A logical statement (or proposition) is a declarative sentence that has exactly one truth value: it is either true or false. Logical statements are the atoms of mathematical proof — every theorem, definition, and conditional rule is built from statements connected by logical operators.
Definition
A logical statement (or proposition) is a declarative sentence that has exactly one truth value: it is either true or false. For example, '7 is prime' is a logical statement (true), while 'Is 7 prime?' is not (it's a question).
💡 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
Formula
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
Related Concepts
🚧 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
Go Deeper
Frequently Asked Questions
What is Logical Statement in Math?
A logical statement (or proposition) is a declarative sentence that has exactly one truth value: it is either true or false. For example, '7 is prime' is a logical statement (true), while 'Is 7 prime?' is not (it's a question).
What is the Logical Statement formula?
P \in \{T, F\} (every statement has exactly one truth value)
When do you use Logical Statement?
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.
Next Steps
Cross-Subject Connections
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.
Watch how others think about this
See a teacher and students work through common confusions — step by step.
Visualization
StaticVisual representation of Logical Statement