Logic & Paradox

Course Description

Logic is a system of rules upon which human reasoning is based. It is a tool that we deploy routinely in our everyday lives. It pervades every academic discipline, from mathematics to the sciences to the humanities. To philosophers, however, logic is a deep and complex subject of study in its own right. This course is devoted in part to exploring this system of rules, which we will build formally from the ground up. We will eventually use our formal system both to better understand the structure of language and to model and assess real-world arguments.

Consider these two statements:
1. All ravens are black.
2. If something isn't black, then it's not a raven.

Though these two statements appear to be saying somewhat different things, in fact they are logically equivalent: Either they're both true, or else they're both false. We can't make exactly one of them true. And this is fact about logic.

Now let's see an example of a paradox:

At a desert oasis, A and B both separately undertake plots to try to kill C. A poisons his canteen, and later B punches a hole in it. C dies of thirst. Who killed him? At the trial, A argues that she can't possibly be the killer, for C never drank the poison. B argues that it couldn't have been him either, for B only deprived C of poisoned water. Both of their arguments seem pretty good, but C was surely killed, and someone should be held accountable. But, whom?

A paradox is a chain of reasoning that starts from seemingly obvious premises and arrives at a conclusion we find unacceptable. The above story is just a simple example, but it illustrates nicely that even very innocent seeming propositions can sometimes lead us into trouble. Part of this course will be an opportunity to investigate some of the most mind-bending and perplexing paradoxes that have ever been discovered, and we will try our hardest to solve them together.

The logic part of the course will be similar to an accelerated math class. We will cover new material every day; there will be problem sets every night. In terms of content, we will cover much of the same material that a college-level introduction to logic course would cover. We will start by formally defining the core concepts (propositions, truth/falsity) as well as the logical operators (conjunction, disjunction, negation, the conditional). We will use truth tables to 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 introduce predicates and quantifiers into our system in order to study first-order logic in all of its depth and rigor.

The paradoxes part of the course will not be similar to any high school course that I know of. Each day, we will explore two new paradoxes together. Students will discuss the paradoxes in small groups, trying to investigate possible ways to resolve the paradoxes of the day. In the evenings, students will have the option to craft a write-up of their preferred solution to a particularly intriguing problem, and occasionally, students will have the opportunity to present their preferred solutions to their peers in class the following day. In past instances of this course, student solutions have, on occasion, been so innovative and insightful that we have published them to the web. Paradoxes we will study include: Zeno’s paradoxes of motion, the Trolley Problem, the Paradox of the Surprise Exam, Newcomb’s Problem, the Two-Envelope Paradox, the Problem of Moral Luck, the Mere Addition Paradox, and many others.

Prerequisites: Prerequisites: Algebra II