Logic and Foundations of Mathematics Commons^™

Three Essays On Substructural Approaches To Semantic Paradoxes, Brian C. Porter

Dissertations, Theses, and Capstone Projects

This thesis consists of three papers on substructural approaches to semantic paradoxes. The first paper introduces a formal system, based on a nontransitive substructural logic, which has exactly the valid and antivalid inferences of classical logic at every level of (meta)inference, but which I argue is still not classical logic. In the second essay, I introduce infinite-premise versions of several semantic paradoxes, and show that noncontractive substructural approaches do not solve these paradoxes. In the third essay, I introduce an infinite metainferential hierarchy of validity curry paradoxes, and argue that providing a uniform solution to the paradoxes in this hierarchy …

Zero, Śūnya And Pūrṇa: A Comparative Analysis, Animisha Tewari

Comparative Philosophy

Due to apparent duality in this world, one has to face a lot of difficulties while searching for the Truth. Our ego is the root cause for perception of duality and this in turn leads to suffering. This suffering can only be extinguished by attainment of the Truth, i.e, non-duality. However, in order to enable the finite intellect to comprehend the incomprehensible non-duality, this undifferentiated whole is sometimes denoted by nothingness (śūnya) or fullness (pūrṇa). Non-duality is usually understood by the numeral ‘1’ which stands for unity or oneness. The main aim of this paper is …

Ineffability, Emptiness And The Aesthetics Of Logic, Andreas Kapsner

Comparative Philosophy

In this essay, I explore the nature of the logical analysis of Buddhist thought that Graham Priest has offered in his book The Fifth Corner of Four (5of4). The paper traces the development of a logical value in- troduced in 5of4, which Priest has called e. The paper points out that certain criticisms I have made earlier still stand, but focuses on a recon- ceptualization of 5of4 in which these arguments carry less weight. This new perspective on the book, inspired by a response to my arguments by Priest himself, sees the logical analysis of Buddhism …

Collation Model For Ljs 457: Loyca Parva ... [Etc.]., Dot Porter

Collation Models

Work on scholastic logic used in universities in the late 15th century by Paolo Veneto, professor of logic in Padua, Siena (1420-1424), and Perugia (1424-1428); member of the community of Hermits of Saint Augustine in Perugia.. Followed by a brief logical work by Paolo della Pergola, a student of Paolo Veneto.

Necessity, Essence And Analyticity: Toward An Analytic Essentialist Account Of Necessity, Dongwoo Kim

Dissertations, Theses, and Capstone Projects

Some truths could not have failed to hold. Such are called metaphysically necessary truths. As Michael Dummett once aptly formulated, the philosophical problem about necessity is twofold: what makes necessary truths necessarily true and how do we recognize them as such? This dissertation aims to address these questions by developing and defending a novel account of necessity, which has the following three main theses: (1) the necessity of a statement about an entity is established as a consequence of a general principle implying that if the entity is a certain way then it is necessarily that way and the fact …

Foundational Mathematical Beliefs And Ethics In Mathematical Practice And Education, Richard Spindler

Journal of Humanistic Mathematics

Foundational philosophical beliefs about mathematics in the mathematical community may have an unappreciated yet profound impact on ethics in mathematical practice and mathematics education, which also affects practice. A philosophical and historical basis of the dominant platonic and formalist views of mathematics are described and evaluated, after which an alternative evidence-based foundation for mathematical thought is outlined. The dualistic nature of the platonic view based on intuition is then compared to parallel historical developments of universalizing ethics in Western thought. These background ideas set the stage for a discussion of the impact of traditional mathematical beliefs on ethics in the …

Ethics And Mathematics – Some Observations Fifty Years Later, Gregor Nickel

Journal of Humanistic Mathematics

Almost exactly fifty years ago, Friedrich Kambartel, in his classic essay “Ethics and Mathematics,” did pioneering work in an intellectual environment that almost self-evidently assumed a strict separation of the two fields. In our first section we summarize and discuss that classical paper. The following two sections are devoted to complement and contrast Kambartel’s picture. In particular, the second section is devoted to ethical aspects of the indirect and direct mathematization of modern societies. The final section gives a short categorization of various philosophical positions with respect to the rationality of ethics and the mutual relation between ethics and mathematics.

