# MATHS 315 : Mathematical Logic

## Science

### Course Prescription

Logic addresses the foundations of mathematical reasoning. It models the process of mathematical proof by providing a setting and the rules of deduction. Builds a basic understanding of first order predicate logic, introduces model theory and demonstrates how models of a first order system relate to mathematical structures. The course is recommended for anyone studying high level computer science or mathematical logic.

### Course Overview

The course content is split into two halves. In the first half, we develop sentential logic (also known as statement logic or propositional logic). We develop this from both semantic and syntactic perspectives, that is, from a study of implication, and from a formal study of proofs. We will also consider the relationship between these notions. In the second half, we develop a richer logical system called predicate (or first order) logic, which allows us to formalise many ideas from mathematics; in particular, this is the system in which set theory is typically developed, and in which axioms for many mathematical systems are expressed. In this course, predicate logic is studied from both the syntactic and semantic perspectives, paralleling the ideas developed for sentential logic in the first half of the course.

The course is designed to provide an understanding of many of the mathematical concepts and methods involved in mathematical logic and computer science.

This course could be of interest to students majoring in mathematics and/or computer science. The skills gained in this course may be of particular use to any students interested in formal systems, or in further study of the foundations of mathematics.

### Course Requirements

Prerequisite: B+ or higher in COMPSCI 225 or MATHS 254 or 255 or PHIL 222

### Capabilities Developed in this Course

 Capability 1: Disciplinary Knowledge and Practice Capability 2: Critical Thinking Capability 3: Solution Seeking Capability 4: Communication and Engagement Capability 5: Independence and Integrity

### Learning Outcomes

By the end of this course, students will be able to:
1. Be able to formalise arguments in sentential logic and predicate logic (Capability 1, 2, 3 and 5)
2. Be able to use semantic methods in sentential and predicate logic (Capability 1, 2, 3 and 5)
3. Be able to prove that formal systems have certain properties (Capability 1, 2, 3 and 5)
4. Be able to find and work with derivations in formal systems (Capability 1, 2, 3 and 5)
5. Demonstrate and apply soundness and adequacy of certain formal systems (Capability 1, 2, 3 and 5)
6. Identify and describe applications of predicate logic (Capability 1, 2 and 5)
7. Identify and describe which notions can and cannot be expressed in first-order arithmetic (Capability 1, 2 and 5)
8. Use appropriate language, terminology and symbolic means to communicate proofs both informally and within formal systems (Capability 1 and 4)

### Assessments

Assessment Type Percentage Classification
Assignments 25% Individual Coursework
Test 20% Individual Coursework
Tutorials 5% Individual Coursework
Final Exam 50% Individual Coursework
1 2 3 4 5 6 7 8
Assignments
Test
Tutorials
Final Exam

To pass this course, a student must earn a score of at least 35% in the final exam

### Tuākana

This course is supported by the Mathematics Tuakana Program. See https://www.auckland.ac.nz/en/science/study-with-us/maori-and-pacific-at-the-faculty/tuakana-programme.html for details.

### Key Topics

Predicate and first order logic, both studied from syntactic and semantic points of view.
Additional topics applying knowledge from first order logic (if time permits)

### Learning Resources

MATHS 315 course book

### Special Requirements

N/A

This course is a standard 15 point course and students are expected to spend 10 hours per week involved in each 15 point course that they are enrolled in.

For this course, you can expect 3 hours of lectures, a 1 hour tutorial, 3 hours of reading and thinking about the content and 3 hours of work on assignments and/or test preparation.

### Digital Resources

Course materials are made available in a learning and collaboration tool called Canvas which also includes reading lists and lecture recordings (where available).

Please remember that the recording of any class on a personal device requires the permission of the instructor.

The content and delivery of content in this course are protected by copyright. Material belonging to others may have been used in this course and copied by and solely for the educational purposes of the University under license.

You may copy the course content for the purposes of private study or research, but you may not upload onto any third party site, make a further copy or sell, alter or further reproduce or distribute any part of the course content to another person.

The University of Auckland will not tolerate cheating, or assisting others to cheat, and views cheating in coursework as a serious academic offence. The work that a student submits for grading must be the student's own work, reflecting their learning. Where work from other sources is used, it must be properly acknowledged and referenced. This requirement also applies to sources on the internet. A student's assessed work may be reviewed against online source material using computerised detection mechanisms.

### Inclusive Learning

All students are asked to discuss any impairment related requirements privately, face to face and/or in written form with the course coordinator, lecturer or tutor.

Student Disability Services also provides support for students with a wide range of impairments, both visible and invisible, to succeed and excel at the University. For more information and contact details, please visit the Student Disability Services’ website at http://disability.auckland.ac.nz

### Special Circumstances

If your ability to complete assessed coursework is affected by illness or other personal circumstances outside of your control, contact a member of teaching staff as soon as possible before the assessment is due.

If your personal circumstances significantly affect your performance, or preparation, for an exam or eligible written test, refer to the University’s aegrotat or compassionate consideration page: https://www.auckland.ac.nz/en/students/academic-information/exams-and-final-results/during-exams/aegrotat-and-compassionate-consideration.html.

This should be done as soon as possible and no later than seven days after the affected test or exam date.

### Student Feedback

During the course Class Representatives in each class can take feedback to the staff responsible for the course and staff-student consultative committees.

At the end of the course students will be invited to give feedback on the course and teaching through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback.

Your feedback helps to improve the course and its delivery for all students.

### Student Charter and Responsibilities

The Student Charter assumes and acknowledges that students are active participants in the learning process and that they have responsibilities to the institution and the international community of scholars. The University expects that students will act at all times in a way that demonstrates respect for the rights of other students and staff so that the learning environment is both safe and productive. For further information visit Student Charter (https://www.auckland.ac.nz/en/students/forms-policies-and-guidelines/student-policies-and-guidelines/student-charter.html).

### Disclaimer

Elements of this outline may be subject to change. The latest information about the course will be available for enrolled students in Canvas.

In this course you may be asked to submit your coursework assessments digitally. The University reserves the right to conduct scheduled tests and examinations for this course online or through the use of computers or other electronic devices. Where tests or examinations are conducted online remote invigilation arrangements may be used. The final decision on the completion mode for a test or examination, and remote invigilation arrangements where applicable, will be advised to students at least 10 days prior to the scheduled date of the assessment, or in the case of an examination when the examination timetable is published.

Published on 09/07/2020 03:21 p.m.