Logical Statement Formula
A logical statement (or proposition) is a declarative sentence that has exactly one truth value: it is either true or false.
The Formula
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
Notation
What This Formula Means
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).
A logical statement is any claim that can be judged definitively as true or false — questions, commands, and paradoxes are not statements.
Formal View
Worked Examples
Example 1
easyAnswer
First step
Full solution
- 2 (b) 'Close the door' is a command, not declarative. It is not a statement.
- 3 (c) has an unspecified variable, so its truth value is undetermined. It is an open sentence, not a statement.
Example 2
mediumExample 3
mediumCommon Mistakes
- Treating a question or command as a statement — only declarative sentences with a truth value qualify.
- Calling an open sentence like '' a statement — it needs a value or quantifier to have a fixed truth value.
- Assuming 'true for me' counts — a statement must be objectively true or false, not a matter of opinion.
Why This Formula Matters
Statements are the raw material of all logic: you cannot negate, combine, or build truth tables from something that is not a statement. A student who treats a question or an undecided opinion as a proposition will try to assign truth values where none exist and corrupt every later proof step. Recognizing it by "Can this sentence, in principle, be labeled either true or false (and only one)?" — rather than by familiar numbers — is what lets a student tell it apart from open sentence / predicate and question or command and paradox in a mixed problem set.
Frequently Asked Questions
What is the Logical Statement formula?
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).
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?
, , denote statements; truth values are (true) and (false)
Why is the Logical Statement formula important in Math?
Statements are the raw material of all logic: you cannot negate, combine, or build truth tables from something that is not a statement. A student who treats a question or an undecided opinion as a proposition will try to assign truth values where none exist and corrupt every later proof step. Recognizing it by "Can this sentence, in principle, be labeled either true or false (and only one)?" — rather than by familiar numbers — is what lets a student tell it apart from open sentence / predicate and question or command and paradox in a mixed problem set.
What do students get wrong about Logical Statement?
The procedure for logical statement is the easy part; the trap is treating a question or command as a statement. Asking "Can this sentence, in principle, be labeled either true or false (and only one)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.