## syllabus

- Instructor: Dr. Max Weiss
- Instructor email:
`max@whythis.net`

- Office hours: Tuesdays 1:30-2:45, or by appointment
- Office location: Stokes n326

## Overview

The study of logic traces a history all the way back to the works of Aristotle. But in the late 19th and early 20th centuries, logic began to draw concerns and methods from the foundations of mathematics. What is a mathematical proof? What guarantees that what is proven is true? What defines 'truth' in the first place? It seems clear that if 'every number has a successor' is true, then infinitely many numbers exist. But are there any infinite collections of objects? Mustn't all reasoning about infinity ultimately collapse into paradox? Or can infinity become intelligible to ordinary people: if so, then by what means?

This course is an introduction to modern logic. Although the subject can be rather theoretical, we will approach it instead as a source of puzzles and problems, developing just enough theory to state and answer further and further questions.

The main prerequisite is a willingness to think rigorously about abstract matters. It may help for you to have seen some logic before. But a better test for whether you will enjoy the course is whether you enjoy solving puzzles.

## Website

The course website is at

`http://whythis.net/logic`

.

Essential resources, including homework assignments and solutions, will be posted here. The website will also include the official version of the course schedule. You will find your marks there too. To reach this stuff you'll need to sign up at the site. You can do this from the course homepage as follows:

- click
`/signup`

- at the
`/signup`

page, enter a username, your email address and a password - at the subsequent
`/affiliate`

page, enter the code`[redacted, ask the instructor!]`

plus your first and last names - check your email, and click on the link in the message from a site robot
- reload the course website.

## Book

The following required text is available at the campus bookstore:

- Smullyan, Raymond:
*Logical Labyrinths*

# Coursework

For the sake of evaluation, coursework will be weighted by units as follows.

- Six homework assignments (two units)
- Three quizzes (three units)
- Final exam (three units)
- Attendance and participation (one unit)

Your final grade will be the average of your highest eight marks on these nine units.

## Homework

Studying logic is like learning a musical instrument: you learn by doing. Although we'll work through material together in class, the homework is intended to be an essential means by which you make progress in this course. Note, also, that the material is cumulative: so keep with it!

As an academic discipline, logic is highly collaborative. People benefit by talking through problems together, until everyone reaches a satisfactory understanding. Naturally, submitted coursework should reflect the understanding reached by you, since if it reflects another student's understanding instead, then the instructor's feedback will be misdirected.

Homework solutions will be posted shortly after the due date, and late assigments will not be accepted.

## Quizzes and exam

Quiz and exam questions will in general resemble questions from preceding homework assignments.

So that everyone is on the same page, the instructor will answer questions of the form "what will be on the quiz/exam" only to the entire class.

The final exam will be given at 12:30pm on 19 December, in the same place as the usual class meetings.

## Attendance and participation

Your mark on this component depends partly on attendance and partly on contributing to class discussion.

# Schedule

Here is a tentative schedule of topics, readings and assignments. We will aim to cover parts I-IV of Smullyan's text *LL*. But I may adjust the dosage as we go along. As noted above, the `/schedule`

page of the course website will give the ultimate version.

#### Unit 1, weeks 1-3.

We'll consider a bunch of puzzles to be solved using only rigorous deductive argumentation, and develop a method of expressing deductive arguments in a way that makes their logical structure clear.

#### Unit 2, weeks 4-9.

We'll examine the classical systems of propositional and first-order logic. Then we'll develop a method of determining the logical validity of formulas for these systems, the so-called tableau method.

#### Unit 3, weeks 10-12.

We'll investigate the elementary mathematical properties of infinite sets and of infinite structures. We will develop a rigorous understanding of the notion of 'number' of elements of an infinite set. Then we consider the fundamental method of reasoning about infinite structures, namely mathematical induction.

#### Unit 4, weeks 13-14.

What is the relationship between proof and truth? A logical system is said to be sound provided that in the system, only valid conclusions can be deduced. Conversely a system is said to be complete if every valid conclusion can be deduced. In conclusion, we'll aim to prove that the systems of propositional and first-order logic are sound and complete. These results will apply the infinitary methods of Unit 3 to illuminate the structure of the formal systems developed in Unit 2.