## The Computational Application of Bilattice Logic to Natural Reasoning

##### Abstract

Chapter 1 looks at natural reasoning. It begins by considering the inferences that people make, particularly in terms of how those inferences differ from what is sanctioned by classical logic. I then consider the role of logic in relation to psychology and compare this relationship with the competence/performance distinction from syntax. I discuss four properties of natural reasoning that I believe are key to any theory: specifically partially, paraconsistancy, relevance and defeasibility. I then discuss whether these are semantic properties or pragmatic ones, and conclude by describing a new view of logic and inference prevalent in some contemporary writings.
Chapter 2 looks at some of the existing formal approaches to the four properties. For each property I present the basic idea in formal terms, and then discuss a number of systems from the literature. Each section concludes with a brief discussion of the importance of the given property in the field of computation.
Chapter 3 develops the formal system used in this thesis. this is an evidential, bilattice-based logic (EBL). I begin by presenting the mathematical preliminaries, and then show how the four properties of natural reasoning can be captured. The details of the logic itself are presented, beginning with the syntax and then moving on to the semantics. The role of pragmatic inferences in the logic is considered and a formal solution is advanced. I conclude by comparing EBL to some of the logics discussed in Chapter 2.
Chapter 4 rounds off Part 1 by considering the implementation of the logic and some of it's computational properties. It begins by considering the application of evidential bilattice logic to logic programming; it extends Fitting's work in this area to construct a programming language, QLOG2. I give some examples of this language in use. The QLOG2 language is then used as a part of the implementation of the EBL system itself: I describe the details of this Implementation and then give some examples of the system in use. The chapter concludes by giving an informal presentation of some basic complexity results for logical closure in EBL, based on the given implementation.
Chapter 5 presents some interesting data from linguistics that reflects some of the principles of natural reasoning; in particular I concentrate on implicatures and presupposition. I begin by describing the data and then consider a number of approaches from both the logical and the linguistic literature.
Chapter 6 uses the logic developed in Chapter 3 to analyse the data presented in Chapter 5. I consider the basic inference cases, and then move on to more complex examples involving contextual interactions. The results are quite successful, and add weight to Mercer's quest for a common logical semantics for entailment and presupposition. All of the examples considered in this chapter can be handled by the implemented system described in Chapter 4.
Finally, Chapter 7 rounds off by presenting some further areas of research that have been raised by this investigation. In particular, the issues of quantification and modality are discussed.