We introduce subatomic logic, a new methodology where by looking inside of atoms we are able to represent a wide variety of proof systems in such a way that every rule is an instance of a single, regular, linear rule scheme. We show the generality of the subatomic approach by presenting how it can be applied to several different systems with very different expressivity. In this thesis we use subatomic logic to study two normalisation procedures: cut-elimination and decomposition. In particular, we study cut-elimination by characterising a whole class of substructural logics and giving a generalised cut-elimination procedure for them, and we study decomposition by providing generalised rewriting rules for derivations that we can then apply to decompose derivations and to eliminate cycles.

Thesis (M.S.)--Boise State University, 2009. / Includes abstract. Includes bibliographical references (leaves 52-53).

Thesis (M.S.)--Boise State University, 2009. / Title from t.p. of PDF file (viewed June 15, 2010). Includes abstract. Includes bibliographical references (leaves 52-53).

M.Ed. / Recently there has been renewed interest in proof and proving in schools worldwide. However, many school students and even teachers of mathematics have only superficial ideas on the nature of proof. Proof is considered the heart of mathematics as individuals explore, make conjectures and try to convince themselves and others about the truth or falsity of their conjectures. There are basically two categories of deductive proof, namely proof by direct argument and indirect proofs. The aim of this study was to examine the structural features common to most of the mathematical proofs for formalised mathematical systems, with the emphasis on indirect proof techniques. The main question was to investigate which mathematical activities and logical principles at secondary school level are necessary for students to become proficient with proof writing. A great deal of specialised language is associated with reasoning. Such words as axiom, theorem, proof, and conjecture are just some of the terms that students must understand as they engage in the proof-making task. The formal aspect of mathematics at secondary school is extremely important. It is inevitable that students become involved with hypothetical arguments. They use among others, proofs by contradiction. Furthermore, necessary and sufficient conditions are related to theorems and their converses. It is therefore apparent that the study of logic is necessary already at secondary school level in order to practise mathematics satisfactorily. An analysis of the mathematics syllabus of the Department of Education has indicated that students should use indirect techniques of proof. According to this syllabus students should be familiar with logical arguments. The conclusion which is reached, gives evidence that students’ background in logic is completely lacking and inadequate. As a result they cannot cope adequately with argumentation and this causes a poor perception of what mathematics entails. Although proof writing can never be reduced to a mechanical process, considerable anxiety and uncertainty can be eliminated from the process if students are exposed to the principles of elementary logic and techniques. Mathematics educators and education researchers have reported students’ difficulties with mathematical proof and point out the conflict between the nature of this essential mathematical activity and current approaches to teaching it. This recent interest has led to an increased effort to teach proof in innovative ways.

Default inheritance reasoning is a propositional approach to non monotonic reasoning designed to model reasoning with natural language generics. Inheritance reasoners model sets of natural language generics as directed acyclicgraphs,and inference corresponds to the specification of paths through those networks. A proliferation of inheritance proof theories exist in the literature along with extensive debate about the most reasonable way to construct inferences, based on intuitions about interpretations of particular inheritance networks. There has not been an accepted semantics for inheritance which unifies the set of possible proof theories, which would help identify truly ill motivated proof theories. This thesis attempts to clarify the inheritance literature in the three ways indicated in the title: psychological plausibility, proof theory and semantics.

Thesis (M. Ed.)--University of Hong Kong, 2001. / Includes bibliographical references (leaves 88-91).

Thesis (Ph. D.)--University of Texas at Austin, 2008. / Vita. Includes bibliographical references.

Univ., Diss.--Freiburg, 2002. / Literaturverz. S. [243] - 247.

Thesis (M.A.)--Brigham Young University. Dept. of Mathematics Education, 2007. / Includes bibliographical references (p. 56-58).

Thesis (M.Ed.)--University of Hong Kong, 2001. / Includes bibliographical references (leaves 88-91). Also available in print.