Logic and Foundations Commons^™

Model-Theoretic Arguments In Philosophy, Peter Susanszky

Dissertations, Theses, and Capstone Projects

This dissertation is on model-theoretic arguments in philosophy, especially those of Quine, Davidson, and Putnam. In the first part, to ground the debate, I give a rigorous introduction to the salient parts of first-order model theory. I start the second part by giving an introduction to Quine's philosophy, and how the model-theoretic arguments fit into it. After considering how Donald Davidson adopted the Quinean lesson, I move on to Putnam's model-theoretic arguments. Putnam's spin on these model-theoretic considerations significantly departs from Quine and Davidson, while retaining many of the core ideas. Most importantly, I argue that the target of Putnam's …

Opening The Lantern: The Leiden Declaration, Mark Huber

Journal of Humanistic Mathematics

The Leiden Declaration on Artificial Intelligence and Mathematics provides insight into how mathematicians view their discipline. However, when positioning mathematics against AI, the declaration falls short in recognizing recent advances. This column introduces the reader who might be unfamiliar with these new methods through the formal language Lean and discusses how these new abilities might change the way journals operate.

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.

Conditionals And Modalities In Constructive Quantum Logics, Juan P. Aguilera, Guillaume Massas

Mathematics, Physics, and Computer Science Faculty Articles and Research

We investigate logics that generalize both intuitionistic logic and quantum logic. In earlier work, we introduced Ex-logic, an extension of Holliday's fundamental logic that coincides with the intersection of orthologic and the implication-free fragment of intuitionistic logic. In this paper, we add an implication connective to Ex-logic and axiomatize iEx-logic, the intersection of full intuitionistic logic and orthomodular logic with the implication connective interpreted as the Sasaki hook. As a consequence, we obtain a characterization of the lattice of logics extending iEx-logic as the product of the lattice of intermediate logics and the lattice of orthomodular logics. We also explore …

(R2177) Further Study On Intuitionistic Fuzzy Matrices, S. Sriram, A. Anitha

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we present a comprehensive study of the distributive law associated with two newly introduced operations addition and multiplication defined on Intuitionistic Fuzzy Matrices (IFMs). The proposed operations are systematically examined to investigate their algebraic properties, with particular emphasis on the behavior of the distributive law under these novel definitions. Furthermore, modal and extended modal operators are employed to establish several theoretical results that reinforce the mathematical foundation of the proposed framework. The validity and practical relevance of these results are further demonstrated through applications in decision-making scenarios, highlighting their effectiveness in addressing complex problems characterized by uncertainty.

(R2170) Delta-Continuous Functions In Interval-Valued Neutrosophic Soft Topological Spaces And An Application Using Distance And Similarity Measures, B. Vijayalakshmi, S. Madhunika

Applications and Applied Mathematics: An International Journal (AAM)

This paper introduces delta-continuous functions in interval-valued neutrosophic soft topological spaces and investigates their fundamental properties. Additionally, delta-irresolute functions are introduced within the same framework. The relationships between these functions and existing function classes are explored. Several theorems, accompanied by illustrative examples, are provided to support the theoretical findings. The study also includes an application of the proposed concepts to distance and similarity measures.

(R2187) Analysis Of Neurological Impairments In Hospitalized Patients Using Cubic Neutrosophic Sets, B. Anitha, M. Lavanya

Applications and Applied Mathematics: An International Journal (AAM)

This study introduces an MCDM-based framework for identifying neurological diseases in hospitalized patients using symptom-based evaluations. A team of interns, guided by the chief doctor, was responsible for determining each patient’s precise condition from the presented neurological symptoms. To enhance diagnostic accuracy, the interns employed the TOPSIS and WASPAS methods to assess and rank the potential disease options. The combined analysis yielded a clear identification of the highest ranked disease for every patient, highlighting the effectiveness of these MCDM techniques in supporting clinical decision making.

