Logic and Foundations of Mathematics 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 …

The Limits Of Universal Logic: A Cross-Cultural Archaeology Of Logical Critique, Murat Kelikli

Comparative Philosophy

This study examines critiques of logic articulated across diverse historical and cultural traditions within a comparative framework. The critiques developed in Islamic, Buddhist, Chinese, Indian, Zen, and modern Western philosophy do not, in the main, reject logic or reasoning as such; rather, they target the conceptualization of logic as a universal, context-independent, and singular normative standard. The analysis shows that objections raised at ontological, epistemic, linguistic, methodological, and political levels converge on a set of shared concerns rooted in the structural features of logic itself, most notably categorical determinacy, universalist assumptions, and claims of context-independence. At the same time, these …

The Modal Catuṣkoṭi: Formalizing Emptiness In Classical Modal Logic, Chance Chapman

Comparative Philosophy

Nāgārjuna’s arguments in the Mūlamadhyamakakārikā rely extensively on modus tollens, yet this inference is invalid in the paraconsistent frameworks typically used to formalize the catuṣkoṭi. We develop a modal extension of Belnap-Dunn semantics to address this, expressed as: ¬□(v(P) = t) ∧ ¬□(v(P) = f) ∧ ¬□(v(P) = b) ∧ ¬□(v(P) = n). This formula applies self-reflexively, resolves longstanding interpretive debates regarding verses within Nāgārjuna’s Mūlamadhyamakakārikā and Vigrahavyāvartanī such as his “no thesis” defense, and bridges interpretive gaps between traditionally designated Eastern and Western logical traditions, providing a Western analytical translation of Indian dialectic into symbolic modal logic that …

Somatic Prosthetics For Planetary Resonance, Disha Dharesh Kumar R

Masters Theses

This thesis investigates how industrial design can transcend extractive technological paradigms in favor of relational interfaces that foster planetary attunement. Framing the climate crisis as a 'crisis of imagination', the research challenges the Western bifurcation of nature and culture by drawing on Indic cosmologies- which recognize stones, plants, and ecosystems as conscious at different levels and participants in a shared cosmic field. By synthesizing research in Biosemiotics, Quantum Information Pansycishm, and Neuroscience, the project redefines intelligence as a distributed, more-than-human phenomenon.

The research materializes as a speculative design artifact: a device that functions as a somatic prosthetic for planetary resonance. …

A Modal Analysis Of Human Dignity, Alexander S. Eberspacher

Theses/Capstones/Creative Projects

This work examines the history of the human rights tradition and philosophically compares it to the natural law tradition as most pre-eminently developed in Catholicism. It then turns to examine how Catholicism offers a philosophically sound base for universal dignity through the light of Saul Kripke’s modal work, while the Universal Declaration of Human Rights cannot account for such.

Redefining Certainty: Non-Euclidean Geometry And Theology Transformation Throughout The Intellectual Unrest Of The Early 1800s, Luke Bensinger

Honors Theses

To bridge the gap between mathematics and theology, it is necessary to explore their intersection and challenge the notion that these fields are incompatible. This study focuses on the 19th century, a period when non-Euclidean geometries emerged and disrupted mathematical certainty, while Protestant theologians such as Barton W. Stone and Alexander Campbell grappled with Calvinism and shifting theological perspectives. By analyzing mathematicians studying geometry, such as Gauss, Lobachevsky, and Riemann, this research examines how both disciplines balance change with enduring truths.

Why Read The Classics (Of Mathematical Proofs)?, Cosimo Perini Brogi

Journal of Humanistic Mathematics

Inspired by Italo Calvino’s (almost) homonymous writing, this short essay suggests a formal perspective to complement the typical categorisation of classical results in mathematics on aesthetic bases. The traditional proof that √2 is not a rational number provides a simple example of proof mining. When studied through the tools of logic, that proof has also led to designing new systems for formal proving based on cyclic proofs. This example thus supports the central thesis of this essay that reading the classics of mathematical proofs enables us to acquire a taste for mathematical beauty and can open new directions in mathematics …

A Cross-System Analysis Of Structural Commensurability Between Axiomatic Foundations Of Set Theory (Zfc) And The Principles Of Transcendent Wisdom, Mohammadjavad Maarefvand

