What are Conditional Statements and When Are They True
At some point in your life, you were likely told the following conditional statement:
“If you work hard, then you will be successful.”
Under what circumstances can we say that we were lied to?
Suppose you have worked hard and are successful. In this case, the statement is certainly true.
Suppose you DIDN’T work hard, and yet you will still be successful. Is the statement above false? Certainly not! We were told nothing about what would happen if you DIDN’T work hard; we were only told about what would happen if you DID work hard.
Suppose you didn’t work hard and you wont be successful. In which case, you cannot necessarily say that the given statement is false because you didn’t meet the condition of “working hard.”
The only way the above conditional statement is false is if you had worked hard, and yet you will NOT be successful. That is the case where you technically would have been lied to.
In general…
Let \(p\) and \(q\) be statements. The conditional statement \(p\rightarrow q\) is false only when \(p\) is true but \(q\) is false. Otherwise, the statement is true. This is expressed in the truth table below.
| \(p\) | \(q\) | \(p\rightarrow q\) |
| F | F | T |
| F | T | T |
| T | F | F |
| T | T | T |
When an implication/conditional statement \(p\rightarrow q\) is asserted to be true, we sometimes say that \(q\) follows from \(p\).
The \(p\rightarrow q\equiv \neg p \vee q\) Equivalence
There is an occasionally useful equivalence between the statement \(p\rightarrow q\) and the statement \(\neg p \vee q\). To illustrate this equivalence, let \(p\) be the statement “you don’t brush your teeth” and \(q\) be the statement “you will get cavities.” Then
\(p\rightarrow q=\) “If you don’t brush your teeth, then you will get cavities.”
\(\neg p\rightarrow q=\) “You brush your teeth, or you get cavities!”
The equivalence, in this case, is pretty easy to see. Be sure to brush your teeth!
To prove that \(p\rightarrow q\equiv (\neg p) \vee q\), note that the associated columns on the following truth table are identical. Therefore, the statements are logically equivalent.
| \(p\) | \(q\) | \(p\rightarrow q\) | \(\neg p\) | \(\neg p \vee q\) |
| F | F | T | T | T |
| F | T | T | T | T |
| T | F | F | F | F |
| T | T | T | F | T |
Negating an Implication/Conditional
The above equivalence, along with De Morgan’s laws, give us an easy way to find the negation of an implication. Note that for statements \(p\) and \(q\),
$$\begin{align}\neg (p \rightarrow q) &\equiv \neg (\neg p \vee q)\\&\equiv (\neg (\neg p))\wedge (\neg q)\\&\equiv p \wedge \neg q\end{align}$$
Therefore, the following theorem is true.
To illustrate this equivalence, suppose we let \(p\) be the statement “you work hard” and let \(q\) be the statement “you will be successful.”
To say the opposite of the conditional statement “If you work hard, then you will be successful” we can use the theorem above to write “You work hard AND you will NOT be successful” (how sad).
Notice that the negation of such a verbal conditional statement seems to highlight the case where the original conditional statement is false. This can help you determine whether you have correctly negated a given conditional statement.
| \(p\) | \(q\) | \(\neg p\) | \(\neg p \vee q\) | \(\neg p\) | \(\neg p \vee q \rightarrow \neg p\) |
| F | F | T | T | T | T |
| F | T | T | T | T | T |
| T | F | F | F | F | T |
| T | T | F | T | F | F |
Note that the only way that \(\neg p \vee q \rightarrow \neg p\) is false is when both \(p\) and \(q\) are true.
| \(p\) | \(q\) | \(r\) | \(p\vee q\) | \((p\vee q)\rightarrow r\) | \(p\rightarrow r\) | \(q\rightarrow r\) | \((p\rightarrow r)\wedge (q\rightarrow r)\) |
| F | F | F | F | T | T | T | T |
| F | F | T | F | T | T | T | T |
| F | T | F | T | F | T | F | F |
| F | T | T | T | T | T | T | T |
| T | F | F | T | F | F | T | F |
| T | F | T | T | T | T | T | T |
| T | T | F | T | F | F | F | F |
| T | T | T | T | T | T | T | T |
Note that the fifth and eighth columns are identical, so the statements are equivalent.
- \(q\rightarrow p\)
- \(\neg p \rightarrow q\)
- \(\neg q \rightarrow \neg p\)
- \(\neg p \rightarrow \neg q\)
- “If I make good grades, then I do my homework”
- “If I do not do my homework, then I will make good grades”
- “If I don’t make good grades, then I don’t do my homework”
- “If I don’t do my homework, then I don’t make good grades”
It is helpful to write the negation of each statement first, before assembling each implication.
Note: \(\neg p\) is “I don’t do my homework” and \(\neg q\) is “I don’t make good grades.”
Replace each letter you see in each statement with the corresponding phrase given (or found, as above with the negations).
- “If I won the lottery, then I am rich”
- “If I haven’t finished this lesson, then I can’t sleep”
- “If I finished this lesson, then I can eat dinner.”
- I haven’t won the lottery or I am rich.
- I have finished this lesson, or I can’t sleep
- I didn’t finish this lesson, or I can’t eat dinner
Yes, some of these don’t sound logically equivalent, but they are nonetheless.
- “If I drive safely, then I will go down in a blaze of glory.”
- “If I don’t eat paint chips, then I am probably going to be okay.”
- “If life gives you lemons, then you know how the saying goes”
- “If eat an omelette, then I like eggs or I just like omelettes”
- “If I own six cats, then I have a problem and I have too many cats”
- “I drive safely and I won’t go down in a blaze of glory.”
- “I eat paint chips and I am probably not going to be okay.”
- “Life gives me lemons, but I don’t know how the saying goes.”
- “I eat an omelette and I don’t like eggs and I don’t (just) like omelettes”
- “I own six cats and I don’t have a problem or I don’t have too many cats”