Classes Of Analytic Functions Defined By Salagean Derivative Operator Associated With Neutrosophic Generalized Poisson Distribution, Soliu O. Opeyemi Okunola, Olushola Adeyemo, Sayo A. Abidemi Gbangbala, Folorunso I. Isola Akinwale

Neutrosophic Systems with Applications

This study introduces and analyses new subclasses of analytic functions by applying the Salagean derivative operator to the Neutrosophic Generalized Poisson Distribution (NGPD) series. We develop a model where the mean parameter is treated as an interval or set to account for indeterminacy in complex systems. By employing Stirling numbers of the second kind and decreasing factorials, we derive necessary and sufficient coefficient inequalities and inclusion relations for these new subclasses. Numerical results and graphical illustrations demonstrate the sensitivity of these functions to orientation and the neutrosophic parameter, providing a framework for applications in fields like medical imaging and network …

Identifiability, Sequentiality And Infinity, Jose L. Menaldi

Mathematics Faculty Research Publications

Abstract: A definition of identifiable-sets is used with sequential analysis to establish a realm of mathematics. Within this imaginary world, a specific consonant between infinite sets and sequentiality is reached.  This consonant allows some mathematical constructions to model pieces of the reality, based on dual philosophy and physics itself.  There is an effort made to render this understandable for the scientific community.

A New Approach To Generate Combinatorial Patterns In Logical Analysis Of Data And Its Application To Predict College Retention, Salihah Ahmed E. Jaafari

Theses and Dissertations

Student retention and degree completion remain central challenges for higher-education institutions, with significant implications for student success, institutional effectiveness, and public accountability. While advances in predictive analytics have enabled earlier identification of students at risk of withdrawal, many commonly used machine learning approaches suffer from limited interpretability, constraining their practical usefulness for advising, intervention, and policy decision making. This dissertation addresses the problem of predicting student persistence by developing and evaluating optimization based, interpretable classification models within the Logical Analysis of Data (LAD) framework. Building on existing LAD formulations, this research introduces two novel pattern generation models, the Best Term …

Objections To The Use Of The Axiom Of Choice To Model The Physical World, Kensey Doughtie

Rose-Hulman Undergraduate Mathematics Journal

We look at platonistic mathematics and the application of this perspective in the physical world. We recognize paradoxes within Zermelo–Fraenkel set theory with the axiom of choice (ZFC) that conflict with physical reality, giving us reason to question if the axiom of choice should be so freely applied in theories of the physical world especially since it appears to enable a deterministic perspective. In theories of quantum physics, the axiom of choice is used to assume noncomputable numbers as initial conditions. This is equivalent to assuming a finite system contains an infinite amount of information at an instant in time; …

Demystifying Hardware Formal Verification For Undergraduate Education: A Risc-V Processor Case Study With Coursework Implementation, Riley A. Peters

Master's Theses

Hardware verification engineers apply formal methods to prove that a digital device always behaves according to its specification. This differs from traditional functional verification, in which engineers establish correctness by repeatedly sending test inputs to the device and comparing the outputs against a reference model. With the growing complexity of integrated circuits, the demand for digital verification engineers with formal methods experience has continued to increase. However, California Polytechnic State University: San Luis Obispo's current curriculum lacks dedicated material to prepare students for these roles.

This thesis seeks to address the lack of formal methods material through two efforts. First, …

A Virtual Community Math Circle, Skona Brittain, Sayonita Ghosh Hajra, Steve Heller, Daniel Hodgins, Peter Petto, Gabriella Pinter, Lauren Rose, A. Gwinn Royal, Asmita Sodhi

Journal of Math Circles

The Julia Robinson Mathematics Festival (JRMF) Community Math Circle is free, volunteer-run, and online. We collaborate with JRMF and use their activities in our events. Started during the pandemic, the Community Math Circle continues to thrive. We have about 40 participants attending every month, typically kids aged 6 to 13, teachers, facilitators, and other adults. This article describes how our event is organized, planned, and executed, and how we train facilitators. We will also offer reflections on our successes and challenges.

Toward Completeness Theorem For Guarded Kleene Algebra With Tests, Hung Pham

Honors Theses

Code refactoring is a fundamental practice in software engineering, in which a program is restructured without changing the actions it performs and the results it produces. To carry out refactoring with confidence, one requires a formal method for verifying that two programs are equivalent. Guarded Kleene Algebra with Tests (GKAT) provides such a framework, an algebraic system designed to reason about a natural class of programs, namely those in which every branch and loop is governed by a Boolean condition, such as if–else and while statements. Central to GKAT is a finite set of algebraic axioms for deriving program equivalences. …

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.