Comparative Philosophy

Building on observed similarities in uniqueness proofs for the empty set and the Necessary Existent (Wājib al-Wujūd), this study engages the meta-methodological issue of cross-system commensurability. It identifies four instances of structural commensurability between Zermelo-Fraenkel set theory with Choice (ZFC) and Mulla Sadra's Transcendent Wisdom (Ḥikmat al-Muta'āliyah). First, identity principles: the Axiom of Extensionality provides a formal operational criterion analogous to the philosophical principle of identity through absence of distinguishing features. Second, multiplicity mechanisms: ZFC's constructor axioms (bottom-up construction) and emanation principles (top-down explanation) address analogous problems with inverse orientations. Third, transcending limitation: the Axiom of Infinity ensuring quantitative extensional …

Challenging The Double-Negation Principle: An Analysis Of Strategic Theories Of Sun Zi, Carnot, And Clausewitz Through Intuitionist Logic, Antonino Drago

Comparative Philosophy

To study a subject similarly represented in Eastern and Western cultures, I examine some classical theories of war strategy according to a new analytical method. This method is based on the recognition that the most important strategic theories are organized as theories essentially not deduced from axioms according to classical logic. Such theories reason through doubly negated propositions (DNPs), whose meanings differ from those of their affirmative counterparts. This corresponds to the use of intuitionist logic instead of classical logic. An inspection of the works of the texts of Sun Zi, Lazare Carnot, and Clausewitz reveals many such propositions, and …

The Influence Of Scale In Modeling Social Vulnerability And Disaster Assistance, Sina Razzaghi Asl, Oronde Drakes, Eric Tate, Samuel Brody, Wesley Highfield, Kayode Atoba

Political Science & Geography Faculty Publications

Understanding how social vulnerability relates to disaster impacts is critical for addressing social equity, yet the role of spatial scale in this relationship is often overlooked. Most studies use aggregated data, risking ecological fallacy-misinterpreting individual outcomes from group-level data. This study examines how spatial scale influences the relationship between social vulnerability and federal disaster assistance after Hurricane Harvey. Using spatial econometric models at both household and census tract levels, we assessed the strength of key vulnerability indicators in explaining disaster assistance. Results show that disability, housing tenure, household size, and income predict assistance at the household level, but their influence …

The Principle Of Tolerance And Logical Pluralism, Teresa Kouri Kissel

Philosophy Faculty Publications

Logical pluralism is the view that there is more than one correct logic. This is in contrast with logical monism, which holds that only one is, and logical nihilism, which holds that none are. Monists are typically folks who defend a single logic, like Michael Dummett (2000) or Alan Ross Anderson and Nuel D. Belnap (1975), as correct for guiding deductive reasoning.

Auseinandersetzung: Heidegger's Black Notebooks And Intercultural Engagement, Steven Burik

Research Collection School of Social Sciences

Heidegger’s relation to Asian philosophy should be guided by the notion of Auseinandersetzung. Often translated as con-frontation, this term plays a pivotal role in understanding Heidegger’s engagements with Asian philosophy. Auseinandersetzung (and related terms) can help us realise the changed perspectives on our own thinking that we derive from engagements with culturally and/or historically other types of thinking. Heidegger warns us against facile endeavours of comparing and contrasting, which, as a form of comparative philosophy, he sees as a kind of dialectics unworthy of true thinking. Thinking through the term Auseinandersetzung allows us to reflect more meaningfully on comparative philosophy.

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, …

Magicicada And Mathematics: What The Parochiality Of Our Explanations Means For The Indispensability Argument, Jadyn Malone

Philosophy Senior Theses

This thesis examines whether the indispensability argument justifies belief in the existence of mathematical entities. I adopt ontological naturalism, arguing that we should believe in entities that feature in our best scientific theories, which are determined through inference to the best explanation. The indispensability argument contends that because mathematical entities are explanatorily indispensable to our best scientific theories – as demonstrated by examples like the prime number explanation of cicada life cycles – we should believe they exist just as we believe in electrons and stars.