Unknowable Truths: The Incompleteness Theorems And The Rise Of Modernism, Caroline Tvardy

Honors Scholars Collaborative Projects

This thesis evaluates the function of the current history of mathematics methodologies and explores ways in which historiographical methodologies could be successfully implemented in the field. Traditional approaches to the history of mathematics often lack either an accurate portrayal of the social and cultural influences of the time, or they lack an effective usage of mathematics discussed. This paper applies a holistic methodology in a case study of Kurt Gödel’s influential work in logic during the Interwar period and the parallel rise of intellectual modernism. In doing so, the proofs for Gödel’s Completeness and Incompleteness theorems will be discussed as …

Semantic Completeness Of Intuitionistic Predicate Logic In A Fully Constructive Meta-Theory, Ian Ray

Masters Theses & Specialist Projects

A constructive proof of the semantic completeness of intuitionistic predicate logic is explored using set-generated complete Heyting Algebra. We work in a constructive set theory that avoids impredicative axioms; for this reason the result is not only intuitionistic but fully constructive. We provide background that makes the thesis accessible to the uninitiated.

Inferring Inferences: Relational Propositions For Argument Mining, Andrew Potter

Proceedings of the Society for Computation in Linguistics

Inferential reasoning is an essential feature of argumentation. Therefore, a method for mining discourse for inferential structures would be of value for argument analysis and assessment. The logic of relational propositions is a procedure for rendering texts as expressions in propositional logic directly from their rhetorical structures. From rhetorical structures, relational propositions are defined, and from these propositions, logical expressions are then generated. There are, however, unsettled issues associated with Rhetorical Structure Theory (RST), some of which are problematic for inference mining. This paper takes a deep dive into some of these issues, with the aim of elucidating the problems …

Russian Logics And The Culture Of Impossible: Part Ii: Reinterpreting Algorithmic Rationality, Ksenia Tatarchenko, Anya Yermakova, Liesbeth De Mol

Research Collection School of Social Sciences

This article reinterprets algorithmic rationality by looking at the interaction between mathematical logic, mechanized reasoning, and, later, computing in the Russian Imperial and Soviet contexts to offer a history of the algorithm as a mathematical object bridging the inner and outer worlds, a humanistic vision that we, following logician Vladimir Uspensky, call the “culture of the impossible.” We unfold the deep roots of this vision as embodied in scientific intelligentsia. In Part I, we examine continuities between the turn-of-the-twentieth-century discussions of poznaniye—an epistemic orientation towards the process of knowledge acquisition—and the postwar rise of the Soviet school of mathematical logic. …

Assessing Risk At The National Strategic Level: Visualization Tools For Military Planners, Wade A. Germann, Heather S. Gregg

The US Army War College Quarterly: Parameters

The reemergence of great power competition, conflict with near-peer competitor states below the level of armed conflict, and persisting threats from nonstate actors with transnational ambitions and global reach pose challenges for strategists planning, executing, and assessing military operations and strategy. Building on current visualization tools, two proposed models—the National Strategic Risk Abacus and the National Strategic Risk Radar Chart—address these challenges and better depict how the US military may inadvertently contribute to risk at the national strategic level.

Patrick Aidan Heelan’S The Observable: Heisenberg’S Philosophy Of Quantum Mechanics, Paul Downes

Research Resources

The publication of Patrick Aidan Heelan’s The Observable, with forewords from Michel Bitbol, editor Babette Babich and the author himself, offers a timely invitation to reconsider the relation between quantum physics and continental philosophy.

Patrick Heelan does so, as a contemporary of and interlocutor with Werner Heisenberg on these issues, as a physicist himself who trained with leading figures of quantum mechanics (QM), Erwin Schrödinger and Eugene Wigner. Moreover, Heelan highlights Heisenberg’s interest in phenomenology as ‘a friend and frequent visitor of Martin Heidegger’ (55). Written originally in 1970 and unpublished then for reasons Babich explicates in her foreword, …

Dimentia: Footnotes Of Time, Zachary Hait

Senior Projects Spring 2021

Time from the physicist's perspective is not inclusive of our lived experience of time; time from the philosopher's perspective is not mathematically engaged, in fact Henri Bergson asserted explicitly that time could not be mathematically engaged whatsoever. What follows is a mathematical engagement of time that is inclusive of our lived experiences, requiring the tools of storytelling.

