Methods in Mathematical Philosophy

Course Mechanics

Course Description

Logic, set theory, and probability theory are critical tools for philosophers, and there continues to be active research, debate, and discovery at the foundations of these disciplines. This course is an introduction to the these three areas of mathematics, with the aim of providing students with the necessary formal skills and domain knowledge to complete the masters degree in logic and philosophy of science at LMU. Although I assume students have some passing familiarity with propositional logic, no background in mathematics is assumed. So please do not worry, and leave your anxiety with respect to mathematics at the door.

Course Goals

The course has two central goals. First, students should be able to define and explain the importance of certain mathematical terms that occur regularly in logic, philosophy of science, and philosophy of mathematics. Such terms include, for example, completeness, decidability, uncountable vs. countable sets, the probability axioms, and conditional probability. Second, students should be able to to explain how logic, set theory, and probability theory been used to address long-standing problems in analytic philosophy, and they should begin to be able to apply these tools to philosophical problems in their own area of research.

Course Requirements

There are problem sets due every week and a final exam. Some of you may find these exercises and the exam difficult. Please do not worry. Many philosophy students are not accustomed to struggling with weekly exercises, some of which they cannot complete. This is because active discussion sessions and writing papers are, perhaps, the best ways to learn most areas of philosophy. In contrast, the best way to acquire formal skills is by repeatedly practicing such skills and by studying definitions and proofs. For the overwhelming majority of people, mathematical ability is learned; it is not innate. If you practice problem-solving and if you study the arguments and definitions that we discuss in class, you will learn the material. By the end of the class, you will be able to accomplish the above outlined goals. Conversely, if you do not solve problems and study, there is a non-trivial chance that you will not learn the material.


Each week, you must turn in your problem set at the beginning in class. Unless you are granted an extension (see below), late work will be penalized 10% for every day that it is late. You are permitted to work with other students on the assignments, and feel free to ask for my help as well. However, you should always write the final copy of your problem set alone; doing so ensures that you understand the material. If you collaborate with other students, please write the names of the other students with whom you worked at the top of your problem set.

Twice during the course of the semester, I will grant you a two-day an extension on your problem set, no questions asked. That is, instead of turning the problem set in on Tuesday, you may turn it in some time on Thursday by bringing it to my office or Maria Csauscher in Room 134 in Ludwigstraße 31. If you need more than two additional days to complete an assignment, please talk to me.


Your final grade will be calculated via a weighted average using the following weights:

If you earn betweem 95% and 100%, your final grade will be a 1.0; between a 90% and 95% will constitute a 1.3, and so on.

Course Files


Below is a table indicating readings and assignments that are due each class. If you are a registered student in the class, then you can download the readings from the link in the "Course Files" section above.

Date Topic Readings Assignment
15/10 Propositional Logic: Semantics

In-Class Exercises
Logic and Proofs. Chapters 1-3. None
22/10 Propositional Logic: Syntax

Logic and Proofs. Chapters 4-6. Problem Set 1
29/10 Propositional Logic: Metamathematics

Logic and Proofs. Chapter 7.
Course notes.
Problem Set 2
5/11 Predicate Logic: Semantics

Logic and Proofs. Chapters 8-9. Problem Set 3
12/11 Predicate Logic: Syntax.

Logic and Proofs. Chapters 10-12. Problem Set 4
19/11 Introduction to Set Theory: Axioms, Boolean Algebra, Construction of the Natural Numbers

Hrbacek and Jech. Introduction to Set Theory. Chapter 1, 3.1-3.2. Problem Set 5
26/11 Orderings, Equivalence Relations, and Functions

Functions and Computations. Chapters 2-3. Problem Set 6
3/12 Injections, Bijections, and Sizes of Sets

Functions and Computations. Chapter 4. Problem Set 7
10/12 Predicate Logic: Metamathematics I

Course Notes. Problem Set 8
17/12 Predicate Logic: Metamathematics II

Course Notes Problem Set 9
7/1 Introduction to Probability: Axioms and Combinatorics

DeGroot. Sections 1.1-1.8 Problem Set 10
14/1 Conditional Probability and Independence

DeGroot. Sections 1.10-1.11, 2.1-2.2 Problem Set 11
21/1 Random Variables and Special Distributions

DeGroot. Sections 3.1-3.5. Problem Set 12
28/1 Expectation, Variance, and Covariance

DeGroot. Sections 4.1-4.3, 4.6, 4.8-4.9. None
4/3 Final Exam