Thesis (D.A.)--Illinois State University, 1988. / Title from title page screen, viewed September 16, 2005. Dissertation Committee: Michael Plantholt (chair), John Dossey, Patricia Klass, Albert Otto, Charles Vanden Eynden. Includes bibliographical references (leaves 172-173) and abstract. Also available in print.

Thesis (Ph.D.) - University of St Andrews, April 2010.

No description available.

The model-theoretic analysis of the concept of logical consequence has come under heavy criticism in the last couple of decades. The present work looks at an alternative approach to logical consequence where the notion of inference takes center stage. Formally, the model-theoretic framework is exchanged for a proof-theoretic framework. It is argued that contrary to the traditional view, proof-theoretic semantics is not revisionary, and should rather be seen as a formal semantics that can supplement model-theory. Specifically, there are formal resources to provide a proof-theoretic semantics for both intuitionistic and classical logic. We develop a new perspective on proof-theoretic harmony for logical constants which incorporates elements from the substructural era of proof-theory. We show that there is a semantic lacuna in the traditional accounts of harmony. A new theory of how inference rules determine the semantic content of logical constants is developed. The theory weds proof-theoretic and model-theoretic semantics by showing how proof-theoretic rules can induce truth-conditional clauses in Boolean and many-valued settings. It is argued that such a new approach to how rules determine meaning will ultimately assist our understanding of the apriori nature of logic.

In this thesis we examine the relationship between hypersequent and some types of labelled sequent calculi for a subset of intermediate logics—logics between intuitionistic (Int), and classical logics—that have geometric Kripke semantics, which we call Int∗/Geo. We introduce a novel calculus for a fragment of first-order classical logic, which we call partially-shielded formulae (or PSF for short), that is adequate for expressing the semantic validity of formulae in Int∗/Geo, and apply techniques from correspondence theory to provide translations of hypersequents, simply labelled sequents and relational sequents (simply labelled sequents with relational formulae) into PSF. Using these translations, we show that hypersequents and simply labelled sequents for calculi in Int∗/Geo share the same models. We also use these translations to justify various techniques that we introduce for translating simply labelled sequents into relational sequents and vice versa. In particular, we introduce a technique called "transitive unfolding" for translating relational sequents into simply labelled sequents (and by extension, hypersequents) which preserves linear models in Int∗/Geo. We introduce syntactic translations between hypersequent calculi and simply labelled sequent calculi. We apply these translations to a novel hypersequent framework HG3ipm∗ for some logics in Int∗/Geo to obtain a corresponding simply labelled sequent framework LG3ipm∗, and to an existing simply labelled calculus for Int from the literature to obtain a novel hypersequent calculus for Int. We introduce methods for translating a simply labelled sequent calculus into a cor- responding relational calculus, and apply these methods to LG3ipm∗ to obtain a novel relational framework RG3ipm∗ that bears similarities to existing calculi from the literature. We use transitive unfolding to translate proofs in RG3ipm∗ into proofs in LG3ipm∗ and HG3ipm∗ with the communication rule, which corresponds to the semantic restriction to linear models.

In contemporary formal theory of truth, model-theoretic and non-classical approaches have been dominant. I rather pursue the so-called classical axiomatic approaches toward truth and my dissertation begins by arguing for the classical axiomatic approach and against the others. The classical axiomatic approach inevitably leads to abandonment of the nave conception of truth and revision of the basic principles of truth derived from that nave conception such as the full T-schema. In the absence of the general guiding principles based on that nave conception, we need to conduct tedious but down-to-earth eld works' of various theories of truth by examining and comparing them from various points of view in searching for satisfactory theories of truth. As such attempt, I raise two new criteria for comparison of truth theories, make a proof-theoretic study of them in connection to the foundation of mathematics.

This thesis is about automated reasoning in quantified modal and temporal logics, with an application to formal methods. Quantified modal and temporal logics are extensions of classical first-order logic in which the notion of truth is extended to take into account its necessity or equivalently, in the temporal setting, its persistence through time. Due to their high complexity, these logics are less widely known and studied than their propositional counterparts. Moreover, little so far is known about their mechanisability and usefulness for formal methods. The relevant contributions of this thesis are threefold: firstly, we devise a sound and complete set of sequent calculi for quantified modal logics; secondly, we extend the approach to the quantified temporal logic of linear, discrete time and develop a framework for doing automated reasoning via Proof Planning in it; thirdly, we show a set of experimental results obtained by applying the framework to the problem of Feature Interactions in telecommunication systems. These results indicate that (a) the problem can be concisely and effectively modeled in the aforementioned logic, (b) proof planning actually captures common structures in the related proofs, and (c) the approach is viable also from the point of view of efficiency.

