Recent Doctoral Dissertations in Mathematical Logic

Since the 1970s, considerable progress has been made in point-free topology using locale theory, in both the classical and the impredicative constructive context. In contrast, the predicative approach to point-free topology, known as formal topology, remains some way behind impredicative locale theory.
One of the principal aims of this thesis is to show how some of the key ideas and results from locale theory can be transferred to formal topology, working in Constructive Zermelo-Fraenkel Set Theory (CZF). Building on the work of [7], we also examine, in the constructive predicative case, the relationship between various structures in point-set and point-free topology.
In Chapter 6 we study uniform formal topologies; as well as making a contribution to the predicative theory, some of this work goes beyond what was previously known about uniform locales in the impredicative constructive case.
Chapter 7 explores the relatively new field of the Basic Picture, including the study of formal topologies with a binary positivity predicate. We prove some results which shed some light on the similarities and differences between this and the traditional approach to point-free topology.

We address the problem of characterizing the choice processes of two like-minded yet non-communicating agents who intend to select, from a finite set of options, the same possible world. Hence, we call the resulting framework "Rationality-as-conformity"
Within the scope of our formalisation, in which a choice problem is defined on a non-empty subset of maps from a finite set $A$ to the binary set ${ 0,1}$, we introduce and investigate three distinct logico-mathematical characterisations of Rationality-as-conformity. Finally, we discuss the applicability of our framework to problems such as pure coordination games and radical interpretation which are traditionally related to "rationality".
The key characterisation results presented throughout Chapters 3--6 appear in \cite{hp05}, whereas parts of Chapters 7--8 have been submitted for publication.
Keywords: Rationality, reasons, coordination, choice functions, radical interpretation, selection of multiple-Nash equilibria, social choice.

The task of inductive reasoning in the first order predicate calculus is considered. A natural generalisation of the Principle of Instantial Relevance is given and a complete characterisation of the functions satisfying this principle, together with other naturally occurring principles, is shown to be a family of discrete probability functions parameterised by a single real variable in [0,1). This continuum bears similarities to Carnap's celebrated lambda-continuum, to which it is compared.
Also considered is the task of induction in first order languages containing binary predicates. We formulate an approach by which sentences in a binary language are interpreted as sentences of some suitable large unary language. This class of functions is shown to depend only on the sizes of classes of 'indistinguishable' individuals. We tread towards a representation theorem for such functions in general.

This thesis introduces a generalization of epsilon probability functions and a new correspondence with rational consequence relations. The rational consequence relations corresponding to the generalized epsilon probability functions of maximum entropy are investigated and it is shown that in this system the analogous notion of `maximum entropy closure' equals the rational closure, thus reconfirming the rational closure as the simplest, least prejudiced rational consequence relation satisfying that knowledge base. Comparisons are drawn with the work of Bamber and a new, more conservative closure is developed which corresponds to Bamber's strong entailment with near surety, thus further validating the approach given.

In classical logic, all inconsistencies are equal and meaningless. In reality, however, some are worse than others. Indeed, there are certain inconsistent belief sets which are quite rational and which have meaningful content. This thesis provides a measure for inconsistency, some proof theories based on that measure, and two measures for inconsistent information. Parts of Chapters 2 and 3 appeared in a shortened form as [29], parts of Chapter 5 are to appear as [30], and parts of Chapter 4 and Appendix B have been submitted for publication.

The main objective of the thesis was to investigate using propositional inference processes to solve some problems of induction in the predicate calculus by identifying finite predicate knoweldge bases with infinite propositional knowledge bases and taking the limiting solutions of theses. In a number of cases it was verified that these limiting solutions exist, and their appropriateness as solutions to the original problem of induction was considered. In the course of this investigation a number of new results were proved on the satisfiability of various principles of uncertain reasoning (in particular the Relativisation Principle) by certain well-known propositional inference processes.

We define the language and some principles of a system of predicate uncertain reasoning. We investigate some aspects of induction, including some generalisations of the principle of instantial relevance. We introduce and look at some consequences of the principle of duality and dual priors, including a characterisation of the duals of some priors satisfying some elementary symmetry conditions.