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Section 1.9 Conditionals

A conditional is a statement of the form “if \(\phi\) then \(\psi\text{,}\)” or “\(\phi\) only if \(\psi\text{,}\)” or “\(\phi\) implies \(\psi\text{.}\)” We write this symbolically as

\begin{equation*} \quine{\phi \implies \psi}\text{.} \end{equation*}

This statement says that whenever \(\phi\) is true, we can conclude \(\psi\) is true. Statement \(\phi\) is called the premise of the conditional, and \(\psi\) is its conclusion. We can also write this backwards as \(\quine{\psi \Longleftarrow \phi}\text{,}\) which is read “\(\psi\) if \(\phi\text{.}\)”

Here is the truth table for the conditional

\(\phi\) \(\psi\) \(\quine{\phi \implies \psi}\)
T T T
T F F
F T T
F F T

Notice that if the statement \(\psi\) is true, then the statement \(\quine{\phi \implies \psi}\) is true. If \(\phi\) is false, then the statement \(\quine{\phi \implies \psi}\) is again true. The only way \(\quine{\phi \implies \psi}\) ends up false is when \(\phi\) is true but \(\psi\) isn't.

What are the truth-values of the following statements?

  1. “If 2 is even, then 14 is even”

  2. “If 2 is even, then 15 is even”

  3. “If 57 is prime, then 937 is prime”

  4. “If 57 is prime, then 2 is odd”

Solution.

  1. “This is true, because the conclusion are true, so it doesn't matter what the premise is.”

  2. “This one is false. The premise is true, but the conclusion is false.”

  3. “This is true, because the premise is false, so it doesn' t matter what the truth-value of the conclusion is. Or, another perfectly good explanation in this case is that since the conclusion is true, it doesn't matter what the premise is!”

  4. “This is true, because the premise is false, so it doesn' t matter what the truth-value of the conclusion is.”

This sometimes surprises people a little, because in English (and in other natural languages), we are a little more casual in our use of the words “if \(\phi\text{,}\) then \(\psi\text{.}\)” In mathematics, the statement \(/quine{\phi \implies \psi}\) can be re-expressed as the following:

“either \(\psi\) is true or \(\phi\) is false.”

Indeed, \(\quine{\phi \implies \psi}\) has exactly the same truth table as \(\quine{\neg{\phi} \vee \psi}\text{.}\)

To prove a conditional statement \(\quine{\phi\Longrightarrow \psi}\) directly, you proceed as follows:

  1. Start your proof by assuming the premise \(\phi\text{.}\)

  2. Then deduce the conclusion \(\psi\text{.}\)

Consider the following statement

“If \(n\) is an odd natural number, then \(n^2\) is odd.”

The premise in this example is “\(n\) is an odd natural number”, and the conclusion is “\(n^2\) is odd.” How does the proof go?

Solution.

Assume that \(n\) is an odd number. By definition this means \(n=2k+1\) for some integer \(k\text{.}\) Then

\begin{equation*} n^2=(2k+1)^2=4k^2+4k+1=2(2k^2+2k)+1\text{.} \end{equation*}

Consequently \(n^2\) is odd.

If \(\phi\implies \psi\) and \(\psi\implies \chi\text{,}\) then \(\phi\implies \chi\text{.}\) In other words, \(((\phi \implies \psi) \wedge (\psi \implies \chi)) \implies (\phi \implies \chi)\text{.}\)

The premise is \(\quine{(\phi\implies \psi)\wedge(\psi\implies \chi)}\text{.}\) The conclusion is \(\quine{\phi\implies \chi}\text{.}\) How does the proof go?

Solution.

Assume \((\phi\implies \psi)\wedge(\psi\implies \chi)\text{.}\) That is, both \(\quine{\phi \implies \psi}\) and \(\quine{\psi \implies \chi}\) are true.

We must deduce that \(\phi\implies \chi\text{.}\) To do this, assume \(\phi\) is true. We aim to show that \(\chi\) is true.

Since \(\phi\) is true, and since \(\phi\implies \psi\text{,}\) we conclude \(\psi\) is true. Since \(\psi\) is true, and since \(\psi\implies \chi\text{,}\) we conclude \(\chi\) is true.

We have now shown that if \(\phi\) is true, then \(\chi\) is true, thus \(\phi\implies \chi\text{.}\)

Definition 1.9.4.

The converse to the conditional \(\quine{\phi \implies \psi}\) is the statement \(\quine{\psi \implies \phi}\text{.}\)

Note that the converse of a conditional is not the same statement. Compare the truth tables:

\(\phi\) \(\psi\) \(\quine{\phi \implies \psi}\) \(\quine{\psi \implies \phi}\)
T T T T
T F F T
F T T F
F F T T

The conditional “if \(2\) is odd, then \(937\) is prime” is true (if a little strange). The converse “if \(937\) is prime, then \(2\) is odd” is false!

Definition 1.9.5.

The contrapositive of a conditional \(\quine{\phi \implies \psi}\) is the statement \(\quine{\neg{\psi} \implies \neg{\phi}}\text{.}\)

The contrapositive really is just a repackaging of the original conditional. It has exactly the same truth table:

\(\phi\) \(\psi\) \(\quine{\phi \implies \psi}\) \(\quine{\neg{\phi}}\) \(\quine{\neg{\psi}}\) \(\quine{\neg{\psi} \implies \neg{\phi}}\)
T T T F F T
T F F F T F
F T T T F T
F F T T T T