Statements in Logic

The basic unit of logic is the statement (also known as proposition). A statement is a declaration about which it can be unambiguously determined whether it is true or false. This distinguishes logical statements from all other linguistic forms.

Examples of Statements

• "2 + 2 = 4" → true statement.
• "7 is a prime number" → true statement.
• "5 is divisible by 3" → false statement.

Non-Statements

Not every sentence is a statement. If a declaration has no unambiguous truth value, it is not considered a logical statement.

• "What is your name?" → question, not a statement.
• "Go out to the yard!" → command, no truth value.
• "This is beautiful." → subjective opinion, not unambiguously true or false.

Truth Value

Every statement is assigned a truth value. In classical logic, there are only two possibilities: true (1) or false (0).

Here p denotes a statement, ∈ means “is an element of”, and {0,1} is the set of truth values. Thus p ∈ {0,1} expresses that the truth value of a statement is always either 0 (false) or 1 (true).

Statements in Mathematics

The entire system of mathematics is built on statements. The formulation of a theorem is a statement that we support with proof. Example: “Every prime number is greater than 1.” This is a general statement that is true in every case.

Summary

A statement is every declaration that has an unambiguous truth value. In classical logic, this truth value is either true (1) or false (0). Non-statements, such as questions or commands, do not form the subject of logical investigation.

Practice Exercise