Model-Theoretic Approaches to Theoretical Equivalence and Reduction

This dissertation centers around the following three questions: 1. What are formal theories about? 2. What does it mean to say that two formal theories are equivalent? 3. What does it mean to say that a formal theory is reducible to another?

This dissertation seeks to better understand possible answers to these questions by employing methods and tools from model theory, a branch of mathematical logic that studies the relationship between formal languages and the mathematical structures that provide such languages with meaning. At the same time, this dissertation evaluates the philosophical import of such methods and tools.

Chapter 1 provides technical preliminaries. In Chapter 2, I motivate a a distinctively 'semantic' view of mathematical theories according to which formalized mathematical statements are taken to specify conditions which describe a variety of possibly non-isomorphic structures.

In Chapters 3 and 4, I argue that there is good reason to privilege bi-interpretability as a criterion for theoretical equivalence. In particular, Chapter 3 argues that bi-interpretability offers the most promising notion of theoretical equivalence, especially when compared to its contrast class. Chapter 4 critically discusses the notion of Morita equivalence, a criterion for comparing many-sorted theories recently defined in the philosophical literature.

Chapter 5 advances a 'model-theoretic' approach to logicism which consists of two parts: (1) a model-theoretic conception of reduction; and (2) a model-theoretic conception of logicality. I discuss how this model-theoretic approach to logicism aims to realize a broadened version of the classical logicist position (as originally conceived by Gottlob Frege and Richard Dedekind) that is still worth pursuing.