Logic and Foundations Commons^™

Riding The Rails Of Reason: A Dialogue On Truth, Logic, And Proof, Surinder Pal Singh Kainth

Journal of Humanistic Mathematics

On a quiet train ride, I found myself in conversation with Noor, an inquisitive teenager with sharp questions about truth, logic, and mathematical proof. As we talked, I used the train itself as a metaphor to explain how mathematical proofs provide certainty, far beyond what repetitive verification alone can offer. Our discussion ranged from common misconceptions about the foundations of logic to the need for clear definitions and axioms. We also touched on fundamental ideas such as the challenges posed by the Axiom of Choice and the limitations revealed by Gödel’s incompleteness theorem.

A Nonstandard Exploration Of Approximate Identity And Unitization, Tong (Nicole) Wu

HMC Senior Theses

The goal of this senior thesis is to explore general nonstandard analysis and some possible applications to 𝐶^*-algebras in functional analysis. More specifically, we shall define an approximate identity of a 𝐶^*-algebra using nonstandard analysis and study nonstandard hulls of internal 𝐶^*-algebra in the context of different unitizations. We shall also prove a few results for ideals in 𝐶^*-algebra using nonstandard definitions of approximate identities. We shall also briefly discuss the history and developments of nonstandard analysis.

Modeling Deductive Inference: A Historico-Philosophical Introduction To First-Order Logic, David J. Buller

Faculty Books & Book Chapters

This book is a companion text for lectures on first-order logic and its elementary metatheory (used in Intermediate Logic at Northern Illinois University). It covers the basic concepts of set theory necessary for a mathematical development of first-order logic; develops a formal language of first-order logic; presents a classical Tarskian semantics for the language and the “semantic” conception of logical consequence; presents a Gentzenian proof system and the “syntactic” conception of logical consequence; develops a partial decision procedure for logical consequence in the language; demonstrates applications of the formal system to modeling deductive inference expressed in natural language; and extends …

Towards Erasing The Distinction Between The Computational And Syntactic Accounts Of Scientific Theories, Timothy Luft

Theses

One of the main goals of philosophy of science is to give a proper account of scientific theories and their structure. One way that accounts of the structure of scientific theories can be distinguished is by the mathematical or logical structures that they involve. For instance, syntactic accounts of scientific theories hold that theories are axioms in a logical framework, whereas semantic accounts are more liberal in the range of mathematical and logical structures they take as pertinent to the structure of scientific theories. Paul Thagard (1988) offers a computational account of scientific theories, which holds that theories are complex …

Introduction To Upside-Down Logic: Its Deep Relation To Neutrosophic Logic And Applications, Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In the study of uncertainty, concepts such as fuzzy sets [113], fuzzy graphs [79], and neutrosophic sets [88] have been extensively investigated. This paper focuses on a novel logical framework known as Upside-Down Logic, which systematically transforms truths into falsehoods and vice versa by altering contexts, meanings, or perspectives. The concept was first introduced by F. Smarandache in [99]. To contribute to the growing interest in this area, this paper presents a mathematical definition of Upside-Down Logic, supported by illustrative examples, including applications related to the Japanese language. Additionally, it introduces and explores Contextual Upside-Down Logic, an advanced extension that …

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.

Reverse Mathematics Of Ramsey's Theorem, Nikolay Maslov

Electronic Theses, Projects, and Dissertations

Reverse mathematics aims to determine which set theoretic axioms are necessary to prove the theorems outside of the set theory. Since the 1970’s, there has been an interest in applying reverse mathematics to study combinatorial principles like Ramsey’s theorem to analyze its strength and relation to other theorems. Ramsey’s theorem for pairs states that for any infinite complete graph with a finite coloring on edges, there is an infinite subset of nodes all of whose edges share one color. In this thesis, we introduce the fundamental terminology and techniques for reverse mathematics, and demonstrate their use in proving Kőnig's lemma …

Generations Of Reason: A Family’S Search For Meaning In Post-Newtonian England (Book Review), Calvin Jongsma

Faculty Work Comprehensive List

Reviewed Title: Generations of Reason: A Family's Search for Meaning in Post-Newtonian England by Joan L. Richards. New Haven, CT: Yale University Press, 2021. 456 pp. ISBN: 9780300255492.

Richard Whately's Revitalization Of Syllogistic Logic, Calvin Jongsma

Faculty Work Comprehensive List

This is an expanded version of the first chapter Richard Whately’s Revitalization of Syllogistic Logic in Aristotle’s Syllogism and the Creation of Modern Logic edited by Lukas M. Verburgt and Matteo Cosci (Bloomsbury, 2023). Drawing upon the author’s 1982 Ph. D. dissertation (https://digitalcollections.dordt.edu/faculty_work/230/ ) and more current scholarship, this essay traces the critical historical background to Whately’s work in more detail than could be done in the published version.

