What is a proposition?

A statement is a claim that is either true or false. Statements are the fundamental building blocks in logic and mathematics, because they give us something we can analyze, combine, and study with systematic rules.

 

Examples of statements

Some simple statements are:

 

• 2 is an even number
• 7 is greater than 10
• All triangles have three sides

 

In these examples, the first and third statements are true, while the second is false. The crucial point is that each statement has an unambiguous truth value.

 

 

Non-statements

Not all sentences are statements. For example:

• What time is it?
• Close the door!
• x + 5 > 7

 

A question or a command cannot be true or false, and is therefore not a statement. The last sentence is an open claim, because it depends on the value of \( \large x \). Only when \( \large x \) is given a concrete value does it become a statement, e.g. "3 + 5 > 7", which is true.

 

 

Notation

In logic, letters are used as symbols for statements. Let us say that:

 

$$ \large p : 2 \text{ is an even number} $$

$$ \large q : 7 \text{ is greater than 10} $$

 

Then the truth values are:

 

$$ \large p = \text{true}, \quad q = \text{false} $$

 

By introducing symbols such as \( \large p, q, r \) we can work in general with the rules of logic without repeating concrete examples each time.

 

 

Statements as a foundation

Statements are the foundation of propositional logic. By combining them with logical connectives such as \( \land \) (and), \( \lor \) (or), and \( \lnot \) (not), we can form more complex statements and establish systematic rules for when they are true or false.

 

In summary: A statement is a claim that is unambiguously true or false. Non-statements such as questions, commands, or open claims do not belong here. This distinction is the basis of all logic and mathematics.