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Math and Logic

False + False = True

Can a statement be true if all of its parts are false? Surprisingly, the answer is yes!

An implication, also known as an "if-then" statement, describes a particular dependency between two claims. For example, the following statement is an implication:

"If Tony goes to the movie, then Jane goes to the movie."

In the example above, we have two different statements inside the implication:

The premise: "Tony goes to the movie."
The conclusion: "Jane goes to the movie."

The implication doesn't tell us anything about whether or not each of these two statements is actually true. Rather, it tells us about a dependency between their respective truths. An if-then statement tells us that when the premise is true, it forces the conclusion to be true.

Because the movie implication involves two different statements (the premise and the conclusion), there are four possible combinations for their truth values.

Which of these combinations disproves the implication's claim? Think through these scenarios, and which of them (if any) would prove that the implication is false?

"If Tony goes to the movie, then Jane goes to the movie."

The implication claims that Tony's presence at the movie will necessarily result in Jane's presence; so, if Tony is at the movie and Jane is not, the implication was definitely false. However, the implication makes no claims about what Jane will do when Tony doesn't go to the movie, so what Jane does when Tony is elsewhere has no bearing on the implication's truth value. The only way for an implication to be false is if the premise is true but the conclusion is false.

When the premise of an implication is false (when Tony does not go to the movie), the implication is true, but it is said to be vacuously true. As another example, the statement, "if Earth is flat, then all elephants can speak fluent French," is logically true (at least in this universe). It is true but vacuously true because the premise that Earth is flat is false.

In our movie scenario, the vacuously true case is a bit less outlandish, but it still exemplifies the strange but logical result that an implication with a false premise and a false conclusion is itself a true statement. Consider this:

  • "If Tony goes to the movie, then Jane goes to the movie."
  • Tony must go to a family reunion instead of the movie.
  • Jane must go to a football game instead of the movie.

In this scenario, both the premise and conclusion of the first statement are false, but the implication itself is still true. This counter-intuitive property of implication and of vacuous truth is key to understanding many logic puzzles, including the one below.

Today's Challenge

On any given day, Sam either always lies or always tells the truth.

Today, Sam says:

  1. "I am exactly five feet tall."
  2. "If I am exactly five feet tall, then I also own a ham radio!"

Does Sam own a ham radio? \\[1em]

Note: Don't rely on the picture for this problem. Try to determine using logic what we can conclude from Sam's statements. \\[1.5em]

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