However, I identify a challenge: our scientific explanations are inherently parochial, shaped by human …

On The Challenges Posed By Non-Machian Solutions To Relationalism In General Relativity, Zachary Matthew Zito

Undergraduate Honors Capstone Projects

The relationship between Mach’s Principle and solutions to Einstein’s field equations is examined, with special attention to Friedmann-Lematre-Robertson-Walker (FLRW) cosmology. Mach’s Principle is outlined, and the extent to which general relativity fulfills Machs vision is assessed. It is argued that several important solutions to Einsteins equations, particularly the FLRW cosmological models, embody key Machian features. In order to elucidate the FLRW cosmological model, the associated energy-momentum tensor for a perfect fluid under the assumptions of large-scale homogeneity and isotropy is derived and used to obtain the Friedmann equations that govern cosmic expansion. It is shown that FLRW cosmology can be …

01. Harry Stottlemeier's Discovery (Novel - Ebook), Matthew Lipman

Middle School Curriculum

One day Harry finds himself giving the wrong answer in science class and begins to wonder where he has gone wrong. This reflection soon involves his classmates, who begin to think together about the nature of thinking, inquiry and knowledge. With the help of their teacher, Harry and his classmates discover rules of formal and informal logic, relational logic and hypothetical thinking—-not as ends in themselves, but as tools in helping them understand themselves and their world. Some of the ideas they begin to explore this way include education, mind, rights, religion, art, cause and effect, causes and reasons, and …

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 …

Unlocking The Power Of Logic: Thinking Smarter Every Day A Beginner's Guide To Logical Thinking, Jesus Diaz

A with Honors Projects

Power Point presented to an 11th and 12th grade high school audience, introducing them to the concept of logic.

Mathematics And Determinism: Chaos, Quantum Mechanics, And The Limits Of Predictive Structure, Jackson T. Salumbides

CMC Senior Theses

This thesis examines the relationship between mathematics and determinism by analyzing how chaos theory, quantum mechanics, and formal mathematical limits challenge traditional conceptions of predictability and causal structure. Chaos theory shows that deterministic systems can exhibit practical unpredictability due to sensitivity to initial conditions. Quantum mechanics introduces probabilistic outcomes that complicate deterministic interpretation, though alternative frameworks such as Bohmian mechanics and superdeterminism attempt to restore determinism at conceptual cost. Additionally, results from mathematical logic, including Gödel’s incompleteness theorems and Turing’s undecidability, demonstrate intrinsic limitations on what can be deduced or computed, even in fully deterministic systems. By synthesizing these areas, …

The Grim Reaper Argument Is An Argument For No One, Daniel Linford, Alex Malpass

Philosophy Faculty Publications

According to a prominent philosophical argument for the beginning of time – the Grim Reaper Argument (GRA) – had it been possible for the past to be infinite, then a provably impossible scenario could have been constructed. Such a scenario is not possible, so, the GRA concludes, the past cannot be infinite. Here, we show that the GRA includes one premise acceptable only for Humeans and another premise acceptable only for anti-Humeans. Since, plausibly, everyone is either a Humean or an anti-Humean, the GRA is an argument for no one. Additionally, we argue that there may not be a way …

Polynomial Invariants For Classical Invariant Groups, John Diller

UNF Graduate Theses and Dissertations

The classical invariant theory of matrix groups plays a fundamental role in modern algebra and geometry, with origins in Hilbert’s Fourteenth Problem concerning the finite generation of rings of invariants. In this thesis, we study polynomial functions on spaces of matrices and investigate their invariants under natural actions of classical groups. After recalling the geometric and algebraic motivations behind Hilbert’s question, we focus on invariants for several complex Lie groups, including the general linear group GL(V), the symplectic group Sp(V), the orthogonal group O(V), and the special orthogonal group SO(V). Building on foundational work of Hermann Weyl and a modern …

Intuitionism, Justification Logic, And Doxastic Reasoning, Vincent A. Peluce

Dissertations, Theses, and Capstone Projects

In this Dissertation, we examine a handful of related themes in the philosophy of logic and mathematics. We take as a starting point the deeply philosophical, and—as we argue, deeply Kantian—views of L.E.J. Brouwer, the founder of intuitionism. We examine his famous first act of intuitionism. Therein, he put forth both a critical and a constructive idea. This critical idea involved digging a philosophical rift between what he thought of himself as doing and what he thought of his contemporaries, specifically Hilbert, as doing. He sought to completely separate mathematics from mathematical language, and thereby logic. In chapter 3, we …