All Games Have Equilibria, M. Ali Khan, Arthur Paul Pedersen, Maxwell B. Stinchcombe

Publications and Research

Research on Nash equilibrium existence for infinite games has grown into a patchwork of technical preconditions and counterexamples. This paper presents a unified program in equilibrium theory by revising the predominant model of mixed strategies based on countable additivity. A game is specified by a nonempty set of players and, for each player, a nonempty action set and a bounded von Neumann-Morgenstern utility function. Every such game is shown to admit a Nash equilibrium in finitely additive mixed strategies. In addition, the equilibrium correspondence for any such game is shown to be nonempty, compact-valued, and upper hemicontinuous, and the same …

All Games Have Equilibria, Arthur Paul Pedersen, M. Ali Khan, Maxwell B. Stinchcombe

Publications and Research

Research on Nash equilibrium existence for infinite games has grown into a patchwork of technical preconditions and counterexamples. This paper presents a unified program in equilibrium theory by revising the predominant model of mixed strategies based on countable additivity. A game is specified by a nonempty set of players and, for each player, a nonempty action set and a bounded von Neumann-Morgenstern utility function. Every such game is shown to admit a Nash equilibrium in finitely additive mixed strategies. In addition, the equilibrium correspondence for any such game is shown to be nonempty, compact-valued, and upper hemicontinuous, and the same …

On Quantum Processes And The Epistemic Constraints, Varun Immanuel Premkumar Immanuel

Electronic Theses & Dissertations (2024 - present)

This doctoral dissertation on the foundations of quantum theory tells the story of a conceptual protagonist I have called “Epistemic Constraint.” Here, epistemic constraints are the definite, intersubjectively agreeable, ordinary-language conditions under which experiments are described.

The usual formulation of the quantum measurement problem, which I call the Schrodingerian measurement problem, has the structure of an anomaly: if we take quantum theory at face value, we expect no definite values, and yet we see definite values in experiments. The responses to this problem have been either to solve it or to dissolve it. These responses, which have taken the form …

(R2127) On 2-Absorbing Hesitant Primary Fuzzy Ideals Of Rings, B. Anitha, M. Vidhya

Applications and Applied Mathematics: An International Journal (AAM)

By presenting 2-absorbing hesitant primary fuzzy ideals, we begin the investigation of a generalisation of hesitant primary fuzzy ideals (HPRFI) in rings in this research. The concepts of a weakly completely 2-absorbing hesitant primary fuzzy ideal (WC2-AHPRFI) and a Weakly completely 2-absorbing hesitant fuzzy ideal (WC2-AHFI) are developed, and their structural features and attributes are examined. We introduce the idea of a 2-absorbing hesitant K-fuzzy ideal (2-AHK-FI), 2-absorbing hesitant K-primary fuzzy ideal (2-AHK-PRFI) and examine a few of its characteristics.

(Si15-017) Modeling Construction Project Uncertainty Using Fuzzy Inference And Nature-Inspired Metaheuristic Knowledge-Based Algorithms, M. Kapoor, B. K. Pathak, R. Kumar

Applications and Applied Mathematics: An International Journal (AAM)

Construction projects are influenced by various uncertain factors, including supervisory knowledge, labor expertise, and weather conditions. These uncertain factors have linguistic properties that are challenging to quantify with the help of traditional set theory. To address this, the study applies fuzzy inference systems, specifically the Mamdani fuzzy inference system, to model and evaluate the impact of these uncertainties on construction projects. This fuzzy inference system provides a more nuanced understanding of these factors compared to traditional set theory methods. Experimental results indicate that uncertainties significantly impact construction projects. These findings establish a valuable framework for developing adaptive strategies, enhancing project …

(Si15-067) Ahp And Moora Decision Making Methods On Bipolar Fuzzy Sets, S. Anita Shanthi, R. Preethi

Applications and Applied Mathematics: An International Journal (AAM)

Decision making, in the present contemporary world, has been inherently complicated. Nowadays, the major challenge is the selection of an appropriate option. The arrival of numerous brands and models makes a purchase challenging. Hence, in this paper, four different models of the same branded laptops are considered, for the selection of an appropriate option, using the Analytic Hierarchy Process (AHP) and Multi-Objective Optimization on the basis of Ratio Analysis (MOORA), with bipolar fuzzy sets, which help decision makers arrive at the most logical choice based on their preferences. The bipolar fuzzy Analytic Hierarchy Process efficaciously supports decision making in cases …

A Dualistic Interpretation Of Mathematical Creation Through Art And Argumentation, Sofia Almpani, Petros Stefaneas, Mihir Chakraborty

