Before reading on to the main content of the section, complete this preview activity to start thinking about the types of questions this section will address.
1.
Which of the following sentences should count as statements? That is, for which of the sentences below could you potentially claim the sentence was either true or false? Select all that apply.
The sum of the first 100 positive integers.
This is not a statement. It is not even a complete sentence (there is no verb).
What is the sum of the first 100 positive integers?
This is a question. It is not a statement.
The sum of the first 100 positive integers is 5050.
This is a statement. It is either true or false (it happens to be true).
Is the sum of the first 100 positive integers 5050?
This is a question. The answer happens to be “yes”, but that is not the same as saying “true”. Questions are never statements.
The sum of the first 100 positive integers is 17.
This is clearly false. But since it is false, it is a statement!
2.
You and your roommate are arguing, and they make the audacious claim that pineapple is good both on pizza and in smoothies. Which of the following are reasonable responses to this claim, from a logical point of view?
The statement is false because even though pineapple is good in smoothies, it is NOT good on pizza.
If you really think that pineapple is not good on pizza, then you would have to say the statement is false. The statement is claiming both are true.
The statement is false because while pineapple is good on pizza and pineapple is good in smoothies, a pizza smoothie is never good.
The claim that pineapple is good both on pizza and in smoothies is not claiming that it is only good at the same time. It is precisely claiming these two separate facts are both true.
The statement is half true because regardless of what you think about pineapple on pizza, we can all agree at least that pineapple is good in smoothies.
Statements are either true or false. There is no “half true.”.
The statement is false because everyone who likes pineapple on pizza does NOT like pineapple in smoothies.
If it were the case that everyone who liked pineapple on pizza didn’t like pineapple in smoothies, then the statement would indeed be false.
3.
Your roommate now makes an even more outrageous claim: If a superhero movie is part of the Marvel Cinematic Universe, then it is good. Which of the following are reasonable responses to this claim, from a logical point of view?
This is false because there are good superhero movies, like Wonder Woman and Dark Knight, that are based on DC Comics, and so not part of the Marvel Cinematic Universe.
The claim was that if a movie was Marvel, then it was good. It is not claiming that if a movie is good, then it is Marvel.
The statement is false because there is at least one superhero movie that is part of the Marvel Cinematic Universe that is also not good.
Exactly. This is what it means for an if-then statement to be false.
The statement is false because, for example, Green Lantern is neither Marvel nor good.
Actually, I’ve never seen that movie, but even if it were bad, that doesn’t say anything about the original statement, since it is not a Marvel movie.
The statement is true because more than half of the Marvel movies are good.
The claim was that this statement held for every superhero movie, not just most of them.
4.
Your roommate just won’t let up with their outrageous claims. Now they claim that either every troll is a knave, or there is at least one troll that is a knight. What can you say to this?
Yes, this is true because every troll is either a knight or a knave. If it is not the case that all trolls are knaves, then there must be some troll that is a knight.
Exactly!
This is false because some trolls are knights and some other trolls are knaves.
If that were the case, there would be at least one knight.
The statement is false because there is no way to verify which of the two options is the case.
Whether it is possible to verify which part of an “or” statement is true does not change whether the statement is true.
The statement is false because no troll could say that all trolls are knaves, since knaves always lie.
It’s true that if a troll said that all trolls are knaves they would have to be a knave and then it would be false that all trolls were knaves. But luckily there is another option.
