Critical Thinking and Logic in Mathematics

Logic

If Judy likes all things round, then Judy will love donuts.

This is a logical statement. Logic is the study of how to critically think about propositions or statements that are either true or false. The statement I just made about Judy came about from thinking critically about the proposition that Judy likes all things round and about donuts. I know the proposition that Judy likes all things round is true, and I know the proposition that donuts are round is true, as well. Because both propositions are true, I can link them together to reach the conclusion that Judy will love donuts because donuts are round. This is the way logic works.

Logic is very useful in the world of mathematics. Mathematicians use logic all the time to prove theorems and other mathematical facts. Everything we know about math right now is based off of these logical proofs. Without these, we wouldn’t have our formulas, like the wonderful quadratic formula or the very useful Pythagorean Theorem.

Using logic in math is about mixing the specific language used in logic with the specific symbols used in math. Let me show you.

Propositions

In logic, propositions are simple statements that can either be true or false. Your propositions don’t have to be complicated. They can be short ones like, ‘All squares are yellow,’ or ‘Judy likes all things pink.’ Your proposition is any statement that can be labeled as either true or false.

Logic propositions in math usually include math symbols. In geometry, you can have a proposition that says, ‘Line AB is the bisector of line CD’ with the corresponding math symbol for lines instead of the word ‘line.’ In algebra, your proposition can be as simple as x = 2. Depending on what kind of math you’re working with, you can have a mixture of words with math symbols or all math symbols. What matters most is that your logic proposition can be labeled as either true or false.

True or False

Usually, your problem will tell you whether a statement is true or false. One thing to keep in mind here is that if your problem says something is true, you have to believe that. Don’t over think the statement. If you see a statement such as 2 + 2 = 5, and the problem says that it is true, then you have to believe that and work with it, but only for that problem. I know it might be hard to do, but what is true and false in logic does not have to make sense in the real world.

Let’s now see how we can apply logic and critical thinking to a problem.

Critical Thinking

Once we are given our propositions, we need to use our critical thinking skills to come up with conclusions. Critical thinking involves creating new connections using what we know is true. For example, let’s say that our problem tells us that x = 5 and y = 1 are true propositions. What kind of new statements and connections can we make?

We can say that, ‘If z = x + y , then z = 6’ because 5 + 1 = 6. We can also say something like, ‘If z = x y , then z = 5.’ Do you see how we are creating new connections from what we know to be true? We use the if-then structure to write our new connections.

Lesson Summary

In review, logic is the study of how to critically think about propositions or statements that are either true or false. In math, the logic statements can involve just words, words and symbols together or just symbols. A logic proposition is simply a statement that can be labeled as either true or false. You use critical thinking to make new connections based on what you know to be true. You write your new connections in the form of an if-then statement.