The Implication

After exploring the logical connectives "and" and "or", we now shift our focus to the logical connective "implies".  Similar to the other logical connectives we used thus far, we shall look at a truth table to help guide us with finding truth values for the implication.

If P, then Q

For two statements P and Q, the implication of P and Q is the statement

P implies Q.  

The implication "P implies Q" is more commonly written as follows

If P, then Q.

with the truth values of the implication given in the following truth table.

P Q P implies Q
T T T

T F F
F T T
F F T

We often refer to P in the compound statement "If P, then Q" as the hypothesis, while Q is commonly called the conclusion.

Upon first glance, the truth values associated with the implication seem to be a bit absurd when looking at the situations where P is false.  We shall explain this using the following two statements.

P:  You mow the lawn.

Q:  You are paid $100.

We shall look at the implication of these two statements from a slightly different perspective.  Instead of determining truth values let us look at this in the context of being a truth teller or a liar.  Suppose a neighbor stated "If you mow the lawn, then you are paid $100."  Notice that the previous sentence is just "P implies Q", and we will look at the different cases of truth values to see whether the neighbor was lying to you or telling the truth.

Case 1:  We shall assume that P and Q are both true.  This means that you did mow the lawn, and you were paid $100.  Notice the neighbor told you the truth, which means the truth value for "P implies Q" would be true in this case.

Case 2:  We shall assume that P is true and Q is false.  This means that you did mow the lawn, but you never received $100.  Notice the neighbor told you a lie in this case, which means the truth value for "P implies Q" would be false.

Case 3:  We shall assume that P is false and Q is true.  This means that you did not mow the lawn, and you were $100.  Notice the neighbor never gave you any indication of what would happen if you did not mow the lawn.  As a result, the neighbor was still telling the truth, which means the truth value for "P implies Q" would be true.

Case 4:  We shall assume that P is false and Q is false.  This means that you did not mow the lawn, and you never received $100.  Notice the neighbor never gave you any indication of what would happen if you did not mow the lawn.  As a result, the neighbor was still telling the truth, which means the truth value for "P implies Q" would be true.

With a slightly deeper understanding of the implication in place, let's take a look at some examples.

Examples of the Implication of Statements

  1.   Consider the statements

P:  Arnold Schwarzenegger created the Olympics.

Q:  The Nile river is located in Africa.

            The implication of the two statements P and Q is given by the following

P implies Q:  If Arnold Schwarzenegger created the Olympics, then the Nile river is located in Africa.

            Another format for this implication could be written as follows

P implies Q:  Arnold Schwarzenegger created the Olympics implies the Nile river is located in Africa.

     Now we can check the truth value, notice that P is a false statement while Q is a true statement.  So, we head to the truth table to find our answer.  

P Q P implies Q
T T T

T F F
F T T
F F T

            The highlighted row above in the truth table indicates our situation for this example, and we immediately find "P implies Q" is a true statement.

      2.    Assume we are given the following statements 

P:  Cancer is a class of diseases.

Q:  Leukemia is cancer of the body's blood forming tissues.

     This leads us to the following implication "P implies Q"

P implies Q:  If cancer is a class of diseases, then leukemia is cancer of the body's blood forming issues.

     Since both P and Q are true statements, we have the following truth table

P Q P implies Q
T T T

T F F
F T T
F F T

     This leads to "P implies Q" being a true statement.

      3.    Now take a look at the following statements

P:  5 < 9

Q: 7 is a composite number

           Then the implication "P implies Q" is stated as follows 

P implies Q:  If 5 < 9, then 7 is a composite number

           This gives us P is true while Q is a false statement.  Once again we can refer to the truth table in search of our answer. 

P Q P implies Q
T T T

T F F
F T T
F F T

     The truth table comes to the rescue again.  As highlighted above, the case where P is true while Q is false leads to "P implies Q" being false.

Learning Activities

In order to further comprehend the material, please attempt the following exercises.

    1. Find the implication "P implies Q" of the following statements.  Then find the truth value of the implication.
    2. Find the implication "P implies Q" of the following statements.  Then find the truth value of the implication.
    3. Find the truth value of the implication "P implies Q" of the following statements.
    4. Find the truth value of the implication "P implies Q" of the following statements.
    5. Find the truth value of the implication "Q implies P" of the following statements.
    6. Find the truth value of the implication "Q implies P" of the following statements.