Recognizing Mathematics Students As Creative: Mathematical Creativity As Community-Based And Possibility-Expanding, Meghan Riling

Journal of Humanistic Mathematics

Although much creativity research has suggested that creativity is influenced by cultural and social factors, these have been minimally explored in the context of mathematics and mathematics learning. This problematically limits who is seen as mathematically creative and who can enter the discipline of mathematics. This paper proposes a framework of creativity that is based in what it means to know or do mathematics and accepts that creativity is something that can be nurtured in all students. Prominent mathematical epistemologies held since the beginning of the twentieth century in the Western mathematics tradition have different implications for promoting creativity in …

Don’T Be So Fast With The Knife: A Reply To Kapsner, Graham Priest

Comparative Philosophy

The is a brief reply to the central objection against the construction of my The Fifth Corner of Four by Andi Kapsner in his “Cutting Corners: A Critical Note on Priest’s Five-Valued Catuṣkoṭi. This concerns the desirability of adding a fifth corner (ineffability) to the four of the catuṣkoṭi.

Cutting Corners: A Critical Note On Priest’S Five-Valued Catuṣkoṭi, Andreas Kapsner

Comparative Philosophy

Graham Priest has offered a rational reconstruction of Buddhist thought that involves, first, modeling the Catuṣkoṭi by a four valued logic, and then later adding a fifth value, read as “ineffability”. This note examines that fifth value and raises some concerns about it that seem grave enough to reject it. It then sketches an alternative to Priest’s account that has no need for the fifth value.

A Russellian Analysis Of Buddhist Catuskoti, Nicholaos Jones

Comparative Philosophy

Names name, but there are no individuals who are named by names. This is the key to an elegant and ideologically parsimonious strategy for analyzing the Buddhist catuṣkoṭi. The strategy is ideologically parsimonious, because it appeals to no analytic resources beyond those of standard predicate logic. The strategy is elegant, because it is, in effect, an application of Bertrand Russell's theory of definite descriptions to Buddhist contexts. The strategy imposes some minor adjustments upon Russell's theory. Attention to familiar catuṣkoṭi from Vacchagotta and Nagarjuna as well as more obscure catuṣkoṭi from Khema, Zhi Yi, and Fa Zang motivates the …

Connecting Ancient Philosophers’ Math Theory To Modern Fractal Mathematics, Colin Mccormack

Parnassus: Classical Journal

No abstract provided.

Between Evidence And Facts: An Argumentative Perspective Of Legal Evidence, Wenjing Du, Minghui Xiong

OSSA Conference Archive

In this paper, we will present an argumentative view of legal evidence. In an argumentation-based litigation game, the only purpose of the suitor (S) or the respondent (R) is to maximize their own legal rights while the purpose of the trier (T) is to maintain judicial fairness and justice. Different selections of evidence and different orders of presenting evidence will lead to different case-facts and even adjudicative results, the purpose of litigation is to reconcile a balance among the three parties - S, R, and T.

Real Possibility: Modality And Responsibility, Julia Gaul

Honors Scholar Theses

Imagine: someone is backing out of a parking space and does not look in their rear view mirror. They subsequently hit a car that was passing by. One could argue that they simply could have avoided the accident had they looked in their mirror. This non-actual possibility, that they could have looked in the mirror, seems legally and morally relevant. One could also argue that they could have avoided the accident had they stuck their feet out of their window and sung La Marseillaise.

My leading questions is: how do we distinguish possibilities that are legally and morally relevant from …

An Evolutionary Approach To Crowdsourcing Mathematics Education, Spencer Ward

Honors College

By combining ideas from evolutionary biology, epistemology, and philosophy of mind, this thesis attempts to derive a new kind of crowdsourcing that could better leverage people’s collective creativity. Following a theory of knowledge presented by David Deutsch, it is argued that knowledge develops through evolutionary competition that organically emerges from a creative dialogue of trial and error. It is also argued that this model of knowledge satisfies the properties of Douglas Hofstadter’s strange loops, implying that self-reflection is a core feature of knowledge evolution. This mix of theories then is used to analyze several existing strategies of crowdsourcing and knowledge …

The Conceptions Of Self-Evidence In The Finnis Reconstruction Of Natural Law, Kevin P. Lee

St. Mary's Law Journal

Finnis claims that his theory proceeds from seven basic principles of practical reason that are self-evidently true. While much has been written about the claim of self-evidence, this article considers it in relation to the rigorous claims of logic and mathematics. It argues that when considered in this light, Finnis equivocates in his use of the concept of self-evidence between the realist Thomistic conception and a purely formal, modern symbolic conception. Given his respect for the modern positivist separation of fact and value, the realism of the Thomistic conception cannot be the foundation for the natural law as Finnis would …

Logical Pluralism And Vicious Regresses, Daniel Boyd

Dissertations, Theses, and Capstone Projects

This material in this dissertation will be divided into two parts. The first part is a preliminary discussion of vicious regress arguments in the philosophy of logic in the 20th century. The second part will focus on three different versions of logical pluralism, i.e., the view that there are many correct logics. In each case an argument will be developed to show that these versions of logical pluralism result in a vicious regress.

