Logic and Proof
We make several kinds of statements every single day. Some of them are simple expressions of fact. You might say, "I am going to Disneyland today!" or, "My favorite ride is Space Mountain!" or, "I have nightmares about Mickey Mouse hacking me to pieces with a butcher knife!" Hopefully that last one isn't an expression of fact.
On the other hand, conditional statements (or implications) are a bit more complicated. They often have the form "If hypothesis, then conclusion." For example, "If Pirates of the Caribbean is closed, then we should try the Haunted Mansion," and, "If we stomp on Mickey's foot, then security will escort us out of the park," are conditional statements.
We may or may not be thrown out of Disneyland, but once the hypothesis (or condition) of stepping on Mickey's foot is met, our fate is sealed. Let's just hope he left his butcher knife at home or our nightmare might become a reality.
These statements play a large role in mathematics as well, and it is important to recognize their precise meaning. Take the following statement: "If a number ends in 0, then it is divisible by 2." This says nothing about numbers that do not end in 0; they may be divisible by 2 (for example, 4) or they might not be (like 5).
More generally, conditional statements say nothing about what happens when the hypothesis fails. In other words, a conditional statement says, "Either the hypothesis is false, or the hypothesis and conclusion are BOTH true." This is the most common pitfall! So remember, if the hypothesis is false, anything can happen and because of this, we say that it's still true.
We can think of conditional statements as rules. When looking at any other statement, we can ask whether or not it abides by the rule. If the statement follows the rule, we say it is consistent. If it doesn't follow the rule, we say it is inconsistent.
Suppose the statement "If you are under 16 years old, then you may not drive" is true. In most states, it is so we might as well go with it. Determine which of the following are consistent or inconsistent with this statement.
(A) Max is 18 and drives a limousine.
(B) Celine is 14 and drives a station wagon.
(C) Jordana is 22 and does not drive.
(D) Rob is 10 and does not drive.
Both (A) and (C) are automatically consistent with the statement, since in both cases the hypothesis of being under 16 years old is not met. They're older than 16, so we don't care about them.
In (D), Rob meets the hypothesis since he is 10 years old, and he also satisfies the conclusion of not driving, so (D) is consistent. Good thing too, since we don't want ten-year-olds on the road. Jordana, on the other hand, is a counterexample because she meets the hypothesis but does not meet the conclusion. Consequently, only (C) is inconsistent with the general statement.