Which is statement will always be true?

Given an if-then statement "if p , then q ," we can create three related statements:

A conditional statement consists of two parts, a hypothesis in the “if” clause and a conclusion in the “then” clause.  For instance, “If it rains, then they cancel school.” 
  "It rains" is the hypothesis.
  "They cancel school" is the conclusion.

To form the converse of the conditional statement, interchange the hypothesis and the conclusion.
      The converse of "If it rains, then they cancel school" is "If they cancel school, then it rains."

To form the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion.
      The inverse of “If it rains, then they cancel school” is “If it does not rain, then they do not cancel school.”

To form the contrapositive of the conditional statement, interchange the hypothesis and the conclusion of the inverse statement. 
      The contrapositive of "If it rains, then they cancel school" is "If they do not cancel school, then it does not rain."

If the statement is true, then the contrapositive is also logically true. If the converse is true, then the inverse is also logically true.

Example 1:

In the above example, since the hypothesis and conclusion are equivalent, all four statements are true. But this will not always be the case!

Example 2:

In general, a mathematical statement consists of two parts: the hypothesis or assumptions, and the conclusion. Most mathematical statements you will see in first year courses have the form "If A, then B" or "A implies B" or "A $\Rightarrow$ B". The conditions that make up "A" are the assumptions we make, and the conditions that make up "B" are the conclusion. If we are going to prove that the statement "If A, then B" is true, we would need to start by making the assumptions "A" and then doing some work to conclude that "B" must also hold. If we want to apply a statement of the form "If A, then B", then we need to make sure that the conditions "A" are met, before we jump to the conclusion "B." For example, if you want to apply the statement "$n$ is even $\Rightarrow$ $\frac{n}{2}$ is an integer", then you need to verify that $n$ is even, before you conclude that $\frac{n}{2}$ is an integer. In mathematics you will often encounter statements of the form "A if and only if B" or "A $\Leftrightarrow$ B". These statements are really two "if/then" statements. The statement "A if and only if B" is equivalent to the statements "If A, then B" and "If B, then A." Another way to think of this sort of statement is as an equivalence between the statements A and B: whenever A holds, B holds, and whenever B hold, A holds. Consider the following example: "$n$ is even $\Leftrightarrow \frac{n}{2}$ is an integer". Here the statement A is "$n$ is even" and the statement B is "$\frac{n}{2}$ is an integer." If we think about what it means to be even (namely that n is a multiple of 2), we see quite easily that these two statements are equivalent: If $n=2k$ is even, then $\frac{n}{2} = \frac{2k}{2} = k$ is an integer, and if $\frac{n}{2} = k$ is an integer, then $n=2k$ so $n$ is even. In everyday use, a statement of the form "If A, then B", sometimes means "A if and only if B." For example, when most people say "If you lend me \$30, then I'll do your chores this week" they typically mean "I'll do your chores if and only if you lend me \$30." In particular, if you don't lend the \$30, they won't be doing your chores.

In mathematics, the statement "A implies B" is very different from "A if and only if B." Consider the following example: Let A be the statement "$n$ is an integer" and B be the statement "$\frac{n}{3}$ is a rational number." The statement "A implies B" is the statement "If $n$ is an integer, then $\frac{n}{3}$ is a rational number." This statement is true. However, the statement "A if and only if B" is the statement "$n$ is an integer if and only if $\frac{n}{3}$ is a rational number," which is false.

Example.

Next let's consider the conclusion. We are concluding that $x$ must satisfy both inequalities $x>0$ and $x > \frac{1}{2}$. If we look more closely, we see that once we satisfy the second inequality, the first is redundant. (If $x>\frac{1}{2}$, then it must already be larger than zero.)

So the statement is true! (Why?)

This kind of statements "A $\Rightarrow$ B" where A is false are called vaccuously true. A statement "A $\Rightarrow$ B" is true when the relation "A implies B" is true, not when A, or B, or A and B are true. It states that "if A is true, then B must also be true". This means that when A is false, the statement doesn't conclude anything.

So whenever the hypothesis A is false, a statement "A $\Rightarrow$ B" is always true! (independently of whether B is true or false)

Let's look at some examples of statements and non-statements to define the kind of sentences that form statements.

To talk about statements, we’ll start with some examples of statements and non-statements.

Statements

So what makes something a statement?

• Definition: Statements are the kind of sentences that are either true or false.

As such, a statement is an assertion that something is or is not the case. A statement is true if what it asserts is the case, and it is false if what it asserts is not the case.

For instance, the statement “The trains are always late” is only true if what it describes is the case, i.e., if it is actually the case that the trains are always late. This is false in Auckland. Sometimes trains are on time, and sometimes they are early. Someone may impatiently complain that the trains are always late to express their exasperation with the train system, but strictly speaking what they say is false.

