Statements and Primitive Statements

Mathematical logic begins with statements and primitive statements because every truth table, argument, theorem, and proof is built from sentences that can be judged true or false. A statement is not just any sentence; it must have a definite truth value. Students often confuse meaningful English sentences with logical statements, but a question, command, or wish is not a statement in propositional logic. In this lesson, statements and primitive statements will be used to separate ordinary language from the precise language needed for symbolic logic.

A \textbf{statement}, also called a \textbf{proposition}, is a declarative sentence that has exactly one truth value, true or false.

The word declarative is important. A declarative sentence asserts something. It may assert a mathematical fact, a scientific fact, or a fact about a situation. The sentence may be true or false, but it must be possible to assign exactly one of these truth values. A sentence is not a statement if it is a question, command, exclamation, request, wish, or expression whose truth value is not yet determined.

Let

\begin{align*} p &: \text{Every square has four sides.}\\ q &: \text{The number }9\text{ is divisible by }3.\\ r &: \text{The capital of Japan is Tokyo.} \end{align*}

Each sentence has a definite truth value. The statement

p

is true,

q

is true, and

r

is true. Hence, each sentence is a statement.

The following sentences are not statements:

\begin{align*} &\text{Please open the window.}\\ &\text{How far is the library from here?}\\ &\text{What a pleasant morning!} \end{align*}

The first sentence is a command, the second is a question, and the third is an exclamation. These sentences do not have definite truth values. Therefore, they are not statements.

A \textbf{primitive statement} is a basic statement that is not formed from smaller statements by using logical connectives. Primitive statements are commonly represented by letters such as

p

q

r

, and

s

A primitive statement is treated as one whole unit. For example, the statement

p:

The library is open today can be assigned a truth value without first breaking it into smaller logical parts. Later, primitive statements are joined by connectives such as not, and, or, if-then, and if and only if. Thus, primitive statements are the atoms from which compound logical statements are formed.

Let

\begin{align*} p &: \text{The library is open today.}\\ q &: \text{The student submitted the assignment.}\\ r &: \text{ }5+4=9. \end{align*}

These are treated as primitive statements. Thus, they may be represented symbolically by

p

q

, and

r

Determine whether each sentence is a statement: (1) The number

18

is even. (2)

x+5

is greater than

10

. (3) Close the classroom door. (4) The moon is a natural satellite of Earth. (5) If the train arrives early, then the meeting will begin on time. (6) What is the value of

x

? (7) Geometry is a branch of mathematics. (8) May this day be successful!

Let a statement mean a declarative sentence with exactly one truth value. [1] The sentence The number

18

is even is a statement, because it is declarative and true. [2] The sentence

x+5

is greater than

10

is not a statement until a value of

x

is assigned. Its truth value depends on

x

. [3] Close the classroom door is not a statement, because it is a command. [4] The moon is a natural satellite of Earth is a statement, because it is declarative and true. [5] If the train arrives early, then the meeting will begin on time is a statement when the situation is fixed, because it has a definite conditional truth value. [6] What is the value of

x

is not a statement, because it is a question. [7] Geometry is a branch of mathematics is a statement, because it is declarative and true. [8] May this day be successful is not a statement, because it expresses a wish. Hence, the statements are (1), (4), (5), and (7).

Use the checker to practise the definition of a statement. A sentence counts as a statement only when it is declarative and has exactly one truth value. Choose a sentence, select the classification you think is correct, and compare your answer with the feedback. Notice that a false declarative sentence is still a statement, while a command, question, wish, or unassigned variable sentence is not yet a statement.

Identify the primitive statements among the statements in the previous solved problem.

Let the statements from the previous problem be examined for logical structure. The sentences The number

18

is even, The moon is a natural satellite of Earth, and Geometry is a branch of mathematics do not contain logical connectives. Hence, they may be treated as primitive statements. The conditional sentence If the train arrives early, then the meeting will begin on time is formed from two simpler statements by the connective if-then. Therefore, it is a compound statement, not a primitive statement. Hence, the primitive statements are (1), (4), and (7).

Classify each sentence as a statement or not a statement: (1)

11

is a prime number. (2) Solve the equation

x+2=7

. (3) Chennai is a city in India. (4) Is

15

divisible by

5

? (5) If it rains, then the ground becomes wet. (6) What a difficult test!

(1) Statement. (2) Not a statement, because it is a command. (3) Statement. (4) Not a statement, because it is a question. (5) Statement. (6) Not a statement, because it is an exclamation.

Yes. A statement may be true or false. The sentence

2+3=8

is false, but it is still a statement because it has a definite truth value.

Its truth value depends on the value assigned to

x

. After assigning a value to

x

, it may become a statement.

No. A statement is primitive only when it is not built from smaller statements using logical connectives.

BMLabs Mathematics

BMLabs Mathematics

Statements and Primitive Statements

Statements and Primitive Statements

WHY STATEMENTS MATTER

STATEMENT OR PROPOSITION

RECOGNISING DECLARATIVE SENTENCES

STATEMENTS

NON-STATEMENTS

PRIMITIVE STATEMENT

PRIMITIVE STATEMENTS AS BUILDING BLOCKS

PRIMITIVE STATEMENT REPRESENTATION

CLASSIFYING SENTENCES BY TRUTH VALUE

STATEMENT CLASSIFICATION CHECKER

FREQUENTLY ASKED QUESTIONS

PRACTISE LOGICAL CLASSIFICATION

Introduction To Discrete Mathematics

Paths and Circuits