Mathematical and Logical Foundations of Cognitive Science

Fall 2015

General information

Main class: Monday and Wednesday 9:30AM - 10:45AM Cedar Hall (AC) C116.
Lab: Friday 9:05AM - 9:55AM Cedar Hall (AC) C116.
Course website:

Instructor: Ruth Eberle, Ph.D.
Office: Eigenmann (EG) 801.
Email: (include "Q350" in the subject line).
Phone: 856-5722
Office hours: MW 10:45am (immediately after class) in EG801 and by appointment.

Associate Instructor: Zachary Tosi.
Email: (include "Q350" in the subject line).
Office hours: TBA and by appointment.

Course description

Cognitive science aims to provide rigorous explanations of the processes that underlie intelligent behavior. To this end, cognitive scientists employ a wide variety of tools and concepts from logic and various branches of mathematics. In this course, we explore a range of topics in math and logic that are particularly important for cognitive science: propositional and predicate logic, automata theory, and probability theory.

For each topic, our goal is twofold: first, to understand the basic mathematical and conceptual underpinnings; second, to understand the relevance for cognitive science. To address the latter, we consider each mathematical topic as it fits within a certain area of cognitive science. Thus, we consider logic and automata theory as they relate to computational models of the mind, and probability theory in the context of Bayesian modeling. Ideally, then, students will come away from this course with two things: (1) a skill set of basic mathematical tools and (2) appreciation and enthusiasm for how these tools can be used to model and understand the mind. The material for the course is self-contained and no prerequisites beyond a sound high school mathematics background are needed.


There is no one, physical textbook for this course. I will instead draw on a number of resources. Most of these resources will be open access, i.e. freely available online., so that I can share them with you all as well. As I come across useful resources I will direct your attention to them here. Also, if anyone else finds additional resources that are not listed here, please let me know so that I can share them with everyone else!

FC: Carol Critchlow & David Eck (2011). Foundations of Computation. Available free online (download PDF). Used for Units 1 and 2.

NN: Rolf Pfeifer, Dana Damian, and Rudolf Fuchslin (2010). Available free online (download PDF). Used for Unit 3.

IP: Charles M. Grinstead & J. Laurie Snell (2006), Introduction to Probability. Available free online (download PDF).Used for Unit 4.


Week Day Topic Reading Assignment (See Canvas for addditional assignments)
Unit 1: Logic and Proof
Week 1 M Introduction, course logistics, and overview

Propositional logic.

FC §1.1 - 1.2
F Practice Exercises HW 1 Due: FC §1.1 Exercises 1-11.
Week 2 M Disjunctive Normal Form. FC §1.3
W Predicate Logic FC §1.4 - 1.5 HW 2 due: FC §1.2 Exercises 1-10.
F Practice Exercises
Week 3 M Labor Day - No class
W Deduction. Proofs Handout 1
Proofs Handout 2
Proofs Handout 3
Proofs Handout 4
F Deduction - Proofs Review 1
Week 4 M Deduction - Proofs
W Exam Review
F Applications of Logic in Cognitive Science
Unit 2: Theory of Computation
Week 5 M Applications of Logic in Cognitive Science
Introduction to Sets
Review 2
W Exam 1: Logic and Proof
F Sets and Functions

FC §2.1, §2.2, §2.4

Week 6 M Set Theory and Functions

FC §2.1, §2.2, §2.4

W Set Theory and Functions

FC §2.1, §2.2, §2.4

Week 7 M Turing machines 1 (Optional: FC §5.1)
W Turing machines 2 (Optional: FC §5.1)
F No classes: Fall break
Week 8 M Language and Regular expressions

Homework 4 - Due Monday 10/19
W Finite State AutomataFC §3.4
F Finite State Automata  
Unit 3: Linear Algebra and Neural Networks
Week 9 M Introduction to neural networks NN: Ch.1, Ch.2
W Perceptron and matrices

Matrix algebra

F Review for Exam 2
Week 10 M Exam 2 (tentative)  
W Neural networksNN: Ch.4
F Learning (Delta rule) Neural Network Learning
Week 11 M Backpropagation Neural Network For Digital Signal Processing
Chapter 26 in pdf form
W Review TBA
Unit 4: Probability Theory and Bayesian Inference
Week 12 M Neural Networks In Cognitive Science
W Review
F Practice

Week 13 M
W Introduction to probability
Random variables
IP: §1.2, IP: §6.1
F Homework 5 Due 11/20/15 - Write an approximately 600 word essay explaining the importance of math in cognitive science. Your audience is 4th graders. You will likely find it effective to use a specific example of math in cognitive science. Be sure to explain the mathematics. Use the ideas discussed in the writing workshop in class on Wednesday, 11/11.
Thanksgiving week
Week 14 M Combinatorics IP: §3.1, §IP: §3.2
W Conditional probability
IP: §4.1, IP: §4.3 Exam 3 Review
F Exam 3
Week 15 M Bayesian inference
Presentations by students
W Presentations by students
F Presentations by students
Final week
Week 16 Final exam - 8:00-10:00 a.m., Wednesday., December 16

Class Policies

All students are expected to attend every class and lab section. Tardiness and unexcused absences will decrease the participation grade. This class will not consist of lectures 2 days a week and practice exercises on Fridays. Instead, the pedogogical techniques and tools will revolve around active learning: individual problem solving and group problem solving, reading outside of class time, explanation, communicating one's level of understanding, applying mathematical and logical techniques.

Lab section is an integral part of the class and is not optional.

Homework assignments will be given weekly or biweekly, with approximately 7 graded assignments in total. Each assignment will typically be due a week from the date that it is assigned.

Exams and quizzes
There will be four exams, covering the major topics of the course:
1. Logic and proofs.
2. Theory of computation.
3. Linear algebra and neural networks.
4. Probability theory and Bayesian models.
All exams will be cumulative. There will also be occasional quizzes at the beginning of class and/or lab section. Quizzes will cover material covered recently in class. The lowest quiz grade will be dropped.

Final project

You have a variety of options for the final project. Pick a topic of interest to you and do a project in which you learn and demonstrate math and logic in cognitive science. You may do an extension of a project from another class. You may put together a powerpoint presentation. You may write a short paper, making sure to discuss how the topic incorporates ideas from math and logic to study cognition. Other options are possible; please check with an instructor. You should include references (at least four from books, journals, or conference proceedings), demonstrating that you have adequately researched the topic.

Student presentation

You are to present the topic that you are writing about for your final project. Essentially, you should show that you understand how a certain set of mathematical tools are being used to model and understand a certain aspect of cognition. You will have about 10 minutes to present. You are encouraged to be creative in your use of media for your presentation.

Teach-the-class something interesting

You will have an opportunity to enrich the class experience by teaching the class something related to math and logic of cognitive science.

10% Participation
20% Homework assignments
48% Exams (12% each, ×4) 3 during class time throughout the semester; one during finals week
10% Quizzes (lowest dropped)
5% Teach-the-class something interesting
7% Final project (4% written report, 3% oral presentation)