Here’s a question:
If we assume that if I win the lottery I will be poor.
And I win the lottery.
Am I poor?
Logically, the answer is “yes.”
It’s not an intuitive answer, but the conclusion does follow the assumptions. This premise—that the conclusions must follow the assumptions—is the basis of “deductive reasoning,” a skill taught to students of formal logic. In formal logic, problems are rendered into an abstract notation, and worked using various tools.
Here’s what the lottery argument looks like when rendered into formal logic:
The process is similar to math (for example, we can express the problem “if I have two cookies and you give me two more cookies, I have four cookies” as “2+2=4″). But unlike math, according to Bram van Heuveln, a professor of cognitive science who regularly teaches an introduction to formal logic, the tools of formal logic are not “a handy tool.” Here’s what he has to say about logic versus math:
People actually use math, right? If an engineer or a mathematician has a problem, they whip out a piece of paper and do a little calculation. You don’t think the problem through first and then transcribe it into symbols, you transcribe it and then use the symbols to work to the answer. But in logic, we hardly ever do this. Even myself, as an expert in this, there’s hardly a time where I’m presented with some kind of logical problem and I whip out the paper, put down all the symbols and work through a truth table or a formal proof.
An understanding of formal logic is a valuable skill, one that could be useful in fields from law to computer science. But van Heuveln jokes that current means of teaching and learning formal logic—rote memorization and repitition—are “BAH” for “boring,” “abstract,” and “hard to use.” The difficulty and abstraction of formal logic is, he believes, a barrier for students who might benefit from an understanding of deductive reasoning. So, with the help of a small grant from the Rensselaer Office of Undergraduate Education, van Heuveln has partnered with Cognitive Science Assistant Professor Mei Si to create a suite of tools to introduce the fundamentals of logic.
Using the grant, van Heuveln and Si will create two sets of teaching tools: one draws on logic-based puzzles that will help students gain an intuitive grasp on some of the functions of logic; the other teaches students a graphical notation for expressing and solving logic problems that is more intuitive than the abstract notation of formal logic.
One problem with logic, says van Heuveln, is that—unlike math, in which cookies can serve as a stand-in for counting and basic arithmetic—logic is hard to demonstrate in the physical world.
(As an aside, van Heuveln says that the difficulty of teaching logic supports the theory of “situated cognition,” whereby knowledge is interconnected with observation and action in the physical world. Under this model, the oldest sections of the human brain—which evolved in a physical world—interact with the newest sections, which exist as a mental “holodeck” where thoughts can be acted out.)
There are no ”cookies” in the world of logic. But van Heuveln said students can grasp principles of logic from a breed of grid-based puzzles that includes Sudoku, Akari (also known as LightUp), and Nurikabe. To improve upon their score in any of these timed games, players must move beyond guesswork and employ elements of deductive reasoning. The games provide a concrete example of the method and power of logic.
Another entry point are “existential graphs,” an alternate notation system (proposed by logician Charles Sanders Peirce) for solving logic problems that involves a more pictorial approach.
Here’s a video in which van Heuveln demonstrates the different between the notational and graphical methods on a problem that asks “if we assume that if p and q then r, and we assume q and p, can we infer r?”:
Van Heuveln said he hopes to have the system up and running by the end of the year.