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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.


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 ...


Constructing A Categorical Framework Of Metamathematical Comparison Between Deductive Systems Of Logic, Alex Gabriel Goodlad 2016 Bard College

Constructing A Categorical Framework Of Metamathematical Comparison Between Deductive Systems Of Logic, Alex Gabriel Goodlad

Senior Projects Spring 2016

The topic of this paper in a broad phrase is “proof theory". It tries to theorize the general

notion of “proving" something using rigorous definitions, inspired by previous less general

theories. The purpose for being this general is to eventually establish a rigorous framework

that can bridge the gap when interrelating different logical systems, particularly ones

that have not been as well defined rigorously, such as sequent calculus. Even as far as

semantics go on more formally defined logic such as classic propositional logic, concepts

like “completeness" and “soundness" between the “semantic" and the “deductive system"

is too arbitrarily defined ...


Sentential Logic, Tony Roy 2015 CSU, San Bernardino

Sentential Logic, Tony Roy

Books

Excerpted from chapters 1 - 7 of Symbolic Logic

Contents

Preface i

Contents v

Named Definitions ix

Quick Reference Guides xvii

I The Elements: Four Notions of Validity 1

1 Logical Validity and Soundness 4

1.1 Consistent Stories . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 The Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Some Consequences . . . . . . . . . . . . . . . . . . . . . . . . . 22

2 Formal Languages 29

2.1 Sentential Languages . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2 Quantificational Languages . . . . . . . . . . . . . . . . . . . . . . 44

3 Axiomatic Deduction 45

3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 Sentential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.3 Quantificational . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4 Semantics 59

4.1 Sentential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2 Quantificational . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5 Translation 76

CONTENTS vi

5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5 ...


Symbolic Logic, Tony Roy 2015 CSU, San Bernardino

Symbolic Logic, Tony Roy

Books

Contents

Preface i

Contents v

Named Definitions ix

Quick Reference Guides xvii

I The Elements: Four Notions of Validity 1

1 Logical Validity and Soundness 4

1.1 Consistent Stories . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 The Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3 Some Consequences . . . . . . . . . . . . . . . . . . . . . . . . . 22

2 Formal Languages 30

2.1 Sentential Languages . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2 Quantificational Languages . . . . . . . . . . . . . . . . . . . . . . 46

3 Axiomatic Deduction 67

3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.2 Sentential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.3 Quantificational . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4 Semantics 96

4.1 Sentential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.2 Quantificational . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5 Translation 137

5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.2 Sentential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.3 Quantificational . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

6 Natural Deduction ...


The Strict Higher Grothendieck Integral, Scott W. Dyer 2015 University of Nebraska-Lincoln

The Strict Higher Grothendieck Integral, Scott W. Dyer

Dissertations, Theses, and Student Research Papers in Mathematics

This thesis generalizes A. Grothendieck’s construction, denoted by an integral, of a fibered category from a contravariant pseudofunctor, to a construction for n- and even ∞-categories. Only strict higher categories are considered, the more difficult theory of weak higher categories being neglected. Using his axioms for a fibered category, Grothendieck produces a contravariant pseudofunctor from which the original fibered category can be reconstituted by integration. In applications, the integral is often most efficient, constructing the fibered category with its structure laid bare. The situation generalizes the external and internal definitions of the semidirect product in group theory: fibration is ...


How Does “Collaboration” Occur At All? Remarks On Epistemological Issues Related To Understanding / Working With ‘The Other’, Don Faust, Judith Puncochar 2015 Northern Michigan University

How Does “Collaboration” Occur At All? Remarks On Epistemological Issues Related To Understanding / Working With ‘The Other’, Don Faust, Judith Puncochar

Conference Presentations

Collaboration, if to occur successfully at all, needs to be based on careful representation and communication of each stakeholder’s knowledge. In this paper, we investigate, from a foundational logical and epistemological point of view, how such representation and communication can be accomplished. What we tentatively conclude, based on a careful delineation of the logical technicalities necessarily involved in such representation and communication, is that a complete representation is not possible. This inference, if correct, is of course rather discouraging with regard to what we can hope to achieve in the knowledge representations that we bring to our collaborations. We ...


Paratodo X: Una Introducción A La Lógica Formal, P.D. Magnus 2015 University at Albany, State University of New York

Paratodo X: Una Introducción A La Lógica Formal, P.D. Magnus

Philosophy Faculty Books

A translation of the logic textbook forall x (v 1.30) into Spanish. Translation by José Ángel Gascón.


