Using the Laws of Inference to Draw Conclusions

Instructor: Laura Golnabi

Laura Golnabi is a Ph.D. student in Mathematics Education at Teachers College, Columbia University. She also teaches undergraduate mathematics courses, and has developed problem solving courses designed for non-STEM majors. Her current research involves if and when students suffering from mathematics anxiety are able to have positive, flow-like experiences in mathematics.

In this lesson you will learn about the laws of inference and how to use them to draw conclusions. Also, you will learn how to test the validity of conclusions.


The Laws of Inference are rules that can be thought of as the main tools for building valid arguments. Before jumping into what the Laws of Inference are, let's recall that an argument is a sequence of statements that end with a conclusion. We say that the argument is valid if the conclusion follows from the preceding statement(s). Note that the preceding statement(s) are also called premises.

Here's a few examples of arguments broken up into their premises and conclusion.

1. Argument: If the sun is out, then it will be a good day. The sun is out. Therefore, it will be a good day.

Premises: If the sun is out, then it will be a good day. The sun is out.

Conclusion: It will be a good day.

2. Argument: All cars are either red or blue. This car is not red. Therefore, this car is blue.

Premises: All cars are either red or blue. This car is not red.

Conclusion: Therefore, this car is blue.

3. Argument: I will laugh. Therefore, I will laugh or I will cry.

Premise: I will laugh.

Conclusion: Therefore, I will laugh or I will cry.

Now that you are a bit more familiar with the structure of an argument. It is important to know that every argument can also be represented using variables as labels for each statement. Usually, the letters p, q, and r are used. Also, symbols are used to replace key logic terms:

Negation ('not'):








If , then:

if then

To clarify this notation, let's rewrite the first example into variable form. We begin by labeling each statement with a letter:

p: The sun is out.

q: It will be a good day.

Now replacing for the variables, we have:

If p then q. p. Therefore, q.

Next, we will replace the key logic terms with their corresponding symbols and arrange the variables into the usual format for representing a complete argument. Notice that this entails putting the premises one above the other, and then placing the conclusion below the line. Here is the entire argument in variable format:

if p then q therefore q

Laws of Inference

Here are the Laws of Inference with their corresponding variable representations.

1. Modus Ponens

modus ponens

2. Modus Tollens

modus tollens

3. Hypothetical Syllogism


4. Disjunctive Syllogism


5. Addition


6. Simplification


7. Conjunction


8. Resolution


Drawing Conclusions

Using the Laws of Inference, you can also draw conclusions based on the premises you are given. Let's see an example of how this works:

Suppose you are given the following premises: If it is cloudy, then it will rain. If it rains, then the ground will be wet.

To draw a conclusion based on these premises, we start by converting them into symbols as shown above.

p: It is cloudy.

q: It rains.

r: The ground is wet.

if then

Now, take a look at the Laws of Inference listed above. Which one has the same premises?

The Hypothetical Syllogism does! So, based on that law, we know that the conclusion must be

q then r

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