Tools for Tutoring Theoretical Computer Science Topics

This thesis introduces COMPLEXITY TUTOR, a tutoring system to assist in learning abstract proof-based topics, which has been specifically targeted towards the population of computer science students studying theoretical computer science. Existing literature has shown tremendous educational benefits produced by active learning techniques, student-centered pedagogy, gamification and intelligent tutoring systems. However, previously, there had been almost no research on adapting these ideas to the domain of theoretical computer science. As a population, computer science students receive immediate feedback from compilers and debuggers, but receive no similar level of guidance for theoretical coursework. One hypothesis of this thesis is that immediate feedback while working on theoretical problems would be particularly well-received by students, and this hypothesis has been supported by the feedback of students who used the system. This thesis makes several contributions to the field. It provides assistance for teaching proof construction in theoretical computer science. A second contribution is a framework that can be readily adapted to many other domains with abstract mathematical content. Exercises can be constructed in natural language and instructors with limited programming knowledge can quickly develop new subject material for COMPLEXITY TUTOR. A third contribution is a platform for writing algorithms in Python code that has been integrated into this framework, for constructive proofs in computer science. A fourth contribution is development of an interactive environment that uses a novel graphical puzzle-like platform and gamification ideas to teach proof concepts. The learning curve for students is reduced, in comparison to other systems that use a formal language or complex interface. A multi-semester evaluation of 101 computer science students using COMPLEXITY TUTOR was conducted. An additional 98 students participated in the study as part of control groups. COMPLEXITY TUTOR was used to help students learn the topics of NP-completeness in algorithms classes and prepositional logic proofs in discrete math classes. Since this is the first significant study of using a computerized tutoring system in theoretical computer science, results from the study not only provide evidence to support the suitability of using tutoring systems in theoretical computer science, but also provide insights for future research directions

Noncomparabilities & Non Standard Logics

Many normative theories set forth in the welfare economics, distributive justice and cognate literatures posit noncomparabilities or incommensurabilities between magnitudes of various kinds. In some cases these gaps are predicated on metaphysical claims, in others upon epistemic claims, and in still others upon political-moral claims. I show that in all such cases they are best given formal expression in nonstandard logics that reject bivalence, excluded middle, or both. I do so by reference to an illustrative case study: a contradiction known to beset John Rawls\u27s selection and characterization of primary goods as the proper distribuendum in any distributively just society. The contradiction is avoided only by reformulating Rawls\u27s claims in a nonstandard form, which form happens also to cohere quite attractively with Rawls\u27s intuitive argumentation on behalf of his claims

Noncomparabilities & Non Standard Logics

Many normative theories set forth in the welfare economics, distributive justice and cognate literatures posit noncomparabilities or incommensurabilities between magnitudes of various kinds. In some cases these gaps are predicated on metaphysical claims, in others upon epistemic claims, and in still others upon political-moral claims. I show that in all such cases they are best given formal expression in nonstandard logics that reject bivalence, excluded middle, or both. I do so by reference to an illustrative case study: a contradiction known to beset John Rawls\u27s selection and characterization of primary goods as the proper distribuendum in any distributively just society. The contradiction is avoided only by reformulating Rawls\u27s claims in a nonstandard form, which form happens also to cohere quite attractively with Rawls\u27s intuitive argumentation on behalf of his claims

Knowledge, lies and vagueness : a minimalist treatment

Minimalism concerning truth is the view that that all there is to be said concerning truth is exhausted by a set of basic platitudes. In the first part of this thesis, I apply this methodology to the concept of knowledge. In so doing, I develop a model of inexact knowledge grounded in what I call minimal margin for error principles. From these basic principles, I derive the controversial result that epistemological internalism and internalism with respect to self-knowledge are untenable doctrines. In the second part of this thesis, I develop a minimal theory of vagueness in which a rigorous but neutral definition of vagueness is shown to be possible. Three dimensions of vagueness are distinguished and a proof is given showing how two of these dimensions are equivalent facets of the same phenomenon. From the axioms of this minimal theory one can also show that there must be higher-order vagueness, contrary to what some have argued. In the final part of this thesis, I return to issues concerning the credentials of truth-minimalism. Is truth-minimalism compatible with the possibility of truth-value gaps? Is it right to say that truth-minimalism is crippled by the liar paradox? With respect to the former question, I develop a novel three-valued logical system which is both proof-theoretically and truth-theoretic ally well-motivated and compatible with at least one form of minimalism. With respect to the second question, a new solution to the liar paradox is developed based on the claim that while the liar sentence is meaningful, it is improper to even suppose that this sentence has a truth-status. On that basis, one can block the paradox by restricting the Rule of Assumptions in Gentzen-style presentations of the sentential sequent calculus. The first lesson of the liar paradox is that not all assumptions are for free. The second lesson of the liar is that, contrary to what has been alleged by many, minimalism concerning truth is far better placed than its rival theories to solve the paradox

Existence, knowledge & truth in mathematics

This thesis offers an overview of some current work in the philosophy of mathematics, in particular of work on the metaphysical, epistemological, and semantic problems associated with mathematics, and it also offers a theory about what type of entities numbers are. Starting with a brief look at the historical and philosophical background to the problems of knowledge of mathematical facts and entities, the thesis then tackles in depth, and ultimately rejects as flawed, the work in this area of Hartry Field, Penelope Maddy, Jonathan Lowe, John Bigelow, and also some aspects of the work of Philip Kitcher and David Armstrong. Rejecting both nominalism and physicalism, but accepting accounts from Bigelow and Armstrong that numbers can be construed as relations, the view taken in this work is that mathematical objects, numbers in particular, are universals, and as such are mind dependent entities. It is important to the arguments leading to this conception of mathematical objects, that there is a notion of aspectual seeing involved in mathematical conception. Another important feature incorporated is the notion, derived from Anscombe, of an intentional object. This study finishes by sketching what appears to be a fruitful line of enquiry with some significant advantages over the other accounts discussed. The line taken is that the natural numbers are mind dependent intentional relations holding between intentional individuals, and that other classes of number - the rationals, the reals, and so on - are mind dependent intentional relations holding between other intentional relations. The distinction in type between the natural numbers and the rest, is the intuitive one that is drawn naturally in language between the objects referred to by the so-called count nouns, and the objects referred to by the so-called mass nouns

Modes of Truth

The aim of this volume is to open up new perspectives and to raise new research questions about a unified approach to truth, modalities, and propositional attitudes. The volume’s essays are grouped thematically around different research questions. The first theme concerns the tension between the theoretical role of the truth predicate in semantics and its expressive function in language. The second theme of the volume concerns the interaction of truth with modal and doxastic notions. The third theme covers higher-order solutions to the semantic and modal paradoxes, providing an alternative to first-order solutions embraced in the first two themes. This book will be of interest to researchers working in epistemology, logic, philosophy of logic, philosophy of language, philosophy of mathematics, and semantics