In logic, we work with sentences whose truth can be judged. This subtopic introduces statements, their truth values, negation and quantifiers.
What is a statement?
A statement is a sentence that is either true or false, but not both. Questions, commands (instructions), exclamations and opinions are not statements, because a fixed truth value cannot be assigned to them.
Example
"$7$ is a prime number" is a statement (true). "Close the window." is a command, so it is not a statement.
Truth value and negation
The truth value of a statement is either true or false. The negation of a statement is formed by inserting "not" (or "it is not true that"), and it always has the opposite truth value.
Key formula
$p \text{ true} \;\Rightarrow\; \sim p \text{ false}, \qquad p \text{ false} \;\Rightarrow\; \sim p \text{ true}$
Quantifiers
A quantifier tells us "how many". The universal quantifier uses words such as all or every; the existential quantifier uses some or there exists.
- "All squares are rectangles" — universal (true).
- "Some prime numbers are even" — existential (true, since $2$ is an even prime).
The negation of "all … are …" is "not all … are …", which is equivalent to "some … are not …".
Remember
A sentence is a statement only if you can label it exactly true or false. Negation flips the truth value.