Updated on May 21, 2018

## Basic Notation

In symbolic logic, modus ponens and modus tollens are two tools used to make conclusions of arguments as well as sets of arguments. We start off with an antecedent, commonly symbolized as the letter *p*, which is our "if" statement. Based off the antecedent, we expect a consequent from it, commonly symbolized as the letter *q,* which is our "then" statement. For example,

"If the sky is blue, then it is not raining."

Is an argument. "The sky is blue" is our antecedent, while "it is not raining" is our consequent. We can symbolize this argument as

*p ---> q*

Which is read as "if *p,* then *q."* A in front of a letter means that the statement is false or negated. So if the statement is *p*, that reads as, "The sky is not blue."

## Modus Ponens

With this technique, we start off with our argument as a true statement. That is,

*p ---> q*

is given. We hold it to be true. Now, if we find that *p* is a true statement, what can we say about *q*? Since we know that *p* implies *q,* if *p* is true, then we know that *q* is true also. This is Modens Ponens (MP), and though it may seem straight-forward, it is often mis-used.

For example, if *p ---> q* and we know that *q* is true, does that mean *p* is true also? If it is not raining, then is the sky blue? It could be, but the sky could also be cloudy. Thus, while *p* could indeed be true in this case, it might not be and we cannot make a conclusion based off of the consequent. When someone tries to confirm the antecedent by using a true consequent, it is a fallacy known as affirming the consequent (AC).

## Modus Tollens

Once again, we have

*p ---> q*

is true. If we know that the consequent is false (*q*), then we can say that the antecedent is false also *(p*). Since we know that *p* implies *q,* if we do not reach a true consequent then our antecedent must also be false. Since it is raining, the sky is not blue. This method is Modus Tollens (MT).

Once again, we must be careful to not misuse this. If we find that *p,* we cannot say that *q* is true also. We know that *p ---> q* but that does not mean that *p ---> q.* Just because the sky is not blue does not mean that it is raining, for it could just be a cloudy day.This fallacy is known as denying the antecedent (DA) and is a common logical trap that people fall into.

**© 2012 Leonard Kelley**

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