Have you ever encountered the following type of statement?
All cats are animals.
Sheldon is a cat.
Sheldon is an animal.
This type of statement is called a syllogism. A syllogism is a form of a logical argument.
In basic terms, mathematics is all about logic and reasoning. While this reasoning generally takes the form of solving math problems or writing proofs, there are also rules of logic that are followed in math.
Let’s dig in and explore some basics of logic.
What is logic?
Logic is the study of reasoning. Formal logic involves starting with statements that are true or assumed to be true and using deductive reasoning to arrive at valid conclusions.
In reasoning, logicians use arguments. An argument is a claim that contains premises which support a conclusion.
A premise is a true or false statement. It’s a declarative statement that says something about a particular subject. A premise is supposed to help support your proof.
Let’s get back to syllogisms for a moment. A syllogism is an argument where the truth of two or more premises lead to a conclusion.
A syllogism uses deductive reasoning. It starts with some general statements and leads to a specific conclusion.
Syllogisms are commonly used in philosophy and sometimes appear in literature too. The first philosopher to use syllogisms was the ancient Greek philosopher Aristotle in Prior Analytics, around 350 BC. In literature, Shakespeare was known to use variations of syllogisms in some of his works.
In geometry, a syllogism could have this form:
All quadrilaterals have angles that add to 360°.
A rhombus is a quadrilateral.
The angles of a rhombus add to 360°.
In a syllogism, if the two premises are true, then the conclusion must also be true.
In our geometry example, the major premise is that all quadrilaterals have angles that sum to 360°. The minor premise is that a rhombus is a quadrilateral.
These are both true statements. Therefore, the conclusion that the angles of a rhombus add to 360° is also true. This is a valid argument!
We have to be very careful when analyzing arguments. Consider the following syllogism.
If it rains today, then we’ll go to the mall.
We went to the mall.
Therefore, it rained today.
The first statement, “if it rains today, then we’ll go to the mall,” tells us that if it rains, we go to the mall but it doesn’t say that’s the only condition in which we’ll go to the mall.
We went to the mall, but there are other reasons we could have gone. Maybe there was a teacher professional day at school so students flocked to the mall on their day off.
It doesn’t have to be raining in order to go to the mall. So, this reasoning is incorrect – the conclusion is wrong!
In logic, one of the primary goals is to determine the truth or validity of an argument. When analyzing logical arguments, it’s important to understand the language of logic.
In logic, we work with simple statements and more complex statements. To combine two or more statements in logic, we use logical connectives. Some of the most common logical connectives are listed here.
and the symbols associated with them.
Let’s look at some examples of using logical connectives to represent statements. Suppose we have the following statements and their corresponding labels A, B, and C:
|The black raspberry ice
cream is in the freezer.
|The hot fudge sauce is
in the cabinet.
|Mary makes an ice
Translate the following sentences into logical symbols.
- The black raspberry ice cream is in the freezer and the hot fudge sauce is in the cabinet.
- If the black raspberry ice cream is in the freezer and the hot fudge sauce is in the cabinet, then Mary makes an ice cream sundae.
- If the hot fudge sauce is not in the cabinet then Mary does not make an ice cream sundae.
- The black raspberry ice cream is in the freezer or the hot fudge sauce is in the cabinet.
- If the black raspberry ice cream isn’t in the freezer and the hot fudge sauce is in the cabinet then Mary makes an ice cream sundae.
- It is not the case that the hot fudge sauce is not in the cabinet.
- A ⋀ B (this is a simple “and” statement)
- (A ⋀ B) → C
- ~B → ~C
- A ⋁ B (this is a simple “or” statement)
- (~A ⋀ B) → C
- ~(~B) (double negation – this statement means the same thing as “the hot fudge sauce is in the cabinet!”)
Let’s examine conditional statements. The if-then statement comes up a LOT in mathematics so it’s important to understand the ins and outs of such statements!
There are a few equivalent ways of reading the conditional statement A → B. This can be read as, “if A then B,” or “A implies B.”
Sometimes, an if-then statement is reversed as in “B if A.” The best thing to do in this case is to rewrite the conditional in if-then form.
In the conditional statement A → B, A is the hypothesis and B is the conclusion. Using our ice cream example, A → B means that if the black raspberry ice cream is in the freezer then the hot fudge sauce is in the cabinet.
The conditional statement A → B is logically equivalent to its contrapositive, which is formed by negating both parts and reversing the conditional statement. So the contrapositive of A → B is ~B → ~A.
In our ice cream example, this means that if the hot fudge sauce is not in the cabinet then the black raspberry ice cream is not in the freezer.
To convince ourselves that the contrapositive is equivalent to the original conditional statement, it’s helpful to consider a simple example from math.
