Truth value of a statement

Since a statement can only be true or false, we say it carries a truth value: the truth value is written as T when the statement is true, and F when the statement is false. Notice that a statement's truth value being F does not disqualify it from being a statement — a false declarative sentence is still a statement, just a false one; what matters is that it has a definite, single truth status, not that it happens to be true.

Statements — some illustrations. Sentences like "The Sun rises in the East," "Every triangle has three sides," "Mumbai is the capital of Maharashtra," "Every equilateral triangle is equiangular," and "A natural number is an integer" are all statements, because each one can be judged definitely true or false (most of these happen to be true, though a sentence such as $5 \times 2 = 11$ is also a statement — just a false one, since $5\times 2 = 10$).

Sentences that are NOT statements. Sentences that are requests ("Please give your pen"), questions ("What is your name?", "Do you like to play tennis?"), exclamations ("What a beautiful place it is!"), or commands/suggestions ("Open the window," "Sit down," "Let us go for tea") are never statements, because there is no meaningful sense in which they are "true" or "false." As a rule of thumb: interrogative, exclamatory, imperative (command/order/request), and suggestion-type sentences are excluded from the class of statements.

Open sentences. A third category causes the most confusion: sentences whose truth depends on something unspecified — a variable, an unnamed person, or a subjective judgement. For example, $x^2 - 9 = 0$ is true only when $x = 6$ and false for every other value of $x$; "He is tall" cannot be evaluated because we don't know who "he" is; and opinions such as "Mathematics is an interesting subject" or "It is black in colour" (again, colour of what?) vary from person to person or situation to situation. Because their truth value keeps shifting depending on the unstated context, such sentences are called open sentences, and an open sentence is explicitly not a statement.

Worked example — classifying sentences. Consider the following nine sentences and decide which are statements (with their truth value) and which are not:

1. $6 \times 4 = 25$ — this is a declarative sentence with a definite (if wrong) truth status: since $6\times4=24\neq25$, it is a statement, truth value F.
2. $x + 6 = 9$ — the truth of this depends on the unspecified value of $x$, so it is an open sentence, not a statement.
3. "What are you doing?" — this is a question, so it is interrogative, not a statement.
4. "The quadratic equation $x^2 - 5x + 6 = 0$ has 2 real roots" — factoring gives $(x-2)(x-3)=0$, so the roots are $2$ and $3$, both real; the claim is correct, so it is a statement, truth value T.
5. "Please sit down" — this is a request, so it is not a statement. …