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Logic & Paradox

This course is expected to run but has not yet been scheduled.

Course Description

Logic is the system of rules upon which all reasoning is based. It pervades absolutely every academic discipline, from mathematics to the sciences, to the humanities as well. It also is a tool that all of us deploy in our everyday lives. And if that is not enough, to logicians, logic is a deep and complex subject of study in its own right. Part of this course will be devoted to exploring this system of rules, which we will build from the ground up. No prior exposure to logic is necessary to take this course.

A paradox is an argument, or a chain of reasoning, that starts from seemingly obvious premises and arrives at a conclusion we find unacceptable. A paradox can provide us with a window into complexities of logic and can show us where our everyday intuitions break down. Part of this course will be an opportunity to investigate some of the most mind-bending and perplexing paradoxes that have been discovered. If we’re lucky, we will be able to take what we learn about logic and use it to solve some of the paradoxes we encounter.

In terms of content, we will start from the very basics, and we will move quickly. We will define some of the core concepts (propositions, truth/falsity) as well as the logical operators (conjunction, disjunction, negation, the conditional) and we will use truth tables examine how these operators affect the truth of sentences that contain them. We will work our way toward definitions of satisfiability, implication, and validity. In the second half of the course, we will bring predicates and quantifiers into our system in order to study first-order logic in all of its complexity and rigor. Along the way, we will discuss some of the subtle but useful distinctions that studying logic allows us to understand. Accordingly, we will be able to answer such questions as: How do we distinguish a valid argument from a sound one? What is the difference between using a word and mentioning it? If no elephant is purple, is this the same as there not existing a purple elephant, or there existing an elephant that is not purple?

Along the way, we will have the chance to grapple with some of the most accessible yet surprising paradoxes. Many have a very close relationship to many of the logical notions we will be investigating, such as Russell's paradox (does the list of all lists that do not contain themselves contain itself?), the Liar Paradox (is the statement "This statement is false" true or false?), and the knight-knave puzzles (if there are two guides at a fork in the road, one of whom will only lie, one of whom will only tell the truth, and you want to figure out which road to take, what question can you ask to figure out which way to go?). We will discuss many, many more paradoxes throughout the course, including: The Sleeping Beauty Paradox, Zeno’s paradoxes of motion, Newcomb’s Paradox, the Two-Envelope Paradox, the Ship of Theseus, the Monty Hall Problem, Hilbert’s Hotel, and many others. Student groups will have the opportunity to find a paradox of their choosing and present it to their peers.

Along the way, we will have the chance to grapple with some of the most accessible yet surprising paradoxes. Many have a very close relationship to many of the logical notions we will be investigating, such as Russell's paradox (does the list of all lists that do not contain themselves contain itself?), the Liar Paradox (is the statement "This statement is false" true or false?), and the knight-knave puzzles (if there are two guides at a fork in the road, one of whom will only lie, one of whom will only tell the truth, and you want to figure out which road to take, what question can you ask to figure out which way to go?). We will discuss many, many more paradoxes throughout the course, including: The Sleeping Beauty Paradox, Zeno’s paradoxes of motion, Newcomb’s Paradox, the Two-Envelope Paradox, the Ship of Theseus, the Monty Hall Problem, Hilbert’s Hotel, and many others. Student groups will have the opportunity to find a paradox of their choosing and present it to their peers.


Upon completion of the course, students should be able to: Have a more thorough and intuitive understanding of general reasoning; be able to complete rigorous elementary proofs; have a better understanding of the universal importance and applicability of logic; be able to clearly describe some of the complexities inherent in the system of logic. Learning formal logic above all is practice in a particular kind of clear and concise thought. We hope that students will feel clearer about their thinking and their ability to convey their thoughts.

The course requires no previous knowledge, though a passion for puzzles and difficult reasoning may be helpful.