A Closer Look at the Implication
Out of the two conditional-style connectives we’ve seen — the implication $p \to q$ and the biconditional $p \leftrightarrow q$ — we’ve given the biconditional a fairly thorough treatment already. Now we turn back to the implication, to see what else it has to offer.
Recall from its definition that the implication $p \to q$ is false exactly when $p$ is true and $q$ is false. In other words, $\text{true} \to \text{false}$ is a false proposition. This deserves special emphasis:
The reason $\text{true} \to \text{false}$ is a false proposition is that we don’t want true statements leading to false ones in a logical system.
Components of an Implication
Before going any further, it’s worth giving names to the two propositions that make up an implication.
Consider an implication $p \to q$. The proposition $p$ is called the hypothesis of the implication, and the proposition $q$ is called the conclusion of the implication.
The hypothesis is the proposition we’re assuming to be true; the conclusion is what we’re claiming follows once that assumption holds. This is exactly the vocabulary we need to describe the table above in plain English: an implication is false only when its hypothesis is true but its conclusion is false. Whenever the hypothesis is false, we get to declare the implication true no matter what the conclusion happens to be — there’s nothing false about the implication if the assumption driving it never held in the first place.
We’ll lean on this vocabulary constantly going forward, so it’s worth getting comfortable with it now, before we start examining implications more closely.
Trvially True Implications
Curiously, we consider both $\text{false} \to \text{false}$ and $\text{false} \to \text{true}$ to be true propositions. This is because if we start with a false hypothesis, the truth of the conclusion is irrelevant.
Implications of the form
\[ \begin{align*} &\text{false} \to \text{false} \\ &\text{false} \to \text{true} \end{align*} \]are called trivially true.
Suppose Ricardo wants to buy two front-row tickets to a rock concert so he can take a friend. He decides the easiest way to buy the tickets is to save enough money by working a summer job. Two front-row tickets cost $500.
Consider the following propositions:
\[ \begin{array}{rl} s\text{: } &\text{Ricardo earns \$500 by working a summer job.} \\ t\text{: } &\text{Ricardo buys two front-row tickets to the rock concert.} \end{array} \]Let’s take a closer look at the implication $s \to t$.
Case 1: $\text{false} \to \text{false}$
Here, Ricardo doesn’t save the $500 working a summer job, and doesn’t buy two front-row tickets to the rock concert. Because Ricardo was unable to save the needed money, he didn’t go back on his word. As far as we can tell, Ricardo would have bought the tickets if he’d had the money — he just wasn’t able to save it, and so wasn’t able to follow through.
This is a trivially true implication.
Case 2: $\text{false} \to \text{true}$
Here, Ricardo wasn’t able to save the $500, but still bought two front-row tickets to the rock concert — perhaps he won two front row tickets in a radio contest, or was gifted money by friends or family. In this case, Ricardo didn’t go back on his word to save money to buy tickets. Again, he may have bought the tickets if he had saved the money working a summer job.
This is a trivially true implication.
Case 3: $\text{true} \to \text{false}$
In this case, Ricardo did save the $500 working a summer job, but failed to buy the tickets. Here, Ricardo did go back on his word. This means the proposition $s \to t$ isn’t an accurate description of reality — Ricardo fulfilled the premise, but didn’t follow through with the conclusion.
The implication is a false one.
Case 4: $\text{true} \to \text{true}$
In this case, Ricardo saved the $500 working a summer job, and bought two front-row tickets to the rock concert. Ricardo kept his word, and followed through.
This is a true implication, but not a trivially true implication.