In World War II’s most desperate months, the mathematician Alan Turing assembled a team of codebreakers to decrypt intercepted Nazi messages. He tested applicants with chess puzzles, hired a classicist and steered away from applied mathematicians, instead looking for staff good at solving logical puzzles. His team played a vital, long-secret role in Allied success in the war.
Thank you for reading this post, don't forget to subscribe!“If there was one branch of mathematics which we could be said to be using systematically, it was mathematical logic,” said team member Peter Hilton, years later.
Some of the ability to reason logically, seen at its most refined state in Turing’s Enigma team, develops naturally. But certain aspects of logical reasoning must be taught explicitly using symbolic logic. Symbolic logic has lost its place in modern high school curriculum, however. It is time that we start offering symbolic logic courses in high schools across the country.
On supporting science journalism
If you’re enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.
It’s not just that training in symbolic logic is essential for the kind of intellectual excellence seen at Bletchley Park. It is also critical in different careers and for responsible citizenship, and is an important tool for mental health and wellness. Moreover, it is underexploited as a much-needed antidote to the outlandish conspiracy thinking and disinformation that has taken hold of much of the public.
Symbolic logic is a form of mathematics that lets us check whether certain conclusions correctly follow from a set of facts. Consider a crime that involved a shooting in the city of Philadelphia. If a defendant was not in Philadelphia on the day of that shooting, we can correctly conclude that they could not be the shooter. On the other hand, if the defendant was present in Philadelphia on the day of the shooting, the defendant may or may not be the shooter. Just because the defendant was in Philadelphia does not imply that they shot someone. Symbolic logic turns this kind of argument into algebra, readily shared, solved and generalized for cracking a continuum of conundrums.
For careers such as electrical engineering and computer science, the value of symbolic logic, usually taught in sophomore year discrete mathematics courses, is incontrovertible. The fundamental building blocks of modern digital computers are circuits representing “AND” and “OR,” which are as common in symbolic logic as the plus sign or division bar in grade school arithmetic. Almost every computer program contains similar “if-then-else” logical conditions. Professional programmers must routinely write and troubleshoot such statements.
But the usefulness of symbolic logic is not solely limited to technical fields.
Everyone can benefit by learning to think computationally, argue computer scientists such as Columbia University’s Jeanette Wing, a view that has influenced K-12 curricula in the last two decades. Computational thinking involves breaking down problems, pulling out key information and forming solutions. The proof procedures in a symbolic logic course provide an ideal training for such thinking.
Rutgers University mathematician Gary Wenger argues that to teach responsible citizenship through liberal arts education, we must provide students with mathematical reasoning skills.
As a simple example, consider political debate over a study finding that most adults who have not completed their high school diplomas earn lower incomes. A responsible citizen must think through the implications along the lines of the Philadelphia example above. From the study, we can conclude that if a person does not have a high school diploma, then the person is likely to make lower income. If a person is making a low income, however, we cannot conclude that the person does not have a high school diploma. Training in symbolic logic makes these sort of reasonings second nature to voters.
To combat conspiracy thinking, the most promising approach has been found to be prevention. We must either warn people ahead of time about a particular conspiracy theory or explicitly teach them how to spot shoddy evidence. Symbolic logic provides rigorous training to spot inconsistencies and flaws in reasoning. For example, many naysayers to global warming present the evidence of cold weather on a particular day as a proof that global warming is a conspiracy. A logical thinker is quick to spot that weather is what happens on any given day, while climate is what is happening across the globe over decades.
Logical reasoning serves us well throughout life, civic duty aside. In a recent study of 10,318 older individuals in Australia, it was found that the active mental activities that are the playgrounds of symbolic logic, such as cards, chess and other games, and doing crosswords or puzzles, were associated with reduced dementia risk over 10 years.
Traditional approaches to teaching logical semantics are so complex that the subject is often not taught at all, and when it is introduced, it is usually in college or later. Using a new pedagogical approach, however, employing simpler semantics, the difficulty and confusion among students drops drastically, says Michael Genesereth, who has taught logic at Stanford for more than 30 years. Genesereth has successfully tested his approach with high school students, and through his Logic For All initiative, is on a mission to bring the course to secondary schools across the country. His course was also the recommended curriculum in the recently concluded International Logic Olympiad that attracted nearly 3,500 high school students worldwide.
Some might question the need for a standalone course on symbolic logic, arguing other classes fill its role. In writing courses, for example, students learn how to spot fallacies, and elementary proofs are covered in trigonometry. Much can be gained by teaching people logical thinking without using any symbols. Logical rigor, however, is only possible through the use of symbolic logic. Furthermore, the piecemeal approach is insufficient, as a well-rounded coverage of symbolic logic requires a semester-long course. You might score points at cocktail parties bandying about words such as proofs and fallacies, but you won’t really know your proofs are correct, and that you caught all possible fallacies.
A course in symbolic logic is also a way to train students in important skills identified in standards for high school education. For example, the Common Core State Standards, high-quality mathematics and language standards adopted by 41 states, require that the students construct viable arguments. Next Generation Science Standards require that students engage in arguments from evidence. Symbolic logic is a foundation for these skill sets.
California and Texas already grant credit to a logic course at the high school level. What we need next is a sustained and long-term investment to put us on a path to national adoption of symbolic logic courses. We need to disseminate the proven course materials. We must expand the pool of trained teachers. We need to provide the necessary support to school districts and administrators in implementing the course. Also we must continue to do research into connections between educational outcomes and training in symbolic logic.
Symbolic logic is an important aspect of intellectual training that is sorely lacking in our school systems. Making such training systematically available will create pathways for those who seek intellectual excellence. For many it will open doors to well-paying jobs and careers critical for any country to be competitive in the modern-day world. For others it will provide much-needed exercise for the brain. And some may even live longer with the cognitive simulation that comes from logic puzzles.
Most importantly, a nation of logical thinkers will be adept at spotting and challenging the conspiracy theories and disinformation that are so easy to spread using the online media. Logically literate people will know how to ask the right questions of their leaders, how to spot fallacies, and, crucially, how to make decisions that truly align with their values. A logically fluent citizenry is not an option, but a necessity, for any functional democracy. There is so much at stake in thinking systematically, that training for it must be included in the curriculum and cannot be left to chance.
This is an opinion and analysis article, and the views expressed by the author or authors are not necessarily those of Scientific American.
