Statements and Propositions in Mathematical Logic

Statements and propositions form the first language of mathematical logic. In this page, we learn how to decide whether a sentence has a definite truth value, how to name primitive statements, and why variable sentences need extra information before they become statements. The focus keyword statements and propositions is central because every later topic in logic, including connectives and truth tables, begins with this simple test: can the sentence be true or false, but not both? I will use classroom-style examples so that the idea is clear before we move toward symbolic formulas.

A mathematical statement, also called a proposition, is a declarative sentence that has exactly one truth value. It is either true or false, but it cannot be both true and false at the same time. The sentence must assert something; questions, commands, wishes, and exclamations do not qualify. Example from arithmetic: "

2+3=5

" is a statement, and its truth value is true. Example from number theory: "

9

is a prime number" is a statement, and its truth value is false. Example from daily language: "Kolkata is a city in India" is a statement, and its truth value is true. A statement does not need to be mathematical in vocabulary. It only needs to make a claim whose truth value is definite.

2+3=5

" is a statement, and its truth value is true. Example from number theory: "

9

Use the truth value test to decide whether a sentence is a mathematical statement. Choose a sentence, predict its category, and click the button to compare your answer with the logical classification. Notice that a false declarative sentence still passes the test because it has one definite truth value. This matters because logic begins by separating claims from questions, commands, and variable sentences.

A common beginner error is to think that only true sentences are statements. That is not correct. A false declarative sentence is still a statement because it has a definite truth value.

A primitive statement is a statement that we treat as a single unit in a logical discussion. We usually represent primitive statements by lowercase letters such as

p

q

r

s

, and

t

. The letter is not the statement itself; it is a convenient name for the full sentence. Example from a course setting: let

p

mean "Discrete mathematics is a required course." Example from arithmetic: let

q

mean "

4+4=8

." Example from geometry: let

r

mean "Every square is a rectangle." Once a primitive statement is named, we can combine it with other statements without writing the full sentence each time.

A primitive statement symbol is a short name for a full sentence with a definite truth value. Choose one full sentence for each symbol and click the button to build a small symbol dictionary. Notice that the symbol becomes useful only after the sentence behind it is clear. This matters because later logical formulas use letters such as p and q to focus on structure instead of long wording.

Let

p

be "The number

12

is even." Let

q

be "The number

15

is divisible by

4

." Let

r

be "Every integer is a rational number." Then

p

is true,

q

is false, and

r

is true. These three sentences are all statements because each one has a definite truth value. The symbols

p

q

, and

r

allow us to discuss the logical form without being distracted by the English wording.

An open sentence is a sentence containing a variable whose truth value is not fixed until the variable is assigned a value. An open sentence is not a statement until the variable is specified. Example from arithmetic: "

x+2=7

" is not a statement until

x

is known. If

x=5

, it becomes true; if

x=1

, it becomes false. Example from divisibility: "

n

is divisible by

3

" is not a statement until

n

is assigned. If

n=12

, it is true; if

n=14

, it is false. Example from geometry: "The angle

A

is a right angle" is not a statement unless angle

A

is already defined in a particular figure.

An open sentence becomes a statement only after a value is assigned to its variable. Choose an open sentence, enter a value, and click the button to test the resulting truth value. Notice that the same sentence can become true for one value and false for another value. This matters because variables do not carry definite truth values until the mathematical context fixes them.

When deciding whether a sentence is a statement, do not ask first whether it is true. Ask whether it can be assigned one and only one truth value.

Decide whether each sentence is a statement. (a) "

7

is an odd integer." (b) "Close the door." (c) "

x

is greater than

10

." (d) "Every rectangle has four sides."

Sentence (a) is a statement because it is declarative and true. Sentence (b) is not a statement because it is a command. Sentence (c) is not a statement because the value of

x

is not given. Sentence (d) is a statement because it is declarative and true.

Use the same test from the solved problem on new and familiar sentences. Choose a sentence, choose its classification with truth value when needed, and click the button to check your reasoning. Notice the difference between a false proposition and a sentence that has no truth value at all. This matters because many mistakes in logic come from treating questions, commands, or open sentences as propositions.

Is "

10

is a prime number" a statement? Give its truth value. Solution. Yes, it is a statement. It is declarative and has a definite truth value. Since

10

has divisors

1

2

5

, and

10

, it is not prime. Therefore the truth value is false.

Consider the open sentence "

x^2=16

." Give one value of

x

that makes it true. Solution. If

x=4

, then

x^2=4^2=16

. So the open sentence becomes the true statement "

4^2=16

." Also

x=-4

would make it true, but one correct value is enough.

Consider the open sentence "

x+5=9

." Give one value of

x

that makes it false. Solution. If

x=1

, then

x+5=1+5=6

, and

6

is not equal to

9

. So the sentence becomes false. This shows why the original sentence was not a statement before assigning

x

Let

p

denote "The integer

18

is divisible by

6

." Let

q

denote "The integer

18

is odd." State the truth values of

p

and

q

. Solution. Since

18=6\cdot 3

, the statement

p

is true. Since

18

is divisible by

2

, it is even, not odd. Therefore

q

is false.

The words statement and proposition are used in the same basic sense in elementary mathematical logic. A statement may be written in English, in mathematical notation, or in a mixture of both. The logical requirement is always the same: the sentence must have a definite truth value.

Classify each item as a statement or not a statement. If it is a statement, write its truth value. 1. "

3+6=9

" 2. "What is your name?" 3. "

y

is an even integer." 4. "Every prime number greater than

2

is odd." 5. "Please solve this problem."

1. Statement; true. 2. Not a statement; it is a question. 3. Not a statement;

y

is not assigned. 4. Statement; true. 5. Not a statement; it is a request.

Yes. A false declarative sentence is still a statement because it has a definite truth value.

The variable

x

has not been assigned a value. After assigning a value, the sentence may become true or false.

No. They do not assert a claim with a truth value.

The letters let us study logical structure without repeatedly writing long sentences.

BMLabs Mathematics

BMLabs Mathematics

Statements and Propositions in Mathematical Logic

Statements and Propositions in Mathematical Logic

MATHEMATICAL STATEMENT

HOW TO TEST A SENTENCE

STATEMENT SORTING LAB

PRIMITIVE STATEMENT

WHAT NAMING A PROPOSITION DOES

PRIMITIVE STATEMENT MAPPER

NAMING SIMPLE PROPOSITIONS

OPEN SENTENCE

HOW VARIABLES BECOME STATEMENTS

OPEN SENTENCE TRUTH TESTER

CLASSIFYING DECLARATIVE SENTENCES

PRACTICE THE TRUTH VALUE TEST

TRUTH VALUE PRACTICE CHECKER

TRUTH VALUE OF A FALSE PROPOSITION

OPEN SENTENCE MADE TRUE

OPEN SENTENCE MADE FALSE

PRIMITIVE STATEMENT NOTATION

STATEMENTS AND PROPOSITIONS

FREQUENTLY ASKED QUESTIONS

CONTINUE TO LOGICAL CONNECTIVES

Introduction to C Programming

Coordinates and Locus