Journal of Humanistic Mathematics

Creativity, informal reasoning and dynamic exchange of ideas form the pulsating heart of mathematical creation. In this approach, the concept of a mathematical object surpasses conventional boundaries of formal presentation, as they also encompass the intention to prove, significant creative stages within the proving, and the overall experience of prover's journey, which may involve arguments, debates, discovery insights, aesthetic visualizations, and narrative elements. In this paper we present two conceptual frameworks, namely Argumentation-based Proof-Events Calculus (APEC) and Mathematical RUPAs, in order to provide distinct yet interconnected perspectives on informal thinking, knowledge creation, and proving in mathematics. Following the two perspectives, …

Logic Enriched Over A Quantale, Alexander Kurz

Engineering Faculty Articles and Research

Many-valued logics have a long history in mathematical logic as well as in applications to the semantics of programming languages and to engineering more generally. Typically these logics are rich with features motivated by the particular applications they stem from. In his 1973 article "Metric Spaces, Generalized Logic, and Closed Categories", Lawvere argued that any quantale Ω gives rise to a generalized Ω-valued logic that has as its models the categories enriched over the quantale. This suggests developing a uniform framework for many-valued logics parameterized in a quantale. In this talk we will review some previous and ongoing work in …

A Realizability Approach To Constructing Higher Types Via Classifiers, Benjamin Carrick Logsdon

Dartmouth College Ph.D Dissertations

We construct an interpretation of higher types into Peano arithmetic, showing in particular that every model of PA is a model of higher types. This is a reversal of Gödel’s Dialectica construction. We also define the classifier degrees, a degree structure which subsumes the Turing degrees, the enumeration degrees, and the many-one degrees. The classifier degrees boast a rich structure and many well-behaved operations.

Computability Theoretic Aspects Of Profinite Groups And Models Of Presburger Arithmetic, Jason Block

Dissertations, Theses, and Capstone Projects

Profinite groups, which are exactly the Galois groups, are all either finite or uncountable. However, all second countable profinite groups can be presented as the set of paths through a countable tree. We use these tree presentations to find upper bounds on the complexity of the existential theories of profinite groups, as well as to prove sharpness for these bounds. These complexity results enable us to distinguish the class of profinite groups that are isomorphic to a direct product of finite groups, for which we find an upper bound on the complexity of the entire first order theory. Additionally, given …

Logic's Modern British Up-Enders, Defenders, And Extenders: Whately's Revitalization Of Logic, Calvin Jongsma

Faculty Work Comprehensive List

Logic developed dramatically during the last half of the nineteenth century. The baseline for these transformations in Great Britain was the revival of logic by Richard Whately around 1825. Whately successfully defended syllogistic logic as the science of valid reasoning against potent seventeenth and eighteenth-century detractors—Bacon, Locke, Reid, Campbell, Stewart, and others. In so doing, he made logic an intellectually respectable field of investigation for the next generation of logicians to explore and extend. This included John Stuart Mill (inductive logic), Augustus De Morgan (logic of relations), and George Boole (algebraic logic; propositional logic).

Dedekind-Macneille And Related Completions: Subfitness, Regularity, And Booleanness, G. Bezhanishvili, F. Dashiell Jr., M. A. Moshier, Joanne Walters-Wayland

Mathematics, Physics, and Computer Science Faculty Articles and Research

Completions play an important rôle for studying structure by supplying elements that in some sense “ought to be.” Among these, the Dedekind-MacNeille completion is of particular importance. In 1968 Janowitz provided necessary and sufficient conditions for it to be subfit or Boolean. Another natural separation axiom connected to these is regularity. We explore similar characterizations of when closely related completions are subfit, regular, or Boolean. We are mainly interested in the Bruns-Lakser, ideal, and canonical completions, which (unlike the Dedekind-MacNeille completion) satisfy stronger forms of distributivity. The first two are widely used in pointfree topology, while the latter is of …

Belonging, Lawrence M. Lesser

Journal of Humanistic Mathematics

No abstract provided.

The Making Of Mathematics: An Interview With Carlo Cellucci, Marshall Gordon

Journal of Humanistic Mathematics

Carlo Cellucci is professor emeritus of the Sapienza University of Rome, and has written books on logic, mathematics, and philosophy. In the following interview conducted via back-and-forth email correspondence, he discusses his 2022 book, The Making of Mathematics: Heuristic Philosophy of Mathematics [3], which presents a new paradigm for thinking about, doing, and teaching mathematics.

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 …