It is true that bananas have no bones, and I do like bananas, but I like bananas because they are tasty and healthy, not because they have no bones. I would thus say something false if I said “I like bananas because they have no bones.” That’s why “I like bananas because they have no bones” is a statement. It is the kind of sentence that is either true or false – in this case false.

Non-statements

However, it doesn’t make sense to say that the sentence “Welcome to the University of Auckland!” is either true or false. Wouldn’t you be puzzled if someone answered “true” in response to the greetings? It wouldn’t be an appropriate answer. “How can I stop tailgating?” is a question; the sentence doesn’t express something that is either true or false.

Finally, “When the car ahead reaches an object, make sure you can count to four crocodiles before you reach the same object.” is an advice. It advises you to make sure you can count up to four crocodiles (one crocodile, two crocodiles,…,four crocodiles) before you reach the same object as the car preceding you. Try it! You’ll be a much safer driver (that’s a true statement!).

More about Statements

So sentences that can be true or false are statements. Quite simple. But things can get more complicated. Let’s see how.

Something can be a statement even if we don’t know whether it’s true or false. All that matters about statements is that they are the kind of things that can be true or false, not that we know whether they are true or false. For instance:

• Ivan Slotvsky, the famous Irish builder of Madrid, is eating ham steaks and chutney at this very moment.

True or false? I don’t know. But it is the kind of thing that could be true or false.

Here’s another one:

• Sometime in the next 39 years, I will have a creepy next door neighbour.

So far, so good. I’ve been lucky and I haven’t had a creepy next door neighbour. But the statement is true if it will be the case some day that I have a creepy next door neighbour in the next 39 years. Otherwise, the statement is false. But I don’t know whether the statement is true or false. Regardless, what matters is that this sentence is the kind of thing that is true or false.

Another example:

I have no idea what Vero or Promina are. But the sentence expresses something that is either true or false.

The same statement can be true on some occasions and false in others. That is, statements are not always true or always false.

Here’s an example:

This statement is true of Tim, and it’s false of Patrick. Or the statement

• Patrick is a happily married man.

was false before Patrick got married, it is true now (while I’m typing this), and it may become false in the future.

Ambiguous Statements

One difficulty with statements is that they may sometimes express two different things. We call those ambiguous statements. Here’s an example:

• John greeted everybody with a smile.

What are the two states of affairs that this statement may be describing? Try and answer this for yourself before proceeding.

Answer:

1. John was smiling and then he greeted everybody. In this case, it’s true that he greeted everybody with a smile – his smile.
2. Maybe there were smiling people and non-smiling people, and John only greeted the ones that were smiling.

The sentence “John greeted everybody with a smile” can thus be used to describe two different things. That’s what makes it ambiguous.

You need to be careful about this. If you use ambiguous statements, you run the risk of having others misunderstand what you are saying. In some cases, what others think you are saying may be very different from what you were trying to express. For example, Tim is an early bird. He gets up early every morning, and always before his wife. Now, suppose he tried to express this by saying:

• I beat my wife up everyday.

That would most certainly be taken the wrong way. Try and make sure you don’t use ambiguous sentences!

Questions and Commands

In the second list of non-statements, we had questions and commands, and those are typically not statements.

• If humans evolved from monkeys, how come we still have monkeys?

This question does not express something that can be true or false. It makes no sense to respond “true” or “false” when you hear it. It’s not a statement. Notice that a part of the sentence is a statement, namely “humans evolved from monkeys”. This is a false statement. Humans did not evolve from monkeys. Humans, monkeys, and apes in general, have a common ancestor that is no longer around. Even though the question contains as a part a false statement, it doesn’t make the question itself something that is true or false.

Sometimes, however, in a special context, the same question could be used to express a statement. Can you imagine a context in which someone might use this question to express something that is true or false? If so, then presumably the person would be using the question to express that it is false that humans evolved from monkeys, since there are still monkeys. We call questions that are used to express statements rhetorical questions. When you see a rhetorical question, you should always rephrase it as a statement. In our case, the statement would be something like this:

• Humans did not evolve from monkeys, because we still have monkeys.

In summary, statements are the kind of sentences that are either true or false. Sentences are ambiguous when they can be used to express several statements. When you have an ambiguous sentence, you need to decide which statement it is being used to express. Questions, commands and advice are typically not statements, because they do not express something that is either true or false. But sometimes people use them rhetorically to express statements. We saw an example of a question which by itself is not a statement, but can be used to express a statement. When you see rhetorical questions, always rephrase them as statements.