This study of student-to-student discussions focuses on a single inquiry-oriented transition to proof course. Mathematical proof is essential to a strong mathematics education but very often students complete their mathematics studies with limited abilities to construct and validate mathematical proofs (c.f. Harel & Sowder, 1998; Knuth, 2002; Almeida, 2000). The role of mathematical proof in education is to provide explanation and understanding. Both the research on mathematical discourse and the standards of the NCTM claim that participation in mathematical discourse provides opportunities for understanding. Although this link has been established, there is very little research on the role of students and the instructor during discussions on student generated proofs at the undergraduate level -- particularly in inquiry-oriented classes. This research analyzes the types of discussions that occurred in an inquiry-oriented undergraduate mathematics course in which proof was the main content. The discussions of interest involved at least two student participants and at least three separate utterances. These discussions fell along a continuum based on the level of student interaction. As a result of this research, the four main discussion types that were present in this course have been described in detail with a focus on the roles of the instructor and the students. The methodology for this research is qualitative in nature and is an exploratory case study. The data used for this research was video tapes of two to three class sessions per week of an Introduction to Number Theory course taught in the fall of 2005. / text

In “Proof-Theoretic Justification of Logic”, building on work by Dummett and Prawitz, I show how to construct use-based meaning-theories for the logical constants. The assertability-conditional meaning-theory takes the meaning of the logical constants to be given by their introduction rules; the consequence-conditional meaning-theory takes the meaning of the logical constants to be given by their elimination rules. I then consider the question: given a set of introduction (elimination) rules \(\mathcal{R}\), what are the strongest elimination (introduction) rules that are validated by an assertability (consequence) conditional meaning-theory based on \(\mathcal{R}\)? I prove that the intuitionistic introduction (elimination) rules are the strongest rules that are validated by the intuitionistic elimination (introduction) rules. I then prove that intuitionistic logic is the strongest logic that can be given either an assertability-conditional or consequence-conditional meaning-theory. In “Grounding Grounding” I discuss the notion of grounding. My discussion revolves around the problem of iterated grounding-claims. Suppose that \(\Delta\) grounds \(\phi\); what grounds that \(\Delta\) grounds that \(\phi\)? I argue that unless we can get a satisfactory answer to this question the notion of grounding will be useless. I discuss and reject some proposed accounts of iterated grounding claims. I then develop a new way of expressing grounding, propose an account of iterated grounding-claims and show how we can develop logics for grounding. In “Is the Vagueness Argument Valid?” I argue that the Vagueness Argument in favor of unrestricted composition isn’t valid. However, if the premisses of the argument are true and the conclusion false, mereological facts fail to supervene on non-mereological facts. I argue that this failure of supervenience is an artifact of the interplay between the necessity and determinacy operators and that it does not mean that mereological facts fail to depend on non-mereological facts. I sketch a deflationary view of ontology to establish this. / Philosophy

In this dissertation it is shown that abstract data types (ADTs) can be specified by the Classified Model (CM) specification language - a first-order Horn language with equality and sort "classification" assertations. It is shown how these sort assertations generalize the traditional syntactic signatures of ADT specifications, resulting in all of the specification capability of traditional equational specifications, but with the improved expressibility of the Horn-with-equality language and additional theorem proving applications such as program synthesis. This work extends corresponding results from Many Sorted Algebra (MSA), Order Sorted Algebra (OSA) and Order Sorted Model (OSM) specification techniques by promoting their syntactic signatures to assertions in the Classified Model Specification language, yet retaining sorted quantification. It is shown how this solves MSA problems such as error values, polymorphism and subtypes in a way different from the OSA and OSM solutions. However, the CM technique retains the MSA and order sorted approach to parameterization. The CS generalization also suggests the use of CM specifications to axiomatize modules as a generalization of variables within Hoare Logic, with application to a restricted, but safe, use of procedures as state changing operations and functions as value returning operations of a module. CM proof theory and semantics are developed, including theorems for soundness, completeness and the existence of a free model.