Open Access. Powered by Scholars. Published by Universities.®

Logic and Foundations Commons

Open Access. Powered by Scholars. Published by Universities.®

141 Full-Text Articles 107 Authors 33574 Downloads 38 Institutions

All Articles in Logic and Foundations

Faceted Search

141 full-text articles. Page 1 of 4.

From Pythagoreans And Weierstrassians To True Infinitesimal Calculus, Mikhail Katz, Luie Polev 2017 Bar-Ilan University

From Pythagoreans And Weierstrassians To True Infinitesimal Calculus, Mikhail Katz, Luie Polev

Journal of Humanistic Mathematics

In teaching infinitesimal calculus we sought to present basic concepts like continuity and convergence by comparing and contrasting various definitions, rather than presenting “the definition” to the students as a monolithic absolute. We hope that our experiences could be useful to other instructors wishing to follow this method of instruction. A poll run at the conclusion of the course indicates that students tend to favor infinitesimal definitions over epsilon-delta ones.


The Proscriptive Principle And Logics Of Analytic Implication, Thomas M. Ferguson 2017 The Graduate Center, City University of New York

The Proscriptive Principle And Logics Of Analytic Implication, Thomas M. Ferguson

All Graduate Works by Year: 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 ...


Complexity Reduction In State-Based Modeling, Martin Zwick 2016 Portland State University

Complexity Reduction In State-Based Modeling, Martin Zwick

Martin Zwick

For a system described by a relation among qualitative variables (or quantitative variables "binned" into symbolic states), expressed either set-theoretically or as a multivariate joint probability distribution, complexity reduction (compression of representation) is normally achieved by modeling the system with projections of the overall relation. To illustrate, if ABCD is a four variable relation, then models ABC:BCD or AB:BC:CD:DA, specified by two triadic or four dyadic relations, respectively, represent simplifications of the ABCD relation. Simplifications which are lossless are always preferred over the original full relation, while simplifications which lose constraint are still preferred if the ...


Enhancements To Crisp Possibilistic Reconstructability Analysis, Anas Al-Rabadi, Martin Zwick 2016 Portland State University

Enhancements To Crisp Possibilistic Reconstructability Analysis, Anas Al-Rabadi, Martin Zwick

Martin Zwick

Modified Reconstructibility Analysis (MRA), a novel decomposition within the framework of set-theoretic (crisp possibilistic) Reconstructibility Analysis, is presented. It is shown that in some cases while 3-variable NPN-classified Boolean functions are not decomposable using Conventional Reconstructibility Analysis (CRA), they are decomposable using Modified Reconstructibility Analysis (MRA). Also, it is shown that whenever a decomposition of 3-variable NPN-classified Boolean functions exists in both MRA and CRA, MRA yields simpler or equal complexity decompositions. A comparison of the corresponding complexities for Ashenhurst-Curtis decompositions, and Modified Reconstructibility Analysis (MRA) is also presented. While both AC and MRA decompose some but not all NPN-classes ...


Prestructuring Neural Networks Via Extended Dependency Analysis With Application To Pattern Classification, George G. Lendaris, Thaddeus T. Shannon, Martin Zwick 2016 Portland State University

Prestructuring Neural Networks Via Extended Dependency Analysis With Application To Pattern Classification, George G. Lendaris, Thaddeus T. Shannon, Martin Zwick

Martin Zwick

We consider the problem of matching domain-specific statistical structure to neural-network (NN) architecture. In past work we have considered this problem in the function approximation context; here we consider the pattern classification context. General Systems Methodology tools for finding problem-domain structure suffer exponential scaling of computation with respect to the number of variables considered. Therefore we introduce the use of Extended Dependency Analysis (EDA), which scales only polynomially in the number of variables, for the desired analysis. Based on EDA, we demonstrate a number of NN pre-structuring techniques applicable for building neural classifiers. An example is provided in which EDA ...