Symbolic Logic, Tony Roy

Books

Textbook for symbolic logic, beginning at a level appropriate for beginning students, continuing through Godel's completeness and incompleteness theorems. The text naturally divides into two volumes, the first for reasoning in logic, the second for reasoning about it.

The first volume includes parts I and II of the text. Part I introduces the complete classical predicate calculus with equality, including both axiomatic and natural derivation systems. Part II transitions to methods for reasoning about logic, including direct reasoning from definitions and mathematical induction.

The second volume includes parts III and IV of the text. Part III develops basic results in …

How To Guard An Art Gallery: A Simple Mathematical Problem, Natalie Petruzelli

The Review: A Journal of Undergraduate Student Research

The art gallery problem is a geometry question that seeks to find the minimum number of guards necessary to guard an art gallery based on the qualities of the museum’s shape, specifically the number of walls. Solved by Václav Chvátal in 1975, the resulting Art Gallery Theorem dictates that ⌊n/3⌋ guards are always sufficient and sometimes necessary to guard an art gallery with n walls. This theorem, along with the argument that proves it, are accessible and interesting results even to one with little to no mathematical knowledge, introducing readers to common concepts in both geometry and graph …

Inductive Constructions In Logic And Graph Theory, Davis Deaton

Honors Scholars Collaborative Projects

Just as much as mathematics is about results, mathematics is about methods. This thesis focuses on one method: induction. Induction, in short, allows building complex mathemati- cal objects from simple ones. These mathematical objects include the foundational, like logical statements, and the abstract, like cell complexes. Non-mathematicians struggle to find a common thread throughout all of mathematics, but I present induction as such a common thread here. In particular, this thesis discusses everything from the very foundations of mathematics all the way to combina- torial manifolds. I intend to be casual and opinionated while still providing all necessary formal rigor. …

Formally Verifying Peano Arithmetic, Morgan Sinclaire

Boise State University Theses and Dissertations

This work is concerned with implementing Gentzen’s consistency proof in the Coq theorem prover.

In Chapter 1, we summarize the basic philosophical, historical, and mathematical background behind this theorem. This includes the philosophical motivation for attempting to prove the consistency of Peano arithmetic, which traces itself from the first attempted axiomatizations of mathematics to the maturation of Hilbert’s program. We introduce many of the basic concepts in mathematical logic along the way: first-order logic (FOL), Peano arithmetic (PA), primitive recursive arithmetic (PRA), Gödel's 2nd Incompleteness theorem, and the ordinals below ε₀.

In …

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 …

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, …

A Quasi-Classical Logic For Classical Mathematics, Henry Nikogosyan

Honors College 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.

On The Logic Of Reverse Mathematics, Alaeddine Saadaoui

Theses, Dissertations and Capstones

The goal of reverse mathematics is to study the implication and non-implication relationships between theorems. These relationships have their own internal logic, allowing some implications and non-implications to be derived directly from others. The goal of this thesis is to characterize this logic in order to capture the relationships between specific mathematical works. The results of our study are a finite set of rules for this logic and the corresponding soundness and completeness theorems. We also compare our logic with modal logic and strict implication logic. In addition, we explain two applications of S-logic in topology and second order arithmetic.

The Mathematical Landscape, Antonio Collazo

CMC Senior Theses

The intent of this paper is to present the reader will enough information to spark a curiosity in to the subject.  By no means is the following a complete formulation of any of the topics covered.  I want to give the reader a tour of the mathematical landscape.  There are plenty of further details to explore in each section, I have just touched the tip the iceberg.  The work is basically in four sections: Numbers, Geometry, Functions, Sets and Logic, which are the basic building blocks of Math.  The first sections are a exposition into the mathematical objects and their …

A Philosophical Examination Of Proofs In Mathematics, Eric Almeida

Undergraduate Review

No abstract provided.

A Unifying Field In Logics: Neutrosophic Logic Neutrosophy, Neutrosophic Set, Neutrosophic Probability (In Traditional Chinese), Florentin Smarandache, Feng Liu

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

Proof In Law And Science, David H. Kaye

Faculty Scholarship

This article addresses proof in both science and law. Both disciplines utilize proof of facts and proof of theories, but for different purposes and, consequently, in different ways. Some similarities exist, however, in how both disciplines use a series of premises followed by a conclusion to form an argument, and thus constitute a logic. This article analyzes the ways in which legal logic and scientific logic differ. Finding facts in law involves the same logic but quite different procedures than scientific fact-finding. Finding, or rather constructing, the law is also very different from scientific theorizing. But such differences do not …