The material in part one will be divided into three chapters, and there are a few reasons for having a preliminary discussion of vicious regress arguments in …

Are Logic And Math Relevant To Social Debates?, Michael A. Lewis

Journal of Humanistic Mathematics

Social debates, as well as discussions about certain highly charged issues, such as racism, gender identity, and sexuality, usually turn on the uses or mentions of key words. That is, the conclusions we can draw from such discussions depend on how certain terms are used or mentioned in them. Yet participants in social debates may often fail to precisely define their terms or fail to make important distinctions in terms uttered by others. Both logic and mathematics pay attention to the importance of precise definitions when it comes to engaging in discussions, arguments, or proofs. Logic also makes an important …

Engaging The Paradoxical: Zeno's Paradoxes In Three Works Of Interactive Fiction, Michael Z. Spivey

Journal of Humanistic Mathematics

For over two millennia thinkers have wrestled with Zeno's paradoxes on space, time, motion, and the nature of infinity. In this article we compare and contrast representations of Zeno's paradoxes in three works of interactive fiction, Beyond Zork, The Chinese Room, and A Beauty Cold and Austere. Each of these works incorporates one of Zeno's paradoxes as part of a puzzle that the player must solve in order to advance and ultimately complete the story. As such, the reader must engage more deeply with the paradoxes than he or she would in a static work of fiction. …

Rules, Tricks And Emancipation, Jessie Allen

Book Chapters

Rules and tricks are generally seen as different things. Rules produce order and control; tricks produce chaos. Rules help us predict how things will work out. Tricks are deceptive and transgressive, built to surprise us and confound our expectations in ways that can be entertaining or devastating. But rules can be tricky. General prohibitions and prescriptions generate surprising results in particular contexts. In some situations, a rule produces results that seem far from what the rule makers expected and antagonistic to the interests the rule is understood to promote. This contradictory aspect of rules is usually framed as a downside …

Pluralistic Perspectives On Logic: An Introduction, Colin R. Caret, Teresa Kouri Kissel

Philosophy Faculty Publications

(First paragraph) Logical pluralism is the view that there are distinct, but equally good logics. Recent years have witnessed a sharp upswing of interest in this view, resulting in an impressive literature. We only expect this trend to continue in the future. More than one commentator has, however, expressed exasperation at the view: what can it mean to be a pluralist about logic of all things? [see, e.g., Eklund (2017); Goddu (2002); Keefe (2014)]. In this introduction, we aim to set out the basic pluralist position, identify some issues over which pluralists disagree amongst themselves, and highlight the topics at …

The Rhetorical Structure Of Modus Tollens: An Exploration In Logic-Mining, Andrew Potter

Proceedings of the Society for Computation in Linguistics

A general method for mining discourse for occurrences of the rules of inference would be useful in a variety of natural language processing applications. The method described here has its roots in Rhetorical Structure Theory (RST). An RST analysis of a rule of inference can be used as an exemplar to produce a relational complex in the form of a nested relational proposition. This relational complex can be transformed into a logical expression using the logic of relational propositions. The expression can then be generalized as a logical signature for use in logic-mining discourse for instances of the rule. Generalized …

Logic, Thought, And Language In Hegel, Marx, And Rosenzweig, Omar Moreno

Open Access Theses & Dissertations

The objective of this Thesis is to open a conversation regarding the role of grammar in two areas of philosophy: interpretation and normative philosophy. The task is divided into three chapters, each of which focuses on one major issue. The first is a demonstration of the use of grammar in understanding and interpreting works of philosophy, namely those of Hegel and Marx. The second chapter is an interpretation of Franz Rosenzweig's renovated grammar as seen in The Star of Redemption. The last uses an analysis of grammar to challenge the role of empirical knowledge in community building. The last chapter …