We make several kinds of statements every single day. Some of them are simple expressions of fact. You might say, "I'm 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's 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's consistent. If it doesn't follow the rule, we say it's inconsistent.
Suppose the statement "If you are under 16 years old, then you may not drive" is true. It is in most states, 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. Celine, on the other hand, is a counterexample because she meets the hypothesis but does not meet the conclusion. Consequently, only (B) is inconsistent with the general statement.
As your English teacher would say, good writers vary their sentence structure. The same is true of conditional statements: after a while, the If-Then formula becomes a real snoozefest. Some ways to mix it up are: "All things satisfying hypothesis are conclusion" and "Conclusion whenever hypothesis."
However, mathematicians can be drier than the Sahara desert: they tend to write conditional statements as a formula p → q, where p is the hypothesis and q the conclusion. In fact, the old saying, "Mind your p's and q's," has its origins in this sort of mathematical logic.
Identify p and q in the following statements, translating them into p → q form.
(A) If it rains outside, then flowers will grow tomorrow.
(B) I cut off a finger whenever I peel rutabagas.
(C) All dogs go to heaven.
For (A), p = "it rains outside" and q = "flowers will grow tomorrow."
In (B), we may rewrite the statement as "If I peel rutabagas, then I cut off a finger," telling us that p = "I peel rutabagas" and q = "I cut off a finger."
Finally, we may rewrite (C) as "If it is a dog, then it will go to heaven," yielding p = "it is a dog" and q = "it will go to heaven."
The hypothesis and conclusion play very different roles in conditional statements. Duh. In other words, p → q and q → p mean very different things. It's kind of like subtraction: 5 – 3 gives a different answer than 3 – 5. To highlight this distinction, mathematicians have given a special name to the statement q → p: it's called the converse of p → q.
No, not those Converse.
Write the converse of the statement, "If something is a watermelon, then it has seeds."
We want to switch the hypothesis and the conclusion, which will give us: "If something has seeds, then it is a watermelon." Of course, this converse is obviously false, since apples, cucumbers, and sunflowers all have seeds and are not watermelons. At least not during their day jobs.
There are some other special ways of modifying implications. For example, if you negate (that means stick a "not" in front of) both the hypothesis and conclusion, you get the inverse: in symbols, not p → not q is the inverse of p → q. Sometimes mathematicians like to be even more brief than this, so they'll abbreviate "not" with the symbol "~". So we can also write the inverse of p → q as ~p → ~q.
Finally, if you negate everything and flip p and q (taking the inverse of the converse, if you're fond of wordplay) then you get the contrapositive. Again in symbols, the contrapositive of p → q is the statement not q → not p, or ~q → ~p. Fancy.
What is the inverse of the statement "All mirrors are shiny?" What is its contrapositive?
If we abbreviate the first statement as mirror → shiny, then the inverse would be not mirror → not shiny and the contrapositive would be not shiny → not mirror. Written in English, the inverse is, "If it is not a mirror, then it is not shiny," while the contrapositive is, "If it is not shiny, then it is not a mirror."
While we've seen that it's possible for a statement to be true while its converse is false, it turns out that the contrapositive is better behaved. Whenever a conditional statement is true, its contrapositive is also true and vice versa. Similarly, a statement's converse and its inverse are always either both true or both false. (Note that the inverse is the contrapositive of the converse. Can you show that?)
Those words sound awfully fancy, don't they? It should be illegal to say them without a top hat and a monocle. They give two answers to the question, "So, what good are conditional statements, anyway?" They're used as a way of getting new information from information we already have.
The law of detachment allows you to "detach" the hypothesis from the conclusion. More precisely, if we know both p and p → q to be true, then we may conclude that q is true. When the traffic law says, "If the red light is blinking, then come to a full stop," and you see a blinking red light up ahead, it's clear what you'll do next (hint: resist the temptation to slam the gas pedal).
What can you infer from the following two statements: "All chickens hatch from eggs" and "Betsy is a chicken"?
We can write the two statements in shorthand as follows: "chicken → hatch from egg" and "Betsy is a chicken." Since Betsy satisfies the hypothesis of chickenhood, she must by detachment meet the conclusion of egginess. In other words, "Betsy hatched from an egg."
The law of syllogism, on the other hand, allows us to squeeze together conditional statements. If we know both p → q and q → r to be true, we can squeeze them together to get p → r. After all, going through q when we can go straight from p to r would be just plain silly-gism.
"If you make a right turn, you must use a turn signal" and "There is a right turn on the way to school." What can be inferred from these two statements?
As usual, it's a good idea to write the statements in their shorthand to make the structure of the implications clear. The first becomes "right turn → signal" and the second "school → right turn." (That is, if you go to school, then you must make a right turn.) Syllogism lets us write the chain "school → right turn → signal" and then cut out the middle part, leaving us with "school → signal." Translating out of our shorthand, "If you are driving to school, then you will use a turn signal."
We'll list the two laws one last time: