Meaning of Statements

Meaning of statements explained with clear definitions, truth values, and simple everyday examples used in mathematical reasoning.

1. Idea of a Statement

A statement in mathematical reasoning is a sentence that clearly expresses an idea and has a definite truth value. That means it must be either true or false, but not both and not uncertain. The key point is that the truth of a statement does not depend on how someone feels about it—it is logically determined.

Statements form the basic building blocks of all mathematical arguments. More complicated logical forms like implications and compound statements are built using these basic statements.

1.1. Why statements matter in reasoning

Reasoning is based on making clear claims and then drawing correct conclusions from them. If a sentence cannot be judged true or false, it cannot participate in logical reasoning. So mathematics uses statements because their truth can be tested or logically deduced.

2. Formal Definition

A statement is a declarative sentence that is either true or false, but not both at the same time.

It must be precise, unambiguous, and capable of having a definite truth value.

2.1. Conditions for a sentence to be a statement

  • It must declare something.
  • It must have only one truth value.
  • It must not depend on opinions or uncertainty.

3. Examples of Statements

These examples show how to identify statements:

3.1. Simple true statements

Each of these is clearly true, so each is a statement.

3.2. Simple false statements

These are false, but they are still statements because they have a definite truth value.

4. Sentences That Are Not Statements

Some sentences cannot be classified as statements because their truth cannot be determined directly, or because they do not express a clear claim.

4.1. Examples of non-statements

  • "What is your name?" (a question)
  • "Close the door." (a command)
  • "This is a beautiful picture." (depends on personal opinion)
  • "x + 3 = 10" (truth depends on the value of x)

Such sentences cannot be given a definite true or false value without additional information.

5. Truth Values of Statements

Every valid statement has exactly one truth value: it is either true or false. In reasoning, once the truth of a statement is known, it can be used to build more complex logical expressions.

5.1. Testing truth values

To check whether a sentence is a statement, ask: "Can this be judged clearly as true or false?" If yes, it is a statement. The actual truth value can then be determined using facts or logical reasoning.

6. Notes and Observations

Key points to keep in mind:

  • A statement must be a declarative sentence.
  • Questions, commands, and expressions of opinion are not statements.
  • Open sentences containing variables are not statements unless the variables are assigned specific values.
  • Logical arguments rely entirely on statements because they have fixed truth values.