Statements

At the heart of mathematics, and mathematical logic, is the notion of a statement and the relationship between statements. But what can we say about the statements we shall meet in mathematics? We can say that they must be either true or false, but not both. And it does not matter if we cannot work out whether a statement is actually true or false so long as it must be one or the other. We shall use this answer to give us a way of understanding roughly what we shall mean by a “statement” in these notes. For present purposes, we shall make do with the following:

A statement is a sentence which is definitely true or definitely false.
A statement can never be both true and false.

The principle that a statement can only be either true or false but not both is known as the law of the excluded middle. It is fundamental to all the logic and formal mathematics that you will meet in these notes.$^{1}$

It does not matter if we cannot work out whether a statement is actually true or false so long as it must be one or the other. For instance, here is a statement:

The equation $x^3 + y^2 = 88$ has no integer solutions.

This is clearly either true or false but establishing which is not so easy. Here is a second [rather famous] example of a sentence that is NOT a statement:

The only barber in a town shaves each and every man who does not shave himself.

This last sentence is not a statement as it is neither true nor false.$^{2}$

Here are some more examples of statements and non-statements :

(1) “It rained yesterday in Auckland, New Zealand.” Again, this is a statement, as it is either true or false.$^{3}$

(2) “Go home!” and “What is your name?” These are not statements, as it does not make sense to say that they are true or false.

(3) “If $x = 3$, then $x^2 = 9$.” This is certainly true, so it is a statement. We will have more to say about “If … then …” statements later.

(4) “If $x = 3$, then $x^2 = 4$.” This is certainly false, so it is a statement. There is no requirement for statements to be true!

(5) “The sum of two odd numbers is an even number.” This is certainly true, so it is a statement.

From now on, we will only be working with statements and relationships between statements. We shall try to keep to the convention that we write all our logical statements using italics when they are in words and as bold letters when a statement is indicated by a letter [e.g. A, B, etc.]. Later we shall use bold for some terms to help us see how they fit into sentences.

Now we know what we mean by a statement, we shall pause to dig a little deeper into the sorts of statements you might meet and how we discern their truth or falsity. Here is a statement:

24 is divisible by 2

This statement says something that is true and cannot be false, so it is an example of a statement we know to be true just by looking at it. Here is another example:

453653987389875629 is divisible by 987283

This is clearly a statement as it is obviously either true or false. However, whilst it is obvious that it is a statement, it is not so obvious whether it is a true statement or a false statement. To decide that we would need to do some more [somewhat tedious] work.

Here is another statement:

The square root of 2 is irrational

This statement is very much like the first statement [24 is divisible by 2] in that it is true. However, it is not as obvious as the first statement and some work needs to be done to show why it must be true. Later we shall see how we can set out a proof to show that this statement is true.

And here is an expression that has the potential to be a statement:

The positive integer $x$ is divisible by 2

Here we don’t know what $x$ is so we cannot say whether the expression is true or false as it stands; in other words, whether the expression is true or false is conditional on what we are told about $x$ so until we clarify this, we cannot say the expression is a statement according to our definition. We could say it is true or false if we had some more information about $x$. General expressions like this tend to occur in combination with other statements [and, as you will read later, they need to be “quantified” in some way – that is, the set of possible $x$ values to which the statement applies must be clearly stated] and then what is often important is what the combination is saying. For instance, the statement:

if the integer $x$ is divisible by 4 then the integer $x$ is divisible by 2,

is definitely true even though each of the expressions that we have combined to make the bigger statement cannot be said to be true or false by themselves.

So, we shall meet three sorts of basic statement in what follows:

• those that are obviously true [or obviously false];

• those that are true or false, but which need some work to decide which;

• those that are combinations of expressions which are quantified [roughly, that means the range of what the $x$ can be is clearly stated] and are then either true or false [and these will often require some work to decide which they are].

Later in these notes, we shall spend a lot of time building new statements out of basic statements.

In what follows we shall tend to learn how various logical rules work by dealing with statements denoted just by letters – such as A or B or P or Q - but then we shall apply these rules to statements that have definite truth values. This is a little like learning about quadratic equations by studying various things about $ax^2 + bx + c = 0$ and then applying what you discover to specific examples.

In what follows we shall talk a lot about the “truth value” of a statement.$^{4}$ By “truth value” we simply mean whether the statement is true or false. For instance, the truth value of the statement 2 is an even number is “true”, and the truth value of the statement 2 is an odd number is “false”.

We shall often say that two statements are logically equivalent. This will mean that the two statements have the same truth values in the same circumstances.$^{5}$

For instance, the following two statements are logically equivalent:

Today is Tuesday

Today is the day after Monday