Complexity And Decomposability Of Relations, Martin Zwick 2016 Portland State University

Complexity And Decomposability Of Relations, Martin Zwick

Martin Zwick

A discrete multivariate relation, defined set-theoretically, is a subset of a cartesian product of sets which specify the possible values of a number of variables. Where three or more variables are involved, the highest order relation, namely the relation between all the variables, may or may not be decomposable without loss into sets of lower order relations which involve subsets of the variables. In a completely parallel manner, the highest order relation defined information-theoretically, namely the joint probability distribution involving all the variables, may or may not be decomposed without loss into lower-order distributions involving subsets of the variables. Decomposability ...


Exploring Mathematical Strategies For Finding Hidden Features In Multi-Dimensional Big Datasets, Tri Duong, Fang Ren, Apurva Mehta 2016 University of Houston

Exploring Mathematical Strategies For Finding Hidden Features In Multi-Dimensional Big Datasets, Tri Duong, Fang Ren, Apurva Mehta

STAR (STEM Teacher and Researcher) Presentations

With advances in technology in brighter sources and larger and faster detectors, the amount of data generated at national user facilities such as SLAC is increasing exponentially. Humans have a superb ability to recognize patterns in complex and noisy data and therefore, data is still curated and analyzed by humans. However, a human brain is unable to keep up with the accelerated pace of data generation, and as a consequence, the rate of new discoveries hasn't kept pace with the rate of data creation. Therefore, new procedures to quickly assess and analyze the data are needed. Machine learning approaches ...


Mathematical Practice And Human Cognition, Bernd Buldt 2016 Indiana University - Purdue University Fort Wayne

Mathematical Practice And Human Cognition, Bernd Buldt

Philosophy Faculty Presentations

Frank Quinn (of Jaffe-Quinn fame, see [1]) worked out the basics of his own account of mathematical practice, an account that is informed by an analysis of contemporary mathematics and its pedagogy (see [2]). Taking this account as our starting point, we can characterize the current mathematical practice to acquire and work with new concepts as a cognitive adaptation strategy that, first, emerged to meet the challenges posed by the growing abstractness of its objects and which, second, proceeds according to the following three-pronged approach:

  1. (i) sever as many ties to ordinary language as possible and limit ordinary language explanations ...


On Fixed Points, Diagonalization, And Self-Reference, Bernd Buldt 2016 Indiana University - Purdue University Fort Wayne

On Fixed Points, Diagonalization, And Self-Reference, Bernd Buldt

Philosophy Faculty Presentations

We clarify the respective role fixed points, diagonalization, and self- reference play in proofs of G ̈odel’s first incompleteness theorem. We first show that the usual fixed-point construction can be reduced to a double diagonalization; this is done to address widely held views such as that fixed-point are “paradoxical” (Priest), or work by “black magic” (Soare), or that their construction is “intuitively unclear” (Kotlarski). We then discuss three notions of self-reference; this can be seen an extension of a recent study by Halbach and Visser and is meant to show that we do not (yet?) have a robust theory ...


Mathematical Practice And Human Cognition, Bernd Buldt 2016 Indiana University - Purdue University Fort Wayne

Mathematical Practice And Human Cognition, Bernd Buldt

Philosophy Faculty Presentations

Frank Quinn (of Jaffe-Quinn fame, see [1]) worked out the basics of his own account of mathematical practice, an account that is informed by an analysis of contemporary mathematics and its pedagogy (see [2]). Taking this account as our starting point, we can characterize the current mathematical practice to acquire and work with new concepts as a cognitive adaptation strategy that, first, emerged to meet the challenges posed by the growing abstractness of its objects and which, second, proceeds according to the following three-pronged approach:

  1. (i) sever as many ties to ordinary language as possible and limit ordinary language explanations ...


Sentential Logic, Tony Roy 2016 CSU, San Bernardino

Sentential Logic, Tony Roy



Excerpted from the longer Roy, Symbolic Logic, including chapter 1 and just the first parts of chapters 2 - 7.

From the preface: There is, I think, a gap between what many students learn in their first course in formal logic, and what they are expected to know for their second. While courses in mathematical logic with metalogical components often cast only the barest glance at mathematical induction or even the very idea of reasoning from definitions, a first course may also leave these untreated, and fail explicitly to lay down the definitions upon which the second course is based. The ...


