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# Review of The Propositional Logic (Introduction to Logic 1) course by IMACS

LM1: Introduction to Logic I is an online course meant for mathematically precocious middle and high school students. It is offered by IMACS (Institute for Mathematics & Computer Science).

This propositional calculus course introduces students to propositional logic, a branch of modern mathematics which provides a rigorous mathematical analysis of the process of rational argument.

It is actually just the first course in a series of courses called The Elements of Mathematics (EM) curriculum, offered by IMACS.

The EM curriculum was developed for young students gifted in math, with the aim of familiarizing the students with important mathematical problems, ideas, and theories from antiquity to the present day (and not just the 'lame' topics of the school curriculum).

Some other goals of the EM curriculum are:

• Students should be familiar and indeed comfortable with some of the basic ideas and techniques that are typical of mathematicians.
• Students should be able to follow a mathematical argument, and be experienced in inventing and reporting mathematical arguments of their own.
• Students should feel at home with the axiomatic method of twentieth century mathematics, and appreciate what this method does and does not provide.
• should appreciate the role of abstraction in the development of a mathematical theory.
• Students should be brought into contact with non-trivial, relevant applications of mathematics.

But back to the very first course of the Elements of Mathematics curriculum – the propositional logic.

Upon reading the logic course, I quickly noticed it is very well written, very comprehensive, and detailed. And indeed, it is the result of many decades of development by an international group of mathematicians and educators. The quality shows.

The topics proceed in a very logical fashion (of course :). The problems given within the lessons and the complete exercise sets are well designed (the problem sets and tests are automatically graded). They truly exercise your mind and help you learn the material.

It was easy to observe that this course is not for everyone. It might not even be for a student who does well in math. Studying this logic course requires something more than just good grades in math or the fact one likes math. It requires – and also develops – strict habits of mind: diligent studying, persistency in problem-solving, and applying your mind.

In fact, you cannot even take this course unless you pass an online aptitude test first.

## So what is the outcome?

Like we can read in former students' testimonials, IMACS Elements of Mathematics (EM) curriculum has enabled them to breeze through college mathematics, computer science, chemistry, or other science studies, while their peers struggle to understand formal proofs and abstract concepts.

One former student notes that "The EM program enabled me to cultivate excellent study habits: prolonged concentration, the ability to "work through" unknown problems, and a high level of self-efficacy."

This mathematics curriculum provides a gifted child some true mathematical substance, not just the usual school math one year ahead of others. The student gets used to mathematical formalism and abstraction, and learns the way mathematicians present it without having to wait till college.

## The propositional logic course

This course is divided into these subsections:

• Introduction – this starts out with 20 logic puzzles as an introduction. You might have seen some similar puzzles; they involve citizens of an island who always tell the truth (knights) and who always lie (knaves). These citizens say various statements, and one can logically conclude which kind the citizens must be.

For example, one such puzzle says
Three of the island's inhabitants – A, B, and C – were talking together. A said, "All of us are knaves." Then B remarked, "Exactly one of us is a knight."

What are A, B, and C?
If a student finds these kinds of puzzles fascinating (I did!), then that is one indication he or she might enjoy the logic course itself.

• The Formal Language has to do with certain patterns in English language, such as the "if.. then" statement (implication), negation, and certain others. The tecnical name for these is well-formed formulas ('wffs').

This formal language originated with the work of George Boole and Augustus De Morgan in the 1800s, who began the development of what is now known as symbolic logic or modern logic. Their work was developed further by Gottlob Frege, Bertrand Russell, and Alfred North Whitehead.

I really liked how the LM1 course teaches about well-formed formulas: it was rather intriguing! They provide an automatic interactive "wff checker" that checks if your input is indeed a wff (well-formed formula). You just start typing in letters and brackets and trying to figure out by experimentation what kind of combinations make wff.

Over several lessons, the student will learn that P and Q and [~K] and [~[P ⋀ Q]] and [~[[A ⋁ B] ⇒ [A ⋀ B]]] and others are wffs, whereas for example [⋀A] and ~[C]~ are not. The student learns the terms conjunction, disjunction, implication, and the biconditional.

When learning how to build a truth table, there is again an interactive "machine" that builds the truth table template right on the web page. The student can then fill in the truth values and push a button to have the table checked.

The section also teaches about the important topic of tautologies, and ends with a test.

• The next chapter, Introducing Demonstrations, tackles the task of actual proving or writing demonstrations. But first, one has to learn the right way to infer or conclude true statements from other statements: rules of inference.

So the student studies Modus Ponens and Modus Tollens, Conjunctive Inference, The Substitution Principle, and several others. The names of these won't make sense to you unless you've already studied logic. Actually they often match normal, common-sense arguments that we all use.

For example, Modus Ponens is this argument pattern:
1. If Statement A, then Statement B.
2. But Statement A.
3. Therefore, Statement B.
Or for example, "If it rains hard, go inside". "But it's raining hard!" "Therefore let's go inside!"

Modus Tollens has to do with this argumentation: "If it rains hard, go inside." "Jack is not going inside, so it must not be raining." Of course, this is studied formally using variables, so it looks like this: 1) [P ⇒ Q]. 2) [[~Q] ⇒ [~P]].

There are also Syllogistic Inference, Inference by Cases, Biconditional Inference, and various other rules of inference. Every few lessons there is a problem set to complete. The section ends with a graded test.

• The last section in LM1 is called Working With Demonstrations. It mostly deals with the Deduction Theorem and Indirect Inference – two powerful logical principles that are used in proofs.

In this section, the student writes demonstrations using these two principles and all the previouslys studied rules of inference.

Often, the student first fills in a partially completed demonstration outline, and only after that practices writing demonstrations completely on his/her own.

The problems here might look like this: Show that [P ⇒ [R ⇒ [S ⋀ T]]] ⊢ [[P ⋀ R] ⇒ S]. Just a bunch of symbols for the non-initiated, but after you learn what they mean, this becomes similar to those logic puzzles with knights and knaves and truly helps you learn mathematical thinking.

LM1: Introduction to Logic 1, first part of the Elements of Mathematics course by IMACS (Institute for Mathematics & Computer Science.

Price: There is a tuition fee of \$985 per year for each course in the LM1-4 sequence. Each student is assigned a principal IMACS instructor, and is given unlimited access to the material during the 40-week enrollment period. Please see more information here

Review by Maria Miller