Original statement: If a polygon is a triangle then it has three sides.
Contrapositive: If a polygon does not have three sides, then it is not a triangle.
We can tell pretty easily that both of these statements are true. In logic, we can also make use of truth tables to help analyze statements and arguments.
A truth table summarizes all of the possibilities of a given statement in order to determine its truth values, that is the statement’s truth or falseness.
Let’s work with a specific case. Suppose we have the two statements:
|Josh gets an A in
calculus this quarter.
|Josh takes a trip to
We’ll use truth tables to determine the possibilities for the connectors “and,” “or,” “not,” and the conditional statement “if-then.” To set up a truth table, we systematically list in the first column all of the possibilities for true/false.
This is generally done by listing the first half of the first column with T (true) and the second half of the first column with F (false). Then, we can alternate T and F in the second column.
Let’s start with the truth table for the negation ~C, which means that Josh did not get an A in calculus this quarter. There aren’t a lot of possibilities here for the truth table. If C is true then ~C is false and vice versa. This truth table is a simple one.
Breaking this down, there are two possibilities: Josh got an A in calculus this quarter or he did not get an A in calculus this quarter.
If it’s true that Josh got an A in calculus this quarter (see the first row), then ~C is false because Josh got an A this quarter. Make sense? The notation can be cumbersome at first, but it becomes easier!
Let’s evaluate an “or” statement. We’ll work through the “or” statement C ⋁ H.
This means Josh got an A in calculus this quarter or Josh takes a trip to Hawaii. There is a difference in the math interpretation of the word “or” versus the English interpretation of the word “or.”
In English, “or” generally means one or the other, but not both. In math, however, “or” means one or the other, or both. (credit: spot.pcc.edu)
For this truth table, once again we list all of the possibilities for true or false in the first two columns. The third column C ⋁ H is true when either C is true or H is true or both are true!
So, the only situation where an “or” statement with two propositions is false is if both C and H are false, which is shown in the last row of the table.
|C ⋁ H
Now, we’ll put together the truth table for the “and” statement C ⋀ H. This means that Josh got an A in calculus this quarter and Josh takes a trip to Hawaii.
The only way for an “and” statement to be true is if both parts are true. So, our truth table will look very different from the previous one.
|C ⋀ H
Finally, let’s work with the conditional statement, C → H, which means if Josh got an A in calculus, then he takes a trip to Hawaii.
Before we write out the truth table for a conditional statement, we need to think about this a bit. It’s obvious that if the hypothesis C is true and the conclusion H is true, then C → H is true.
Here’s where it gets interesting. If the hypothesis is not true but the conclusion is true, then the implication C → H is still true! Why? Let’s look at all the possibilities in this case.
- If Josh gets an A in calculus this quarter, then he takes a trip to Hawaii.
- If Josh gets an A in calculus this quarter, then he does not take a trip to Hawaii.
- If Josh does not get an A in calculus this quarter, then he takes a trip to Hawaii.
- If Josh does not get an A in calculus this quarter, then he does not take a trip to Hawaii.
To figure out which of the above statements are false, imagine the scenario where Josh’s parents promised him, “If you get an A in calculus this quarter, then you can take a trip to Hawaii.”
In which of the four possibilities did Josh’s parents actually break their promise?
Choice 1 is the example where he got the A so he goes on the trip so clearly they kept their promise here.
Choice 2 is the case where the parents broke their promise – Josh earned an A in calculus but didn’t take a trip to Hawaii. Broken promise!
Choice 3 isn’t breaking a promise – there could be other reasons Josh took a trip to Hawaii.
Choice 4 the parents didn’t break their promise either – he didn’t get the A in calculus and he didn’t go to Hawaii.
Logically, the only statement that is false is 2. Now let’s look at the truth table that illustrates this.
Once again, the first two columns will look like the previous two truth tables. The only time the conditional is false is when the hypothesis C is satisfied, but H does not occur!
So, the second row, which corresponds to statement 2 above, is the only one that produces a false statement in the last column.
|C → H
Create a truth table for the contrapositive ~H → ~C to show that it is logically equivalent to the conditional statement C → H.
If done correctly, the last column should have the same true/false statements as our truth table above for C → H. Try it!
|~H → ~C
That concludes our brief lesson on logic! To be a successful math student, it’s imperative that we understand mathematical reasoning when working through proofs.
It’s helpful to remember Spock’s words from Star Trek, “Logic is the beginning of wisdom…not the end.” (credit: screenrant.com)
About the author:
Jean-Marie Gard is an independent math teacher and tutor based in Massachusetts. You can get in touch with Jean-Marie at https://testpreptoday.com/.