Symbolic Logic, Tony Roy 2016 CSU, San Bernardino

Symbolic Logic, Tony Roy



From the preface: There is, I think, a gap between what many students learn in their first course in formal logic, and what they are expected to know for their second. While courses in mathematical logic with metalogical components often cast only the barest glance at mathematical induction or even the very idea of reasoning from definitions, a first course may also leave these untreated, and fail explicitly to lay down the definitions upon which the second course is based. The aim of this text is to integrate material from these courses and, in particular, to make serious mathematical logic ...


On Fixed Points, Diagonalization, And Self-Reference, Bernd Buldt 2016 Indiana University - Purdue University Fort Wayne

On Fixed Points, Diagonalization, And Self-Reference, Bernd Buldt

Philosophy Faculty Publications

We clarify the respective roles fixed points, diagonalization, and self- reference play in proofs of Gödel’s first incompleteness theorem.


The Logic Of Uncertain Justifications, Robert Milnikel 2016 Kenyon College

The Logic Of Uncertain Justifications, Robert Milnikel

Robert Milnikel

No abstract provided.


The Logic Of Uncertain Justifications, Robert Milnikel 2016 Kenyon College

The Logic Of Uncertain Justifications, Robert Milnikel

Robert Milnikel

No abstract provided.


Conservativity For Logics Of Justified Belief: Two Approaches, Robert Milnikel 2016 Kenyon College

Conservativity For Logics Of Justified Belief: Two Approaches, Robert Milnikel

Robert Milnikel

No abstract provided.


Exploring Argumentation, Objectivity, And Bias: The Case Of Mathematical Infinity, Ami Mamolo 2016 University of Ontario Institute of Technology

Exploring Argumentation, Objectivity, And Bias: The Case Of Mathematical Infinity, Ami Mamolo

OSSA Conference Archive

This paper presents an overview of several years of my research into individuals’ reasoning, argumentation, and bias when addressing problems, scenarios, and symbols related to mathematical infinity. There is a long history of debate around what constitutes “objective truth” in the realm of mathematical infinity, dating back to ancient Greece (e.g., Dubinsky et al., 2005). Modes of argumentation, hindrances, and intuitions have been largely consistent over the years and across levels of expertise (e.g., Brown et al., 2010; Fischbein et al., 1979, Tsamir, 1999). This presentation examines the interrelated complexities of notions of objectivity, bias, and argumentation as ...


On The Conjugacy Problem For Automorphisms Of Trees, Kyle Douglas Beserra 2016 Boise State University

On The Conjugacy Problem For Automorphisms Of Trees, Kyle Douglas Beserra

Boise State University Theses and Dissertations

In this thesis we identify the complexity of the conjugacy problem of automorphisms of regular trees. We expand on the results of Kechris, Louveau, and Friedman on the complexities of the isomorphism problem of classes of countable trees. We see in nearly all cases that the complexity of isomorphism of subtrees of a given regular countable tree is the same as the complexity of conjugacy of automorphisms of the same tree, though we present an example for which this does not hold.


On A Multiple-Choice Guessing Game, Ryan Cushman, Adam J. Hammett 2016 Bethel College - Mishawaka

On A Multiple-Choice Guessing Game, Ryan Cushman, Adam J. Hammett

The Research and Scholarship Symposium

We consider the following game (a generalization of a binary version explored by Hammett and Oman): the first player (“Ann”) chooses a (uniformly) random integer from the first n positive integers, which is not revealed to the second player (“Gus”). Then, Gus presents Ann with a k-option multiple choice question concerning the number she chose, to which Ann truthfully replies. After a predetermined number m of these questions have been asked, Gus attempts to guess the number chosen by Ann. Gus wins if he guesses Ann’s number. Our goal is to determine every m-question algorithm which maximizes the probability ...


Epistemic Considerations On Extensive-Form Games, Cagil Tasdemir 2016 Graduate Center, City University of New York

Epistemic Considerations On Extensive-Form Games, Cagil Tasdemir

All Graduate Works by Year: 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 ...


Digital Commons powered by bepress