Logical Statement

Logic
definition

Also known as: proposition

Grade 9-12

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A declarative sentence that has exactly one definite truth value — either true (T) or false (F), never both and never neither. Logical statements are the atoms of mathematical proof — every theorem, definition, and conditional rule is built from statements connected by logical operators.

Definition

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

💡 Intuition

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

🎯 Core Idea

Every statement has exactly one truth value: T or F. This binary nature is what makes logical reasoning systematic and checkable.

Example

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

Formula

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

Notation

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

🌟 Why It Matters

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

💭 Hint When Stuck

Try assigning T or F to the sentence. If you can do exactly one, it is a valid statement. If neither works or both work, it is not.

Formal View

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

🚧 Common Stuck Point

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

⚠️ 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

Frequently Asked Questions

What is Logical Statement in Math?

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

Why is Logical Statement important?

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 usually get wrong about Logical Statement?

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

How Logical Statement Connects to Other Ideas

Once you have a solid grasp of logical statement, you can move on to negation, and statement and or statement.

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Visualization

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Visual representation of Logical Statement