Games, Cagil Tasdemir
Dissertations, Theses, and Capstone Projects
In this thesis, we study several topics in extensive-form games. First, we consider perfect information games with belief revision with players who are tolerant of each other’s hypothetical errors. We bound the number of hypothetical non-rational moves of a player that will be tolerated by other players without revising the belief on that player’s rationality on future moves, and investigate which games yield the backward induction solution.
Second, we consider players who have no way of assigning probabilities to various possible outcomes, and define players as conservative, moderate and aggressive depending on the way they choose, and show that all …