Gödel’S Second Incompleteness Theorem, Bernd Buldt 2015 Indiana University - Purdue University Fort Wayne

Gödel’S Second Incompleteness Theorem, Bernd Buldt

Philosophy Faculty Presentations

Slides for the third tutorial on Gödel's incompleteness theorems, held at UniLog 5 Summer School, Istanbul, June 24, 2015


Gödel’S First Incompleteness Theorem, Bernd Buldt 2015 Indiana University - Purdue University Fort Wayne

Gödel’S First Incompleteness Theorem, Bernd Buldt

Philosophy Faculty Presentations

Slides for the second tutorial on Gödel's incompleteness theorems, held at UniLog 5 Summer School, Istanbul, June 24, 2015


Fixed Points, Diagonalization, Self-Reference, Paradox, Bernd Buldt 2015 Indiana University - Purdue University Fort Wayne

Fixed Points, Diagonalization, Self-Reference, Paradox, Bernd Buldt

Philosophy Faculty Presentations

Slides for the first tutorial on Gödel's incompleteness theorems, held at UniLog 5 Summer School, Istanbul, June 24, 2015


Transition To Higher Mathematics: Structure And Proof (Second Edition), Bob A. Dumas, John E. McCarthy 2015 University of Washington - Seattle Campus

Transition To Higher Mathematics: Structure And Proof (Second Edition), Bob A. Dumas, John E. Mccarthy

Books and Monographs

This book is written for students who have taken calculus and want to learn what “real mathematics" is. We hope you will find the material engaging and interesting, and that you will be encouraged to learn more advanced mathematics.

This is the second edition of our text. It is intended for students who have taken a calculus course, and are interested in learning what higher mathematics is all about. It can be used as a textbook for an "Introduction to Proofs" course, or for self-study.

Chapter 1: Preliminaries, Chapter 2: Relations, Chapter 3: Proofs, Chapter 4: Principles of Induction, Chapter ...


Philosophy Of Mathematics: Theories And Defense, Amy E. Maffit 2015 University of Akron Main Campus

Philosophy Of Mathematics: Theories And Defense, Amy E. Maffit

Honors Research Projects

In this paper I discuss six philosophical theories of mathematics including logicism, intuitionism, formalism, platonism, structuralism, and moderate realism. I also discuss problems that arise within these theories and attempts to solve them. Finally, I attempt to harmonize the best features of moderate realism and structuralism, presenting a theory that I take to best describe current mathematical practice.


Some Observations On Scientific Epistemology With Applications To Conflict Resolution And Constructive Controversy, Judith Puncochar, Don Faust 2014 Northern Michigan University

Some Observations On Scientific Epistemology With Applications To Conflict Resolution And Constructive Controversy, Judith Puncochar, Don Faust

Other Presentations

An overview, by Judy and Don (published in 2013 in the BULLETIN OF SYMBOLIC LOGIC):

Explorationism is a perspective wherein all of our knowledge is (so far) less than certain, and naturally would come equipped with a base logic entailing machinery for representing and processing evidential knowledge. One such base logic is Evidence Logic, which strives to deal with the phenomenon of the gradational presence of both confirmatory and refutatory evidence. From this perspective, we will address questions surrounding sociological problem areas that we see as deeply infused with substantial epistemological factors. By defining a framework as any theory, in ...


The Scope Of Gödel’S First Incompleteness Theorem, Bernd Buldt 2014 Indiana University - Purdue University Fort Wayne

The Scope Of Gödel’S First Incompleteness Theorem, Bernd Buldt

Philosophy Faculty Publications

Guided by questions of scope, this paper provides an overview of what is known about both the scope and, consequently, the limits of Gödel’s famous first incompleteness theorem.


A Quasi-Classical Logic For Classical Mathematics, Henry Nikogosyan 2014 University of Nevada, Las Vegas

A Quasi-Classical Logic For Classical Mathematics, Henry Nikogosyan

Theses

Classical mathematics is a form of mathematics that has a large range of application; however, its application has boundaries. In this paper, I show that Sperber and Wilson’s concept of relevance can demarcate classical mathematics’ range of applicability by demarcating classical logic’s range of applicability. Furthermore, I introduce how to systematize Sperber and Wilson’s concept of relevance into a quasi-classical logic that can explain classical logic’s and classical mathematics’ range of applicability.


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