On Fixed Points, Diagonalization, And Self-Reference, 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, 2016 Kenyon College

#### The Logic Of Uncertain Justifications, Robert Milnikel

*Robert Milnikel*

No abstract provided.

The Logic Of Uncertain Justifications, 2016 Kenyon College

#### The Logic Of Uncertain Justifications, Robert Milnikel

*Robert Milnikel*

No abstract provided.

Conservativity For Logics Of Justified Belief: Two Approaches, 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, 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, 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, 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, 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, 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, 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’, 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, 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, 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, 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, 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), 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, 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, 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, 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, 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.