Avoiding Pragmatic Oddity: A Bottom-Up Defeasible Deontic Logic, Guido Governatori, Silvano Colombo Tosatto, Antonio Rotolo

Centre for Computational Law (2022-2025)

This paper presents an extension of Defeasible Deontic Logic to deal with the Pragmatic Oddity problem. The logic applies three general principles: (i) the Pragmatic Oddity problem must be solved within a general logical treatment of contrary-to-duty (CTD) reasoning; (ii) non-monotonic methods must be adopted to handle CTD reasoning; (iii) logical models of CTD reasoning must be computationally feasible and, if possible, efficient. The proposed extension of Defeasible Deontic Logic elaborates a preliminary version of the model proposed by Governatori and Rotolo [15]. The previous solution was based on particular characteristics of the (constructive, top-down) proof theory of the logic. …

A Thesis, Or Digressions On Sculptural Practice: In Which, Concepts & Influences Thereof Are Explained, Set Forth, Catalogued, Or Divulged By Way Of Commentaries To A Poem, First Conceived By The Artist, Fed Through Chatg.P.T., And Re-Edited By The Artist, To Which Are Added, Annotated References, Impressions And Ruminations Thereof, Also Including Private Thoughts & Personal Accounts Of The Artist, Jaimie An

Masters Theses

This thesis is an exercise in, perhaps a futile, attempt to trace just some of the ideas, stories, and musings I might meander through in my process. It’s not quite a map, nor is it a neat catalogue; it is a haphazard collection of tickets and receipts from a travel abroad, carelessly tossed in a carry-on, only to be stashed upon returning home. These ideas are derived from much greater thinkers and authors than myself; I am a mere collector or a translator, if that, and not a very good one, for much is lost. I do not claim comprehensive …

Canonical Extensions Of Quantale Enriched Categories, Alexander Kurz

MPP Research Seminar

No abstract provided.

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 …

“What Line Can’T Be Measured With A Ruler?” Riddles And Concept-Formation In Mathematics And Aesthetics, William H. Brenner, Samuel J. Wheeler

Philosophy Faculty Publications

We analyze two problems in mathematics – the first (stated in our title) is extracted from Wittgenstein’s “Philosophy for Mathematicians”; the second (“What set of numbers is non-denumerable?”) is taken from Cantor. We then consider, by way of comparison, a problem in musical aesthetics concerning a Brahms variation on a theme by Haydn. Our aim is twofold: first, to bring out and elucidate the essentially riddle-like character of these problems; second, to show that the comparison with riddles does not reduce their solution to an exercise in bare subjectivity

Deontic Meta-Rules, Francesco Olivieri, Guido Governatori, Matteo Cristani, Antonino Rotolo, Abdul Sattar

Centre for Computational Law (2022-2025)

The use of meta-rules in logic, i.e., rules whose content includes other rules, has recently gained attention in the setting of non-monotonic reasoning: a first logical formalisation and efficient algorithms to compute the (meta)-extensions of such theories were proposed in Olivieri et al. (2021, Computing defeasible meta-logic. In JELIA 2021, LNCS, vol. 12678, pp. 69-84. Springer.). This work extends such a logical framework by considering the deontic aspect. The resulting logic will not just be able to model policies but also tackle well-known aspects that occur in numerous legal systems. The use of Defeasible Logic to model meta-rules in the …

Spatial Relations Between Conscious And Unconscious Thought, John Shannon Hendrix

Architecture, Art, and Historic Preservation Faculty Publications

No abstract provided.

Alternative Ontology For Ante Rem Structuralism Based On Huayan Buddhist Metaphysics, Chul Soon Hwang

CMC Senior Theses

Ante rem structuralism is a version of mathematical structuralism presented by Shapiro (1997) that asserts the existence of mathematical objects. It stands out amongst other structuralist views in that it secures a face value semantics for mathematical statements. In this paper, I propose an alternative ontology where structures are fully explained by relations. I argue that this alternative ontology avoids arguments against ante rem structuralism’s endorsement of indiscernible entities while retaining the convenience of a face value semantics for mathematical expressions.

This ontology is based on mereological nihilism and the Huayan Buddhist argument that an object can be fully described …