Logical Statement Formula

The Formula

P \in \{T, F\} (every statement has exactly one truth value)

When to use: A logical statement is any claim that can be judged definitively as true or false — questions, commands, and paradoxes are not statements.

Quick Example

'2 + 2 = 4' (true). 'The moon is made of cheese' (false). 'What time is it?' (not a statement).

Notation

P, Q, R denote statements; truth values are T (true) and F (false)

What This Formula Means

A declarative sentence that has exactly one definite truth value — either true (T) or false (F), never both and never neither.

A logical statement is any claim that can be judged definitively as true or false — questions, commands, and paradoxes are not statements.

Formal View

P \in \{\top, \bot\} for every proposition P; \forall P\,(P = \top \lor P = \bot) (law of excluded middle)

Worked Examples

Example 1

easy
Classify each as a statement (proposition) or not: (a) 3 + 5 = 8 (b) 'Close the door.' (c) x > 2.

Solution

  1. 1
    (a) 3 + 5 = 8 is a declarative sentence that is true. It is a statement.
  2. 2
    (b) 'Close the door' is a command, not declarative. It is not a statement.
  3. 3
    (c) x > 2 has an unspecified variable, so its truth value is undetermined. It is an open sentence, not a statement.

Answer

\text{(a) Statement (true), (b) Not a statement, (c) Not a statement (open sentence)}
A logical statement must be a declarative sentence with a definite truth value—either true or false, not both and not indeterminate.

Example 2

medium
Negate the statement: 'All prime numbers are odd.'

Example 3

medium
Classify each as a statement or not: (a) '7 + 3 = 11' (b) 'Is 5 prime?' (c) 'Every square is a rectangle.'

Common Mistakes

  • Treating questions or commands as logical statements — 'Close the door' has no truth value
  • Thinking opinions are logical statements — 'Pizza is delicious' is subjective, not definitively true or false
  • Confusing a statement being false with it not being a statement — 'The Earth is flat' is a perfectly valid (false) statement

Why This Formula Matters

Logical statements are the atoms of mathematical proof — every theorem, definition, and conditional rule is built from statements connected by logical operators.

Frequently Asked Questions

What is the Logical Statement formula?

A declarative sentence that has exactly one definite truth value — either true (T) or false (F), never both and never neither.

How do you use the Logical Statement formula?

A logical statement is any claim that can be judged definitively as true or false — questions, commands, and paradoxes are not statements.

What do the symbols mean in the Logical Statement formula?

P, Q, R denote statements; truth values are T (true) and F (false)

Why is the Logical Statement formula important in Math?

Logical statements are the atoms of mathematical proof — every theorem, definition, and conditional rule is built from statements connected by logical operators.

What do students get wrong about Logical Statement?

'This statement is false' is a paradox—it's not a proper statement.

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